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9The r mody na mic Pr ope r t i e s o f Ide a l

Gases and Idea l Gas Mix tures o fConstant Compos i t ion

I t h a s b e e n s h o w n t h a t it is n e c e s s a r y t o u s e q u i t e s o p h i s t ic a t e d e q u a t i o n s o f s t a te t o d e f i n e

t h e p r o p e r t i e s o f v a p o u r s w h i c h a r e c l o s e t h e s a t u r a te d v a p o u r l in e . H o w e v e r , f o r g a s e s i n

t h e s u p e r h e a t r e g i o n , t h e i d e a l g a s e q u a t i o n g i v e s s u f f i c i e n t a c c u r a c y f o r m o s t p u r p o s e s .

T h e e q u a t i o n o f s ta t e f o r a n i d e a l g a s , i n t e r m s o f m a s s , i s

w h e r e

pV= mRT ( 9 . 1 )

p = p r e ss u r e ( N / m 2)

V = v o l u m e ( m 3 )

m = m a s s ( k g )

R = s p e c i f i c g a s c o n s t a n t ( k J / k g K )

T = a b s o lu t e ( o r t h e r m o d y n a m i c ) t e m p e r a t u r e ( K )

T h i s c a n be w r i tt e n in m o r e g e n e r a l te r m s u s i n g t h e a m o u n t o f s u b s t a n ce , w h e n

pV = ng~T

w h e r e

n - a m o u n t o f s u b s t a n c e , o r c h e m i c a l a m o u n t ( k m o l )

9~ = u n i v e r s a l g a s c o n s t a n t ( k J / k m o l K )

( 9 . 2 )

I t is f o u n d t h a t e q n ( 9 . 2 ) i s m o r e u s e f u l t h a n e q n ( 9 . 1 ) f o r c o m b u s t i o n c a l c u l a t i o n s

b e c a u s e t h e c o m b u s t i o n p r o c e s s t a k e s p l a c e o n a m o l a r b a s i s . T o b e a b l e t o w o r k i n a

m o l a r b a si s it is n e c e s sa r y t o k n o w t h e m o l e c u l a r w e i g h t s ( o r re l at iv e m o l e c u l a r m a s s e s )

o f th e e le m e n t s a n d c o m p o u n d s i n v o l v e d i n a r e a c ti o n .

9.1 Molecular w eigh ts

T h e m o l e c u l a r w e i g h t ( o r r e l a t i v e m o l e c u l a r m a s s ) o f a s u b s t a n c e i s t h e m a s s o f i t s

m o l e c u l e s r e l a t i v e t o t h a t o f o t h e r m o l e c u l e s . T h e d a t u m f o r m o l e c u l a r w e i g h t s i s c a r b o n -

1 2 , a n d t h i s i s g i v e n a m o l e c u l a r w e i g h t o f 1 2 . A l l o t h e r e l e m e n t s a n d c o m p o u n d s h a v e

m o l e c u l a r w e i g h t s r e l a t i v e t o t h i s , a n d t h e i r m o l e c u l a r w e i g h t s a r e n o t i n t e g e r s . T o b e a b l e

t o p e r f o r m c o m b u s t i o n c a l c u l a t i o n s i t i s n e c e s s a r y t o k n o w t h e a t o m i c o r m o l e c u l a r

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State equa tion fo r ideal gases 159

w e i g h t s o f c o m m o n l y e n c o u n t e r e d e l em e n t s ; t he s e c a n b e c o m b i n e d t o g i v e o t h e r

co m p o u n d s . T ab l e 9 . 1 g i v e s t h e d a t a f o r i n d i v i d u a l e l em en t s o r co m p o u n d s i n i n t eg r a l

n u m b er s ( ex cep t a i r w h i ch i s a m i x t u r e o f g a s e s ) ; i n r ea li ty o n l y ca r b o n - 1 2 ( u s ed a s t h e

b as i s f o r a t o m i c / m o l ecu l a r w e i g h t s ) h a s an i n t eg r a l v a l u e , b u t m o s t v a l u e s a r e v e r y c l o s e

t o i n t eg r a l o n es an d w i l l b e q u o t ed a s s uch .

T a b l e 9.1 M olecular weights of elements and comp ound s co mm onlyencountered in com bustion

Atmospheric

A ir 0 2 N2 N2 1-12 CO CO 2 1-120 C

m,, 28.97 32 28 28 .17 2 28 4 4 18 12

9.1.1 AIR

A s s t a te d p r e v i o u s ly , m o s t c o m b u s t i o n t ak e s p la c e b e t w e e n a h y d r o c a r b o n f u e l a n d a ir . A i r

i s a m i x t u r e o f g a s e s , t h e m o s t a b u n d a n t b e i n g o x y g e n a n d n i t r o g e n w i t h s m a l l p r o p o r ti o n s

o f o t h e r g a se s . I n f ac t , i n p r ac t i c e , a i r is a m i x t u r e o f a ll e l e m en t s a n d co m p o u n d s b ecau s e

ev e r y t h i n g w i l l ev ap o r a t e i n a i r u n t i l t h e p a r t i a l p r e s s u r e o f i t s a t o m s , o r m o l ecu l e s ,

a ch i ev es i t s s a t u r a t ed v ap o u r p r e s s u r e . I n r ea l i t y , t h i s ev ap o r a t i o n can u s u a l l y b e

n e g l e c t e d , e x c e p t i n th e c a s e o f w a te r . T a b l e 9 .1 s h o w s t h a t t h e m o l e c u l a r w e i g h t o f

a t m o s p h e r i c n i t r o g en i s h i g h e r t h an th a t o f p u r e n i tr o g en . T h i s i s b ecau s e ' a t m o s p h e r i c

n i t r o g en ' i s t ak en t o b e a m i x t u r e o f n i t r o g en an d ab o u t 1 . 8 % b y m as s o f a r g o n , c a r b o n

d i o x i d e a n d o t h e r g a s e s; t h e m o l e c u l a r w e i g h t o f a tm o s p h e r ic n i t r o g e n i n c l u d e s t h e e f f e c t

o f t h e o t h e r s u b s t an ces .T h e c o m p o s i t i o n o f a i r i s d e fi n e d a s 2 1 % 0 2 a n d 7 9 % N 2 b y v o l u m e ( th i s c a n b e w r i tt e n

2 1 m o l % O 2 an d 7 9 m o l % N 2 ) . T h i s is eq u i v a l en t t o 2 3 . 2 % 0 2 an d 7 6 . 8 % N 2 b y m as s .

9 .2 S ta te eq u a t i on fo r i dea l g ases

T h e eq u a t i o n o f s t a te f o r an i d ea l g a s, a , i s

p V - m , R ~ (9 .3)

I f t h e m as s o f g a s , m o , i s m ad e eq u a l t o t h e m o l ecu l a r w e i g h t o f t h e g a s in t h e ap p r o p r i a t e

u n i t s , t h en t h e am o u n t o f s u b s t an ce a i s k n o w n a s a mol ( i f t h e m as s i s i n k g t h en t h e

am o u n t o f s u b s t an ce i s c a l l ed a kmol ) . I f t h e v o l u m e o c c u p i e d b y t h a t t h i s a m o u n t o fsubs tan ce i s deno ted Vm the n

Vm = (9 . 4 )

w h e r e Vm i s t h e m o l a r s p e c i fi c v o l u m e , a n d h a s t h e u n it s o f m 3 / m o l o r m 3 / k m o l . N o w

A v o g ad r o ' s h y p o t h es i s s t a t e s t h a t

eq u a l v o l u m es o f a l l id ea l g a se s a t a p a r ti cu l a r t em p e r a t u r e an d p r e s s u r e co n t a i n t h e

s a m e n u m b e r o f m o l e c u l e s ( a n d h e n c e t h e sa m e a m o u n t o f s u b st a n c e) .

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160 Thermodynamic propert ies of ideal gases and ideal ga s mixtures

H e n c e , a n y o t h e r g a s , b , a t t h e s a m e p r e s s u r e a n d t e m p e r a t u r e w i l l o c c u p y t h e s a m e

v o l u m e a s g a s a , i .e .

PlUm --_ mw Ra m mw~b--- - . . . mw iR i m . . . -_ ~ (9 . 5 )

T

I f a s y st e m c o n t ai n s a n a m o u n t o f s u b s t a n c e n o o f g a s a , t h e n e q n ( 9 .5 ) m a y b e w r i t te n

pV m nom,oRoT= nog~T (9 .6 )

I t c an be seen t ha t fo r i dea l ga s e s t h e p rod uc t mwi R i i s t he sam e fo r a l l ga se s : i t i s c a l l ed

the un iversal gas constant, 9~. Th e v a l u e s o f t h e u n i v e r s a l g a s c o n s t a n t , 9 ~ , t o g e t h e r w i t h

i ts v a r io u s u n i t s a r e s h o w n i n T a b l e 9 . 2 .

Table 9 .2 Values of the universa l gas constant ,

9~, in va rious u nits

8314 J/kmol K

1.985 kca l /km ol K

1.985 Bm /lb-mol K

1 .985 C HU / l b-mol R

2782 f t l b t /l b -mol K

1545 f t lbf / lb-mol R

9 .2 .1 IDEAL GAS EQU ATION

T h e e v a l u a ti o n o f th e p r o p e r t ie s o f a n i d e a l g a s w i ll n o w b e c o n s i d e r e d . I t h a s b e e n s h o w nt h a t t h e i n t e r n a l e n e r g y a n d e n t h a l p y o f a n i d e a l g a s a r e n o t f u n c t i o n s o f t h e v o l u m e o r

p r e s s u r e , a n d h e n c e t h e s e p r o p e r t i e s a r e s i m p l y f u n c t i o n s o f t e m p e r a t u r e a l o n e . T h i s

m e a n s t h a t t h e s p e c i f i c h e a t s a t c o n s t a n t v o l u m e a n d c o n s t a n t p r e s s u r e a r e n o t p a r t i a l

d e r i v a ti v e s o f t e m p e r a t u r e , b u t c a n b e w r i t t e n

C,= , dT(9 .7 )

%= OT

Al so , i f c~ and cp i n mo l a r q uan t i t i e s a re de no t ed by C ~.m and C p.~ t hen , fo r i dea l and

pe r fec t ga se s

Cp.m- r = ~ (9 .8 )

I t i s p o s s i b l e t o e v a l u a t e t h e p r o p e r t i e s o f s u b s t a n c e s i n t e r m s o f u n i t m a s s ( s p e c i fi c

p r o p e rt ie s ) o r u n i t a m o u n t o f s u b s t a n c e ( m o l a r p r o p e r t ie s ) . T h e f o r m e r w i l l b e d e n o t e d b y

l o w e r c a se l et te r s ( e . g. v , u , h , g , e t c ) , a n d t h e l a t te r w i ll b e d e n o t e d b y l o w e r c a s e l e t te r s

an d a su ffix m (e.g. Vm, Urn, hm, gin, etc). T h e m o l a r p r o p e r t i e s a r e m o r e u s e f u l f o r

c o m b u s t i o n c a lc u l a t io n s , a n d t h e s e w i l l b e c o n s i d e r e d h e r e .

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S t a te e q u a t i o n f o r i d ea l g a s e s 1 61

T h e m o l a r i n t e r n a l e n e r g y a n d m o l a r e n t h a l p y c a n b e e v a l u a t e d b y i n t e g r a t i n g e q n s

(9 .7) , g iv ing

I r % m d T +Um= ro U0,m

(9 .9)

ITCp. m d T +h m - r o h o , m

wh ere U0.m and h0 .m are the va lues o f Um and hm a t the d a tu m tem pera tu re To. N ow , by

d e f i n i t i o n

h m -- IA m + pvm

an d f o r an i d ea l g a s pyre = ~RT; thus

hm =Um + ~RT (9.10)

He nc e, at T = 0, Um= h~ , i .e . U0.m= h0 .m. (A shn i l a r r e la t io nsh ip ex i s t s fo r the spec i f i c

proper t i es , u and h . )

T o b e ab l e t o ev a l u a t e th e i n te r n a l en e r g y o r en t h a l p y o f an i d ea l g a s u s in g eq n ( 9 . 1 0 ) i t i s

necessary to know the var ia t ion of spec i fi c hea t w i th t em pera ture . I t i s poss ib le to der i ve such

a f u n c t i o n f r o m em p i r i ca l d a t a b y cu r v e - f it t in g t e ch n i q u es , i f s u ch d a t a is av a il ab l e. I n s o m e

r eg i o n s i t i s n ece s s a r y t o u s e q u an t u m m e ch an i c s t o ev a l u a t e t h e d a t a , b u t t h is i s b e y o n d t h e

scope o f th i s cour se . I t w i l l be as sum ed here tha t the va lues of c~ are know n in the fo rm

C,.m - a + b T + c T 2 + d T 3 ( 9 . 1 1 )

w h e r e a , b , c an d d h a v e b e en ev a l u a t ed f r o m ex p e r i m e n t a l d a t a . S i n ce C p .m - C ~.m= 9 t

t h en an ex p r e s s i o n f o r Cp.m can b e ea s i ly o b t a i n ed . H en ce i t i s p o s s i b l e t o f i n d t h e v a l u e s o f

i n t e rn a l en e r g y an d en t h a l p y a t an y t em p e r a t u r e i f t h e v a l u e s a t To can b e ev a l u a t ed . T h i s

p r o b l e m i s n o t a m a j o r o n e i f t h e c o m p o s i t i o n o f t h e g a s r e m a i n s t h e s a m e , b e c a u s e t h e

d a t u m l ev e l s w i l l c an ce l o u t . H o w ev e r , i f t h e co m p o s i t i o n v a r i e s d u r i n g t h e p r o ces s i t i s

n e c e s sa r y t o k n o w t h e i n d i v i d u a l d a tu m v a l u e s . T h e y c a n b e m e a s u r e d b y c a l o r i m e t r i c o r

s p ec t r o g r ap h i c t e ch n i q u es . T h e ' t h e r m a l ' p a r t o f t h e i n t e r n a l en e r g y an d en t h a l p y , i . e . t h a t

w h i ch i s a f u n c t i o n o f t em p e r a t u r e , w i l l b e d en o t ed

f T C v m d Tum (T) = ro '(9 .12)

I r c p . m d Th m ( T ) - To

an d t h en eq n s ( 9 . 9 ) b eco m e

Um = urn(T ) + Uo.m (9 .1 3 )

h m= hm (T ) + h0,mas s u m i n g t h a t t h e b a s e t em p e r a t u r e i s ab s o l u t e z e r o .

E n t h a l p y o r i n t e r n a l en e r g y d a t a a r e g en e r a l l y p r e s en t ed i n t ab u l a r o r g r ap h i ca l f o r m .

T h e t w o co m m o n es t ap p r o ach es a r e d e s c r i b ed b e l o w . F i r s t , i t i s u s e f u l t o co n s i d e r f u r t h e r

t h e t e r m s i n v o l v ed .

9 . 2 .2 T H E S I G N I F I C A N C E O F UO.m A N D h o.m

As p rev ious ly d i scussed , U0.m and h0.m are the va lu es o f m ol ar in te rna l ene rgy , Um, and

m ola r en tha lpy , hm, a t the r e ferenc e t em per a ture , To. I f To = 0 then , fo r an idea l g a s ,

u0 .m ---- 0 . m (9 .14)

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16 2 Thermo dynamic properties of ideal gases and ideal gas mixtures

If To* 0 , then U0,m and ho. m al e d i f fe ren t . Mos t ca lcu la t ions invo lve changes in en tha lpy

or in te rna l ene rgy , a nd i f the compos i t ion dur ing a p rocess i s inva r ian t the va lu es o f ho.m o r

U0,m w ill ca nc el.

H o w e v e r , i f t h e c o m p o s i t i o n c h a n g e s d u r i n g a p r o c e s s i t i s n e c e s s a r y t o k n o w t h e

d i f fe rence be tw een the va lues o f U0,m fo r the d i f fe ren t spec ies a t the re fe ren ce tempe ra tu re .

Th is i s d i scussed be low.

Obviously U0,mand ho.m a re consequences o f the idea l gas a ssumpt ion and the e qua t ion

h m f fi hm (T ) + ho.m

con ta ins the a ssu m pt ion tha t the idea l g as law app l ie s dow n to To. I f To ffi O, o r a va lue

o u t si d e t h e s u p e r h e a t r e g i o n f o r t h e g a s b e i n g c o n s i d e r e d , t h e n th e g a s c e a s e s t o b e i d e a l ,

o f ten becoming e i the r l iqu id o r so l id . To a l low fo r th i s i t i s necessa ry to inc lude la ten t

h e a t s . T h i s w i l l n o t b e d e a l t w i t h h e r e , b u t t h e p u b l i s h e d d a t a d o i n c l u d e t h e s e

cor rec t ions .

9 .2 .3 ENTROPY OF AN IDEAL GAS - THIRD LAW OF THERM ODYNAM ICS

The change in en t ropy dur ing a p rocess i s de f ined as

dh.dsm = (9.1 5)

T p

I f the func t iona l re la t ionsh ip h m - - h m ( T ) i s k n o w n t h e n e q n ( 9 .1 5) m a y b e e v a l u a t e d ,

giving

I r dh ~ 9~ In p (9.16)5'm--SO'm---- To T po

where So.m is the va lue of Sm at To and Po.

I t i s conve n ien t to take the re fe rence temp era tu re To as abso lu te ze ro . I t w as p rev iou s lyshown tha t

dh m = Cp, m(T ) d T and hence dh~ = Cp.m(T ) d T (9 .17)T T

Cons ide r

T'---O

I f Cp .m(T0)~0 then the express ion i s e i the r in f in i te o r inde te rmina te . H ow ev er , be fo re

reach ing abso lu te ze ro the subs tance wi l l cease m be an idea l gas and wi l l become a so l id .

I t can be show n by the D ebye theory and ex per im en t tha t the spec i fic he a t o f a so l id i s

g iven by the law Cp-- aT 3. H e n c e

l i m ( d -~ ~ T ) = l i m ( a T T d T ) - li m ( a T 2 d T ) - -- > O (9 .18)T--~O T- ~0 T-->0

To in teg ra te f rom abso lu te ze ro to T i t is necessa ry to inc lude the la ten t he a ts , bu t s t i l l i t i s

poss ib le to eva lua te ~d h m / T .

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S t a te eq u a t i o n f o r i d ea l g a se s 16 3

I f s in (T) is de f in ed as

I r dhm d Ts (r) = r

t h e n e n t r o p y

(9 .19)

s~ -- sin(T) - 9~ In P--- + S O , m (9 .20)Po

The te rm S0.m i s the cons tan t o f in teg ra t ion and th i s can be co mp ared wi th //0 .m an d h0. m.

I f the com pos i t ion i s inva r ian t then the va lue o f S0.m wi l l cance l ou t w hen eva lu a t ing

changes in s and i t i s no t necessa ry to know i t s va lue . I f the compos i t ion va r ie s then i t i s

necess a ry to know , a t l eas t , the d i f fe rence be tween So va lues fo r the subs tances invo lv ed . I t

i s no t poss ib le to ob ta in any in fo rmat ion abou t So f rom c lass ica l (m acros cop ic )

t h e r m o d y n a m i c s b u t s t a t i s t i c a l t h e r m o d y n a m i c s s h o w s t h a t ' f o r a n i s o t h e r m a l p r o c e s s

i n v o l v i n g o n l y p h a s e s i n i n t e rn a l eq u i l ib r i u m t h e ch a n g e i n en t ro p y a p p ro a ch es z e ro a ta b so l u t e z e ro ' . This m eans tha t fo r subs tances that ex is t in c rys ta l line o r l iqu id fo rm a t low

temp era tu res i t i s poss ib le to eva lua te en t ropy changes by assum ing tha t d i f fe rences in S0.m

a r e z e r o .

I t shou ld a l so be no ted tha t the p ressu re te rm, p , in eqn (9 .20) i s the p a r t ia l p r e s s u r e o f

the gas i f i t i s con ta ined in a m ix tu re , and P0 i s a da tum pressu re (o f ten chosen as 1 ba r o r ,

in the pas t , 1 a tmo sphere ) .

9 .2 .4 T H E G I B B S E N E R G Y ( F U N C T I O N ) O F A N I D E A L G A S

T h e G i b b s e n e r g y w i l l n o w b e d e r i v e d f o r u s e l a t e r w h e n c o n s i d e r i n g t h e e q u i l i b r i u m

com pos i t ion o f mix tu res (d i s soc ia t ion ) . B y de f in i tion ,g m= h m - T s m ( 9 . 2 1 )

a n d

hm = h m(T)+ h0 .m (9 .22)

I f the p ressu re ra t io , P / P o , i s deno ted b y the symbol P r , i .e . Pr = P/PO, then

$ m = s m ( T ) - ~ I n P r + S 0.m

a n d

g - h (T ) + h o - T ( s (T ) - 9~ In Pr + SO)

-- ( h o - T So ) + ( h ( T ) - T s ( T ) ) + ~ T In p~

I f the te rms a t To a re com bin ed to g ive

g0.m = h 0 , m - T$o.m

a n d t h e t e m p e r a t u r e - d e p e n d e n t t e r m s a r e c o m b i n ed t o g i v e

g m ( T ) = h m ( T ) - Tsm( r )

then

gm = gin(T) + 9~T In Pr + go.m

(9 .23)

(9 .24)

(9 .25)

(9 .26)

(9 .27)

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164 Thermod ynamic properties of ideal gases and ideal gas mixtures

O f t en g .( T ) and go, , . are co m b i n ed t o g i v e g O w h i ch i s the pressure-indepen dent portionof the Gibbs energy, i .e. gO ffi gin(T) + g 0,. , a nd the n

g m = g O + ~ T I n P r ( 9 . 2 8 )

g O i s t h e v a l u e o f t h e m o l a r G i b b s en e r g y a t a t em p e r a t u r e T and a pressure o fp o, and i t i sa f u n c t i o n o f t em p e r a t u r e a l o n e . T h e d a t u m p r e s s u r e , P 0 , i s u s u a l l y ch o s en a s 1 b a r

n o w a d a y s ( a l t h o u g h m u c h d a t a a r e p u b l is h e d b a s e d o n a d a t u m p r e s s u re o f 1 a t m ; t h e

d i f f e r en ce i s n o t u s u a l l y s i g n i f i c an t i n en g i n ee r i n g p r o b l em s , b u t i t i s p o s s i b l e t o co n v e r t

s o m e o f t h e d a t a , e . g . eq u i l i b r i u m co n s t an t s ) .

G i b b s e n e r g y p r e s e n t s t h e s a m e d i f f i c u l t y w h e n d e a l i n g w i t h m i x t u r e s o f v a r y i n g

co m p o s i t i o n a s u o , h0 an d S o. I f th e co m p o s i t i o n i s in v a r i an t , ch an g es i n G i b b s en e r g y a r e

e a s il y c a l c u l a te d b e c a u s e t h e g o t e rm s c a n c e l. F o r m i x t u re s o f v a r y i n g c o m p o s i t i o n g o m u s t

b e k n o w n . I t c an b e s een f r o m e q n ( 9 .2 6 ) t h a t i f t h e r e f e r en ce t em p e r a t u r e , T o, eq u a l s z e r o

t h en

g 0 , m = h o , m - U 0 , m ( 9 . 2 9 )

9 .3 T a b l e s o f u(T) an d h(T) ag a i ns t T

T h e s e t a b l e s a r e b a s e d o n p o l y n o m i a l e q u a t i o n s d e f i n i n g t h e e n t h a l p y o f t h e g a s . T h e

n u m b e r o f te r m s c a n v a r y d e p e n d i n g o n t h e re q u i r e d a c c u r ac y a n d t h e t e m p e r a t u r e r a n g e t o

b e c o v e re d . T h i s s e c t i o n w i l l li m i t t h e n u m b e r o f c o e f f ic i e n t s to s i x , b a s e d o n B e n s o n

( 1 9 7 7 ). T h e eq u a t i o n u s ed i s o f t h e fo r m

h m ( T ) h m - ho.m= = al + a2T + a3T 2 + a4T 3 + asT 4 (9.30 )

9~T 9~T

T h e v a l u e s o f t h e co e f f i c i en t s f o r v a r i o u s g a s e s a r e li s ted i n T ab l e 9 . 3 , f o r t h e r an g e 5 0 0 t o

3 0 0 0 K . I f u s ed o u t s i d e t h e s e ran g es t h e accu r acy o f t h e ca l cu l a t i o n w i l l d i m i n i s h .

Table 9.3 Enthalpy coeff icients for selected gase s found in combust ion processes (based on kJ /km ol, with the temperature in K)

h 0 .~ a 0

S ubs t ance a s a4 a3 a2 a i a6 k J / k m o l

1-12 0 . 0 0 1 3 0 - 1 . 4 4 3 9 2 e - I I 9 . 6 6 9 9 0 e - 8

C O 0 . 0 0 0 0 - 2 . 1 9 4 5 0 e - 1 2 - 3 . 2 2 0 8 0 e - 8

N 2 0 . 0 0 0 0 - 6 . 5 7 4 7 0 e - 1 2 1 . 9 5 30 0 e - 9

N O 0 . 0 0 0 0 - 4 . 9 0 3 6 0 e - 1 2 - 9 . 5 8 8 0 0 e - 9

CO2 0.O(KI0 8 .6600 2e- 11 -7 . 88 54 2e -7

0 2 0 . 0 0 1 3 0 1 . 53 8 9 7 e- 1 1 - 1 . 4 9 52 4 e -7

H20 0 . ~ - - 1 . 81802e - 11 4 . 95240e - 8

CI'I4 - 8 . 5 86 1 l e - 15 1 . 62497e - 10 - 1 . 24402e - 6

O 0 . 0000 - 1 . 38670e - 11 1 . 00187e - 7

- 8 . 1 8 1 0 0 e - 6 3 . 4 3 3 2 8

3 . 7 6 9 7 0 e - 4 3 . 3 1 7 0 0 0

2 . 94260e - 4 3 . 34435

2 . 99380e - 4 3 . 50174

2 . 73114e - 3 3 . 09590

6 . 52350e - 4 3 . 25304

5 . 65590e - 4 3 . 74292

4 . 96462e - 3 1 . 93529

- 2 . 5 1 4 2 7 e - 4 2 . 7 6 4 0 3

- 3 . 8 4 4 7 0 0 . 0 0 0 0

4 . 6 3 2 8 4 - 1 . 1 3 8 82 e 5

3 . 7 5 8 6 3 0 . 0 0 0 0

5 . 1 1 3 4 6 8 . 9 9 1 4 7 0 4

6 . 5 8 3 9 3 - 3 . 9 3 4 0 5 e 5

5 . 7 1 2 4 3 0 . ( D ~

9 . 6 5 1 4 0 e - 1 - 2 . 3 9 0 8 2 e 5

8 . 1 5 3 0 0 - 6 . 6 9 3 0 0 4

3 . 7 3 3 0 9 2 . 4 6 9 2 3 e 5

H e n c e th e e n t h a l p y , i n te r n a l e n e r g y , e n t r o p y a n d G i b b s e n e r g y c a n b e e v a l u a t e d a s

fo l lows .

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Tables o f u (T) an d h (T) aga ins t T 165

Enthalpy

hm(T) = ~T ( a l + a2T + a3T 2 + a4T 3 + asT 4)

= ~ ( a lT + a2 T2 + a3T 3 + a4T 4 + a5 T5)

5

= ~ ~ a i T i

i - i

(9.31)

Internal energy

um (T) ffi f f tT( (a l - 1) + a2T + a3T 2 + a4T 3 + a5 T4)

= 9~((a l - 1)T + a2 T 2 + a3T 3 + a4 T 4 + a5 TS)

5

= 9~ ~,, t~T t - 9 ~ T = h (T ) - 9~Ti= l

(9.32)

Entropy

This is def ined by eqn (9 .15) as

d s m = d h m _ ~ d p

T p

Integrat ing eqn (9.15) g ives

I r---dhm 9 t _ In( p )sin(T) = S m - So,m = ro r

Equat ion (9 .20) provides some problems in solut ion. The f i rs t i s that

I rdhm ffi I r e P , m d rro T ro T

and this results in In(0) w hen T ffi To. For tun ate ly these p rob lem s can be o ver com e by useof the Van ' t Hof f equa t ion , which wi l l be de r ived in Chap te r 12 , where d i s soc ia t ion i sin t roduced. This s ta tes that

hm =-T2[d--~-~dT l~~ (9.33)

wh ere, for To = 0 K

/~o = che m ical po tentia l = gm (T)+ g0.m = gm (T )+ ho.m (9.34)

and

hm = hm (T) + ho.m

(Note: the term/~, is s imilar to gO, which was in t roduced in eqn (9 .28) . At th is s tage i t i s

suff ic ient to note that the chemical potent ia l has the same numerical value as the specif ic

Gibbs energy. The chemical potent ia l wi l l be def ine d in Sect ion 12.2 . )

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166 Thermodynamicproperties o f ideal gases and ideal gas mixtures

B y d e f i n i t io n , n e q n ( 9 . 3 ) ,

h . ( r ) __ -h0 .ffi az + a2T + as T 2 + a4T3 + as T4

9~T 9iT

giving

h m ( T ) a l= - - + a 2 + a s T + a 4 T 2 + a s 3

9 T T

Subst i tuting eqns (9.34) , (9.13) and (9.36) into eqn (9.33) and rearranging gives

- " -- + a z + a s T + a 4 T 2 + a s T ' + " ~ ~ 2 d r = - -9t

which can be integrated to

a s T 2 a 4 T 3 a s T 4 ~ T },, e.,(r) ho.~,- a ~ I n T + a z T + , - + . . . . . + . . . . . . + A = .................

2 3 4 ~RT

It is convention al to let

A f fi l - a s

e . ( r )

a n d t h e n

9 ~ T

ffi l ( - I n T ) - [ a 2 T +a3T2 a4T3 a4T4

....... + . . . . . + ... 06

2 3

The value of entropy can the n be obtained from

(9.35)

giving

(9.36)

(9.37)

(9 .38)

(9.39)

(9.4O)

~ ( T ) = h m ( T ) - g i n (T ) ( 9 . 4 1 )

T

a 3 4 5 6 )s i n( T ) = 9 ~ l I n T + 2 a 2 T + - - a s T 2 + - - a 4 T s + - - a s T 4 + a

2 3 4

(9.42)

Gibbs energy

The Gibbs energy can be evaluated from eqn (9.40) or eqn (9.26):

gin(T) = hm(T)- Tsm(T)

The values obtained from this approach, using eqns (9.31) to (9.42) are given in Tables9.4 for oxygen, ni t rogen, hydrogen, water , carbon monoxide, carbon dioxide, ni t r ic oxide

and methane. T he tables hav e been evaluated up to 350 0 k, s light ly bey on d the ran gestated in relat ion to Table 9.3, but the error is not s ignif icant . This enables most

combust ion problems to be solved using this data . Other commonly used tables are those

of JANA F (1971).

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T a b l e 9 . 4

M o l e c u l a r o x y s e n , 0 2 " /110= 0 . 0 0 0 0 U / k m o l M o l e c u l a r n i t r o g e n , N 2 " h 0 = 0 . 0 0 0 0 k J / k n m l

....T ~(:D50 1365 .74

100 2757.641150 4174.922O0 5616 .5625 0 ? 0 8 1 . 7 5

298 &~Of. ; 'e300 8569.64350 1OO79.40400 11610.23450 13161.33500 14731.93550 16321.3000O 17928.6tl650 19553.39700 21194.7175O 221151.99

2452435850 26211 . 78900 27913.05

,m .m ns~ .u ++2+ .+1 9 2 6 . 2 5 1 7 3 . 1 2 - I , 5 3 . 92 ~ 7 . 7 7 t u . s o .235n5 .43 9 5 3 . 7 0 1 9 2 . 8 9 - 3 2 9 6 2 . 15003 .18 1 9 9 . 4 3 - 4 2 7 7 5 . 9

+e , .u .~ , ~ .+ , . s . -.~?+vv..,eo~s .~s 2o~.ss .5 2m .87 1 6 9. 40 2 0 9 . 5 1 - 63 248 . 6t 2 u . ~ l 2 t ~ . e o .7 3 82 t29419 .89 2 1 7 . 2 5 414601.0

10574 .711 220 .56 -93547 .5117411 .43 2 2 3 . 5 9 - I06652 .212940.10 2 2 6 . 3 9 - I17902 .414149.09 2 211 .99 -129287.415374.70 23 1 .4 2 -14079~.216616.26 233.71 -152426.111'11173.11 235.116 -164166.619144 .62 237 .91 -170011 .42 0 43 0 .1 8 2 3 9 . 8 5 - I r l 95 5 . 11

9 5 4 ) 2 9 6 2 7 . ' / 6 2 1 " / 2 9 . 1 8 2 4 1 . 7 1 - 1 99 9 9 5 .3" 1 0 0 0 ' 3 1 3 5 5 .3 3 2 3 0 4 1 . 0 5 2 4 3 . 4 8 - 2 i 2 1 2 5 . 3

1 0 5 0 3 3 0 9 5 . 2 5 2 4 3 6 5 . 2 3 2 4 5 . 1 1 1 -2 2 43 4 2. 1

I1 00 341146.92 257 01 .1 9 246 .111 -256642 .0115 0 36 609 .14 270411 .40 24 t l .311 -249021 .91 2 0 0 3 L t 8 3 . 5 2 2 1 14 0 6. 36 2 4 9 . 8 9 -2 61 47 11 .61 2 5 0 4 0 1 6 7 . 4 7 2 9 7 7 4 . 3 9 2 5 1 . 3 4 - 27 40 09 .51 3 0 0 4 1 9 6 1 . 2 2 3 1 1 5 2 . 6 3 2 5 2 . ? 5 - 28 6 61 2 .01 3 5 0 4 3 7 6 4 . 3 4 3 2 . 5 4 0. 0 4 2 5 4 . 1 1 - 29 92 11 3.61 4 0 0 4 5 5 7 6 . 4 1 3 3 9 3 6 . 3 9 2 5 5 . 4 3 - 31 2O 2 2.21450

i

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245 O 1~1411.14 64778.110 276 .462 5 0 0 8 7 0 8 9 . 2 ~25502600265O2700

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9 6 8 5 9 . 1 1 0 7 3 0 i ~ . 4 ~ 2 8 0 . 9 698828.11 755411.07 281.6"7100801.43 77105.611 2112.3710277 9.90 ?116611.43 2113.06~ o 4 7 6 3 . ~ 8 o 2 3 e . 3 2 2. 3 .7 4106753.02 8 1 8 1 0 . 1 2 2114.411011748.05 83389.44 285.0 '1110749.05 114974.72 2115.72112756 .24 110566 . 20 286 . 3 6114769 . 90 88164 . 14 2116 . 99116'790.32 1197611.85 2117.62118817.110 9i380.61 288.241201152.67 92999.76 2118.85122995 .26 94626 .64 289 .46124945.95 96 261 .62 290 .06

r ~ . o 6 2 g o . 6 5

T50

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~ r ~ . . . . . . ,< 7) ~ r ~ _ ~ .13%41 9~10 .~ 140 .27 .5617 :221105Ay7 1 9 7 3 . 6 4 1 5 9 . 7 9 . 1 3 1 7 4 . 04 2 2 5 .9 6 2 9 7 8 . 8 2 I 7 1 . 3 1 - 2 1 47 0 .55 6 5O .0 9 3 9 9 6 . 2 3 1 7 9 . 5 5 -3 0 2 51 .7~ t 0 ~ . 4 ~ S 0 2 S . ~ S m 6 .0 0 . ~ 9 5 . 4

1 0 03 1 . 6 6 7 1 2 1 . 6 5 1 9 5 . 8 5 -5 11 51 5.21 1 5 1 3 . 46 8 1 1 1 7 . 7 4 1 9 9 . 8 1 -6114041.6

45 05OO55 06OO65 0

75 0

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11001150120012501300135014001450

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4 1 8 3 5 . 2 6 3 0 6 3 0 . 9 5 2 3 8 . 1 4 - 2" 19 63 7. 04 3 5 58 .1 3 3 1 9 1 8 . 1 1 2 3 9 . 3 8 .29|575.24 5 2 ~ 0 . ~ s ~ 3 2 n 4 . 6 n u o . s 8 . ~ o ~ s ? t s4 6 9 91 .7 3 3 4 5 2 0 . 2 8 2 4 1 . 7 5 -3 1 5 63 2 .94 1 1 ~ 2 . 0 1 3 5 8 3 4 . 8 3 2 4 2 . t m . 3 2 7 7 4 1 1 . 95 0 4 6 0 .9 6 3 7 1 5 8 .0 1 1 2 4 3 . 9 9 -3 3 9 9 2 0 .95 2 2 0 8 . 3 3 3 1 4 1 1 9 . 7 4 2 4 5 . 0 6 -352147 . 35 3 9 6 3 .8 7 3 9 1 1 2 9. 5 6 2 4 6 . 1 1 -3 6 4 4 2 6 .85 57 27 .3 2 4 1 t 7 7 . 3 0 2 4 7 . 1 3 - 3 7 6 75 8 . 15 7 4 9 8 .4 0 4 2 5 3 2 . 6 6 24 1 1 .1 3 - 3 8 9 13 9 . 959276.1t5 4 3 1 E 1 ~ . 3 9 2 4 9 . 1 1 -401570 .96 1 0 6 2 .3 6 4 5 2 6 3 . 1 9 2 5 0 . 0 6 - 4 1 40 5 0 .2621154.64 46 64 1 . 7 6 25039 . 42~ 76 . $6 4 6 5 3. 40 4 8 0 2 4 . 8 0 2 5 1 . 9 0 -4 3 9 14 8 .966458 .32 4 9 4 1 4 . 0 0 2 5 2 . 7 9 4 5 1 7 6 6 . 3s lm 9 . 07 5 0 5 0 1 9 4 2 3 7 . 6 7 - a~ 2 7. 11? 0 0 8 3 .3 4 3 2 2 0 9 . 6 0 2 3 4 . 3 2 - 47 7 t3 2 .57 .m6 . 79 526 13 . 33 25 5 . 5 6 41m79 . 573733.06 55 02 5.5 9 256.111 . s026 es .o7 5 5 6 3 .8 1 5 6 4 4 0 . 9 2 2 5 6 . 9 8 - S 15 4 97 .177395 .~ 5 - m eo . o s 25 7 . 7 7 . 525366 . 0- ~ n 3 7 . 3 0 39282 . 9s 2311 . 33 -541274 .181r , 60709 .25 , 25 9 .3 1 -554220 .48 2 9 2 4 . 2 6 6 2 1 3 8 . 5 1 2 6 0 . ~ " ~ ~ . 4847 '71 .83 63 57 0 .3 6 260 .711 -51t0225 .31 16 62 1.3 8 6 5 0 0 4 . 4 0 2 6 1 . 5 0 .5 9 3 2 8 2 .58 8 4 7 3 .1 2 6 6 4 4 O . 2 2 2 6 2 . 2 1 . 6 0 6 3 73 . 39 0 32 6 .0 1 6 7 8 7 7 . 4 0 2 6 2 . 9 0 - 6 19 5 0 3. 0

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, , , . , , .

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Table 9 .4 continued

C a r b o n m o n o x i d e , C O :h o = - 1 1 3 8 8 2 . 3 k J / ~ o l C a r b o n d i o x i d e , C O 2 9 h o = - 3 9 3 4 0 4 . 9 k J / ~ o l

...... ~ r ) .(+3, J ( D ~ D50 1386.73 971.01 146.72 -5949.2

100 27 88 .9 3 1957.50 166.15 -13825.6150 420 6.3 9 295 9.2 4 177. 64 -22439.0200 56 38 .9 0 3976.04 185.88 -31536.3250 70 86 .2 7 5007.69 192.33 -40997.3

2 9 8 5 4 5 9 . ~ I 6 0 1 1 . 5 4 ~ 9 7 . 4 7 . 5 0 3 5 6 . 03 0 o i i 5 4 s . 2 e e o s s . ~ 7 1 9 ~ . ~ - so v sl . i350 10024 .68 7114 .67 20 2. 22 -60750.8400 1151 5.29 8189 .5 7 20 6. 20 -70963.2

450 13019.87 92~ 44, 209.74 -81363.2

500 i45 38. 21 10351.06 21 2. 94 -91931.5-550 160 70. 08 11497. 21 21 5. 86 -102652.5600 17 61 5. 24 12 626 .6 6 218.55 -I 13513.6650 1917 3.47 13769 .17 22 1. 04 -124504.1700 207 44. 52 14924. 51 22 3. 37 -135615.0750 223 28. 17 16092 .44 22 5. 56 -146838.7800 239 24. 16 17272 .72 22 7. 62 -158168.5850 255 32. 26 18465. 11 22 9. 57 -169598.5900 271 52. 23 1966 9.3 6 23 1. 42 -181123.5950 , 28 78 3. 80 20 88 5. 22 233.1 8 ,,, -192738.8

1000 3~1 26.7 4 22'il2.44 23 4. 87 -204440.31050 32080 .79 23350.77 2 36. 48 -216224.31100 33745. 68 24599. 95 23 8.0 3 -228087.31150 35421. 17 25859. 72 23 9.5 2 -240026.31200 37106. 98 27129. 82 24 0.9 5 -252038.41250 38802. 85 28409.98 24 2.3 4 -264120.91300 40508. 52 29699.93 24 3.6 8 -276271.51350 42223. 72 30999.41 24 4.9 7 -288487.91400 43948 .16 32308. 14 24 6.2 3 -300768.01450 45681. 58 33625.84 24 7.4 4 -313109.8

1500 47 42 3. 69 34952.24 ' 248. 62 -3255il.6"1550 49174. 22 36287.06 2 49 .77 -337971.61600 5 0 9 3 2 . 8 8 3 7 6 3 0 . 0 0 2 5 0 . 8 9 -3 50 48 8.31650 52699.38 3 8 9 8 0 . 7 9 251.98 -363060.01 7 0 0 5 4 4 7 3 . 4 4 40339.13 2 5 3 . 0 3 - 3 7 5 6 8 5 . 3

1750 56254.75 4 1 7 0 4 . 7 3 254.07 -388363.01800 58043. 03 43077. 29 255 .0 7 -401091.61850 59837. 98 44456. 53 2 56. 06 .413870.1

1900 6163 9.29 45842. 12 257. 02 426697.19 1950 6344 6.67 47233. 79 257 .9 6 -439571.6

2000 65259.8 1 48631.21 25 8. 88 -452492.62050 67078 .39 50034. 07 259 .77 -465458.92100 68902.I 514 42. 08 26 0. 65 478469.72150 70730. 65 52854. 90 261. 51 -491523.92200 72563. 69 54272. 23 262 .3 6 -504620.82250 74400. 92 55693. 74 263 .18 -517759.2300 76242.01 57119. 12 263 .9 9 -530938.72350 780 86. 64 585 48. 03 264.79 -544 158.22400 79934. 47 59980. 15 265 .56 -557417.0

2450 81785. 19 61415. 15 266 .3 3 -570714.3' ' ~ [ ' J I , , --

2500 83638. 45 62852 .70 26 7.0 8 -584049.42550 85493.91 64292. 45 267 .81 -597421.52600 8735 i.25 65 73 4. 07 26 8. 53 ,.610830.2650 89210. I 671 77. 22 2 69 .2 4 -624274.42700 91070 .16 68621. 55 269 .9 3 -637753.9

2750 92931 .04 70066.71 27 0. 62 -651267.72 8 0 0 94792.40 71512.36 271.29 - - 6 6 4 8 1 5 . 4

2850 96653 .90 72958.15 27 1.9 5 -678396.42900 98515 .18 74403.71 27 2. 59 -692010.02950 , 100375.87 75848 .68 ,, 273.23 .... -705655.7

3000 102235.62 772 92.72 273 .86 -719332.93050 104094.07 78735. 45 2 74. 47 -73304. I3100 105950.84 80176. 51 27 5. 07 -746779.83150 107805.57 81615 .53 2 75 .6 7 -760548.43200 109657.89 83052.13 276 .25 -774346.43250 111507.42 84485 .95 2 76 .8 2 -788173.33300 113353.79 8591 6.60 277 .39 -802028.73350 !15196.61 87343.7 1 277 .9 4 -815912.13400 II 7035.50 887 66. 88 27 8. 49 -829822.93450 118870.08 90185 .75 2 79 .0 2 -843760.73500 120699.96 91599 .91 27 9. 55 457725.0

. i i

r ; , ( D50 1342.97

100 2794.62150 4350.19200 6005.05250 7754.65

298 9519.29300 9594.56350 I 1520.46400 13528.14450 15613.47

i.

500 17772.48550 20001.266 0 0 2 2 2 9 6 . 0 3

650 24653.127 0 0 2 7 0 6 8 . 9 6

750 29540.09800 32063.16850 34634.93900 37252.26950 39912.12

i,

1000 42611.601050 45347.891100 48118.271150 50920.171200 53751.081250 56608.631300 59490.551350 62394.681 4 0 0 65318.96

,1450 68261.44

1500 71220.291550 74193.771600 77180.261650 80178.251 7 0 0 8 3 1 8 6 . 3 31 7 5 0 8 6 2 0 3 . 2 01 8 0 0 8 9 2 2 7 . 6 61850 92258.64

1900 95295.171950 98326.36

2000 101381.472050 104429.8521 O0 107480.952150 110534.332200 113589.672250 116646.752300 119705.462350 122765.'/92400 125827.862450 128891.87

i

2500 131958.14255 0 135027. I2600 138099.312650 141175.382700 144256.08

2750 147342.2828O0 150434.932850 153535.1229OO 156644.022950 159762.95

m r ) J t n m r }927 .25 157.68 -6541.+

1 9 6 3 .1 9 1 7 7 . 7 2 .1 4 9 7 7 .73103.05 190. 31 -24196.34342.19 1 99 .8 2 -33958.65676.08 207.62 -44150.0

7 0 4 1. 63 2 1 4 . 0 7 - $ 4 2 7 3. 9, I I I l l , J I , _ _

7100.27 214.32 -34702.38610.46 220.26 -65569.5

10202.42 225.62 -76718.51 1 87 2 .0 4 2 3 0 . 5 3 - 8 8 1 2 3 .7

. i i i

1 3 6 15 .3 3 2 3 5 . 0 8 -9 9 7 6 5. 11 5 4 2 8 . 3 9 2 3 9 . 3 2 - I 1 1 62 6 .217307.45 243.32 -I 23693.219248.83 24 7. 09 -135954.21248.95 25 0. 67 -148398.823304.37 25 4. 08 -161018.25411.72 25 7. 33 -173804.027567.78 26 0. 45 -186749.229769.39 26 3. 44 -199847.132013.54 26 6. 32 -213091.6

34297.30 ' 269J~9 -226477.3-3 6 6 1 7 . 8 7 2 7 1 . 7 6 - 23 9 9 9 8. 83 8 9 7 2 . 5 4 2 7 4 . 3 4 - 25 3 6 5 1. 641358.72 27 6. 83 -267431.043773.92 27 9. 24 -281332.946215.76 28 1. 57 -295353.468681.96 2 8 3. 8 3 -3094811.7

51170.37 28 6. 02 -323735.353678.94 28 8. 15 -338089.856205.70 29 0. 21 -352549.2

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58748.84 29 2. 22 -367110.361306.60 29 4. 17 -381770.363877.38 29 6. 07 -396526.466459.66 29 7. 91 .411376.169052.02 29 9. 71 .426316.871653.17 30 1. 46 -441346.174261.92 30 3. 16 -456461.7- ~ T ~ . ~9 3 04 . 5 2 . 47~s ~ .4

79498.00 30 6. 44 486943.282123.48 , 308 .02 -502304.984752.87 309.56 - 51 7 7 4 4 . 7

87385.53 31 1. 07 -533260.690020.92 31 2. 54 -548850.992658.58 31 3. 98 -564513.995298.21 31 5. 38 -580248.097939.57 31 6. 75 -596051.5

100582.57 318. I0 -611923.0103227.19 31 9. 42 -627861.0105873.54 32 0. 70 -64386,4.10852 ! .83 321.97 -659931.0

i i

I I 1172.39 323.21113825.64 324.42116482.13 325.62119142.48 326.79121807.47 327.94

124477.95 329.07127154.89 330.19129839.36 331.28132532.55 332.37135235.76 333.43

i i . i

3000 162893.29 137950.39 334.483050 166036.55 140677.93 335.523100 169194.35 143420.02 336.553150 172368.42 146178.38 337.573200 175560.59 148954.83 338.573250 178772.80 151751.32 339.573300 182007.10 154569.91 340.553350 185265.64 157412.74 341.533400 188550.70 160282.08 342.513450 191864.64 163180.30 343.4835,00 19520 9.95 1661,09.90 ,344.44

-67606O.5-692251.4-708502.5-724812.6-741180.9

-757006.3-774087.8-790624.7-807216.0-823861.0

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-1010324.9

7/29/2019 Chap9 Thermodynamics of Ideal Gases

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T a b l e 9 . 4 continued

M o l e c u l a r h y d r o g e n , H 2 9 h 0 = 0 . 0 0 0 0 k J / k m o l W a t e r , H 2 0 " h 0 = - 2 3 9 0 8 1 . 7 k J / k m o l

T

501001502OO250

2 ~3OO3504OO

450

50055060065070O75080085090O

950

1000

IO5011001150120012501300135014001450

15001550160016501700175018001850

19001950

2OOO

205021002150220022502300235O24002 4 5 0

25O0255026OO265027OO

27502800285029002950

3O0O30503100315032O0325033003350340O34503500

~ ~ r ~ . . . . . ~ r7 ~142 7.2 0 1011.4 8 79.70 -2557.8285 4.64 2023 .21 99.49 -7094.24282.92 3035.711 111.07 -12377.65712.58 40 49 .7 2 119.30 -18146.67144.17 50 65. 60 12 5. 68 -24277.0

sJzo. 79 , ~j . l ,.~ ~~o. 7~ . ~ o m . ~

8578.21 60 83. 92 13 0. 91 -30695.910015.20 710 5. 19 13 5. 34 -37355.111455.63 812 9.9 1 13 9. 19 -44220.6121~9.96 , 9!_58.53 142.59 , -51266.8

14348.65 10191.5015802.13 1 1 2 2 9 . 2 6

17260.81 12272.2318725.08 13320.7920195.34 14375.3321671.92 15436.2023155.19 16503.7524645.46 17578.3026143.03 18660.1627648.20 19749.61

29161.23 20~'93

30682.38 21952.3732211.88 23066.1533749.95 24188.5135296.78 25319.6236852.56 26459.6838417.44 27608.8539991.57 28767.2641575.07 29935.0543168.06 31112.33

44770.62 32299.1746382.83 33495.6648004.73 34701.8549636.37 35917.7751277.76 37143.4552928.90 38378.8854589.77 39624.0356260.35 40878.89

57940.56 42143.3959630.34 43417.46

61329.61 44701.0163038.25 45993.9364756.13 47296.1066483. I I 48607.3668219.03 49927.5769963.71 51256.5471716.95 52594.0673478.53 53939.9375248.23 55293.9177025.78 56655.74

i

78810.91 5~)25.1680603.35 59401.8882402.77 6O785.59~208.86 62175.9786021.28 63572.67

87839.67 64975.3489663.63 66383.5991492.79 67797.0493326.72 69215.25

145.65 -58474.1148 .42 -65826.7150.95 -73311.8153.30 -80918.9155.48 -88638.9157.51 -96464.3159.43 -104388.4161.24 -I 12405.4162.95 -120510.4164.58 - 1 2 8 6 9 8 . 8

il l

166.13 -136966.7

167.61 -145310.5169.04 -153726.9170.40 -162213.1171.72 -170766.4172.99 -179384.3174.22 -188064.6175.41 -196805.3176.56 -205604.5177.67 -214460.4

178.76 -223371.51 7 9 . 8 2 -232336.11 8 0 . 8 5 -241352.9181.85 -250420.5182.83 -259537.7183.79 -268703.4184.73 -277916.4185.64 -287175.6

186.54 -296480.187.42 ,, ,-305829.0

188.28 -315221.4189.12 -324656.3189.95 -334133.0190.76 -343650.8191.56 -353208.8192.34 -362806.3193. I I -372442.8193.87 -382117.4194.62 -391829.6195.35 -401578.8

196.07 -411364.3196.78 -421185.6197.48 -431042.1198.17 -440933.31 9 8 . 8 4 -450858.6

199.51 460817.6200.17 470809.6200.82 480834.3201.45 -490891.

95165.00 7 0 6 3 7 . 8 2 , , 2 0 ~ ,: 0 8

97007.17 72064.2798852.76 73494.15

100701.29 74926.96102552.25 76362.21104405.12 77799.36106259.36 79237.89108114.42 80677.23109969.70 82116.80I 11824.63 83556.01113678.59 84994.25115530.94 86430.89

i , , | i

-500979.6

202.70 -511099.3203.31 -521249.7203.91 -531430.3204.51 -541640.9205.09 -551880.8205.66 -562149.6206.23 -572447.206.79 -582772.6207. 34 -593125.8207.88 .603506.3208.41 -613913.6

i | , | , i .

r ~ .~ r t , ~ r ~50 1567 .79 I 132.0 8 130. 24 4944. I

100 315 9.4 0 2327.9 7 152 .28 -12068.9150 477 5.0 0 3527.9 4 165 .38 -20031.6200 64 15 .1 0 4752.24 174.81 -28547.250 8079.6 9 6001. I I 18 2. 24 -37479.7

2 9 8 9 7 0 0 . ~ , , 7 2 2 3 . 3 3 1 8 8 . 1 7 -46373.3300 97 69 .0 4 72 74 .7 5 '188.40 ~ -46749.9350 11483.36 85 73.3 5 193 .68 -56304.94 0 0 1322 2.78 9~7.06 19 8. 33 -66107.3

450 , 14 98 7. 47 11246.03 ,, 202.48 ,, -76127.3

500 167 77. 52 12620.37 20 6. 25 46349.15506OO6507OO75O8008509OO

18593.04 1402 0.18 20 9. 71 -96749.520434.10 1544 5.52 21 2. 92 -107316.222300.74 16 89 6. 45 215.91 -I 18037.624192.99 18372 .98 21 8. 71 -128903.726110.85 19875 .13 22 1. 36 -139905.928054.30, 214 02. 86 223,116 -151036.930023.31 229 56. 15 22 6. 25 -162290.332017.79 24 53 4. 92 22 8.53 -I 73660.3

950 340 37. 68 26 13 9. 09 230.72 ,, 7185141.$

1000 36082.85 " 2"/7fdl.55 23 2. 81 -196730.4

38153.17 294 23 .15 234 .8 3 -2011421.840248.48 3110 2.75 23 6. 78 -220212.542368.62 328 07. 17 23 8. 67 -232099.044513.37 3453 6.21 24 0. 49 -244078.346682.51 36 28 9. 64 242 .26 -256 147.448875.80 3806 7.21 24 3. 98 -268303.851092.97 398 68. 66 24 5. 66 -280545.153333.72 416 93. 70 24 7. 29 -292868.9

105011001150120012501300135014001450

15001550160016501700175018001850

1900

55597.74 43542.00i . i

2 4 8 . 8 8 - 3 0 5 2 7 3 . 2|

571184.69 454 3. 24 2 50 .4 3 -317755.960194.21 473 07. 04 25 1. 94 -330315.362525.92 492 23. 04 25 3. 42 -342949.564879.42 5116 0.8 2 25 4. 87 -355656.967254.27 531 19. 96 25 6. 29 -368436.069650.03 551 00. 00 25 7. 68 -381285.372066.22 571 00. 48 25 9. 04 -394203.374502.34 591 20. 89 26 0. 37 -407188.7

76957.88 6116 0.71 26 1. 68 -420240.21950

2OOO205021002150220022502300235O24O0245O

i i i |

2500255O260026502700

275028002850290O2950

30OO3O5O310031503200325033OO335O3400345035OO

.794.32.30 63219.41 26 2.9 7 -433356.6

81925.o3 6529 6.43 26 4 .2 3 ~ 5 3 6 . 784435.48 673 91. 17 26 5. 47 -459779.48 6 9 6 3. 0 5 6 9 5 0 3 . 0 2 2 6 6 . 6 9 - 47 3 0 83 .489507.10 7163 1.3 6 26 7. 89 -486447.99 ~ . 9 8 7 3 7 7 5 . 5 2 2 6 9 . 0 ~ -~ 9 9s 7~ .79 4 6 4 2. 0 1 7 5 9 3 4 . 8 3 2 7 0 . 2 2 -5 1 3 3 53 . 997231.49 781 08. 60 27 1. 36 -526893.499834.70 802 96. 09 27 2. 48 -540489.4

102450.89 8249 6.5 7 27 3. 58 -554141.0mS079.29 u70926 2 7 4 . 6 6 - s s T u T . !107719.12 869 33. 37 27 5. 73 -581607.1110369.56 891 68. 10 27 6. 78 -595419.9113029.77 914 12. 59 27 7. 81 -609284.8115698.90 936 66. 00 27 8. 83 -623200.9118376.06 9592 7.4 5 27 9. 83 -637167.5

121060.35 981 96. 02 28 0. 82 -651183.8123750.84 100 470.8 0 28 1. 79 -665248.9126446.58 1027 50.82 28 2. 74 -679362.129146.60 10503 5.13 28 3. 68 -693522.6131849.91 1073~.73 , 2 U . 6 0 -707729.7

134555.49 10 9612.5 9 28 5. 51 -721982.7137262.29 il 1903.68 28 6. 41 -736280.8139969.26 11 4194.9 3 28 7. 29 -750623.2142675.31 11 6485. 26 28 8. 15 -765009.3145379.33 118 773.5 7 28 9. 01 -779438.4148080.18 12 1058.71 28 9. 84 -793909.6150776.73 1233 39.54 29 0. 67 -808422.4153467.78 12 5614. 88 29 1. 48 -822976.0156152.15 12 7883.5 3 29 2. 27 -837569.8158828.60 13 0144. 27 29 3. 05 -852202.9161495.91 132395.116 29 3. 82 -1166874.8

, , , , ,

7/29/2019 Chap9 Thermodynamics of Ideal Gases

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T a b l e 9 . 4 continued

N i t r i c o x i d e , N O " h o = 8 9 9 1 4 . 7 I d ~ o l

r , ( r~ . ( D . .. . ~ r ~ ~ r j

50 14 6 1 . 9 4 1046. 22 156 . 66 - 6371 .i10 0 2 9 3 6 . 2 6 2 10 4.8 3 1 7 7 . 0 9 - 1 4 7 7 2. 6150 4 4 2 2 . 8 9 3175.75 1 8 9 . 1 4 - 2 3 9 4 8 . 2200 5 92 1 . 7 7 4258. 91 19 / .76 - 336 30 . 925 0 7 4 3 2 . 8 0 5 35 4.2 2 2 0 4 . 5 1 - 4 36 9 3 . 6

z9 8 . . . . m4.v4 . . . . . ++ZZ.OS ~.55 .~5641.93 00 8 9 5 5 . 8 9 6 4 6 1. 60 2 1 0 . 0 6 - 5 4 0 6 1 .83 50 1 0 4 9 0 . 9 / 7 5 80 .9 7 2 1 4 . 7 9 - 6 4 6 8 5 . 94 00 1 2 0 3 7 . 9 2 8 7 1 2 .2 0 2 1 1 1 . 9 2 - 7 5 5 3 0 . 9

45 0 13596 :65 , ,, 9855. 21 2 2 2 . 5 9 , 46570 .5so o ~ 5 ~ e 7 . 0 3 ~ m 0 9 . s s 2 2 5 . 9 o - 9 7 ~ 4 . 35 50 1 6 7 4 8 . 9 5 1 2 17 6 .0 9 2 2 8 . 9 2 - 1 0 9 1 5 5 .9

~ 5 34 2 .3 0 m 5 3 . 7 2 2 3 L e 9 - n0 ~ 7 2. 06 50 1 9 9 4 6 . 9 3 1 4 5 , 1 2 . 6 3 2 3 4 . 2 6 - 1 3 2 32 1 .57 00 2 1 5 6 2 . 7 1 1 5 7 4 2 . 7 0 2 3 6 . 6 5 -1 4 4 0 95 . 075 0 2 3 1 8 9 . 5 0 1 69 53 .7 7 2 3 8 . 9 0 - 1 5 5 9 84 . 480 0 2 4 8 2 7 . 1 5 1 8 1 7 5 . 7 1 2 4 1 . 0 1 - 1 6 7 9 8 2 . 7s so 2 6 4 7 5 . ~ 0 1 94 0 5 .3 5 2 4 5 . m . t m m . s9oo 2sm.~o 20~5~.s+ 24 4 .m -192252. t9s o 295m. ,~ . 2 t90s . j ,o 24 6 . 71 .204575 .0

1 0 0 0 3 1 4 8 3 . 1 6 2 31 68 .8 6 2 4 8 . 4 4 - 2 1 ~ 5 2 . 01 0 5 0 3 3 1 2 2 . 6 8 2 44 42 .6 6 2 5 0 . 0 8 - 2 2 94 1 5 .2

! 1 00 3 4 8 7 2 . 0 3 2 5 7 26 .3 0 2 5 1 . 6 6 - 2 4 1 9 5 9 .21 1 5 0 3 6 5 8 1 . 0 3 2 20 19 .5 8 2 5 3 . 1 8 - 2 5 45 8 0 .71 2 0 0 3 8 2 9 9 . 4 8 2 83 22 .3 2 2 5 4 . 6 5 -2 6 7 2 7 6. 7125 0 4 00 27 . 19 29634 .31 2 5 6 . 0 0 -211OO44.51 3 0 0 4 1 7 6 3 . 9 3 3 09 55 .3 4 2 5 7 . 4 2 - 2 9 2 88 1 .61 3 5 0 4 3 5 0 9 . 4 9 3 22 85 .1 9 2 5 8 . 7 4 - 3 0 5 78 5 .21 4 0 0 4 5 2 6 3 . 6 6 3 36 23 .6 4 2 6 0 . 0 1 - 3 1 87 5 4 .61 4 5 0 4 7 0 2 6 . 2 0 3 49 70 .4 2 2 6 1 . 2 5 - 3 31 7 8 6 .4

1 5 0 0 4 8 " ~ . i 8 3 63 2 5.4 3 2 6 2 . 4 5 - 34 4 87 9 .01 5 5 0 5 0 5 7 5 . 4 7 3 76 88 .3 0 2 6 3 . 6 2 - 3 5 8 0 30 . 91 6 0 0 5 2 3 6 1 . 7 0 3 90 58 .8 2 2 6 4 . 7 5 - 3 7 1 2 4 0. 21 6 5 0 5 4 1 5 5 . 3 3 4 04 36 .7 4 2 6 5 . 8 6 - 3 8 4 50 5 .51 7 0 0 5 5 9 5 6 . I 4 18 2 I . 8 0 2 6 6 . 9 3 - 3 9 7 8 2 5 .21 7 5 0 5 7 7 6 3 . 2 7 4 32 13 .7 4 2 6 7 . 9 8 - 4 1 1 1 98 . 01 8 0 0 5 9 5 7 8 . 0 3 4 46 12 .2 9 2 6 9 . 0 0 - 4 24 6 2 2 .61 8 5 0 6 1 3 9 8 . 6 2 4 6 0 1 7 . 1 7 2 7 0 . 0 0 438097.71 9 0 0 6 3 2 2 5 . 2 6 4 74 2 8.0 9 2 " / 0 . 9 7 - 4 5 1 6 2 2 .01 9 5 0 6 5 0 5 2 . 6 6 4 8 1 1 4 4 . 7 7 2 7 1 . 9 2 . 465194 . 52 0 0 0 . 6 6 8 9 5. 5 1 M 2 6 6 . w 2 7 2 . 8 5 - 47 8 i1 4 .|2 0 5 0 6 8 7 3 8 . 5 3 5 16 94 .2 2 2 7 3 . 7 6 - 4 9 2 47 9 .72 1 0 0 7 0 5 8 6 . 4 1 5 31 26 .3 8 2 7 4 . 6 6 - 5 0 61 9 0 .32 1 5 0 7 2 4 3 8 . 8 2 5 45 63 .0 8 2 7 5 . 5 3 - 5 1 9 9 4 4. 92 2 0 0 7 4 2 9 5 . 4 6 5 60 04 .0 0 2 2 6 . 3 8 - 5 3 3 7 42 . 22 2 5 0 7 6 1 5 5 . 9 9 5 7 4 4 8 . 8 1 2 7 7 . 2 2 - 5 47 5 ~. 72 3 0 0 7 8 0 2 0 . 0 8 5 88 97 .1 9 2 7 8 . 0 4 - 5 6 1 46 4 .2 3 5 0 7 9 8 8 7 . 4 0 6 03 48 .7 9 2 7 8 . 8 4 - 5 7 53 8 6 .12 4 0 0 8 1 7 5 2 . 5 9 6 18 03 .2 7 2 7 9 . 6 3 - 5 8 9 3 47 . 82 4 5 0 8 3 6 3 0 . 3 2 6 32 60 .2 9 2 8 0 . 4 0 - 6 0 3 3 48 . 6

2 5 OO 8 5 5 0 5 . 2 2 6 47 1 9.4 2 2 8 1 . 1 6 -6 1 7 3 8 7 . 62 5 5 0 8 2 3 8 1 . 9 4 6 61 80 .4 7 2 8 1 . 9 0 - 6 3 1 4 64 . 02 6 0 0 8 9 2 6 0 . 0 9 6 7 6 4 2 . 9 1 2 8 2 . 6 3 ,.6 45 57 2.42 6 5 0 9 1 1 3 9 . 3 2 6 91 0 6.4 2 2 8 3 . 3 5 . .6 5 9 7 26 . 82 7 0 0 9 3 0 1 9 . 2 3 2 05 70 .6 2 2 8 4 . 0 5 - 6 7 3 9 1 1. 7

2 7 5 0 9 4 8 9 9 . 4 4 2 2 0 3 5 . 1 1 2 8 4 . 7 4 -6 11 81 31 .42 8 0 0 9 6 7 7 9 . 5 6 7 34 99 .5 2 2 8 5 . 4 2 - 7 0 2 3 8 5. 42 8 5 0 9 8 6 5 9 . 1 8 7 49 63 .4 3 2 8 6 . 0 8 - 7 1 6 6 7 2. 82 9 0 0 100537 . 91 76426 . 44 2 8 6 . 7 3 - 730993 . 3

, , 29 50 10241 5 . 33 77U8., 15 28] .38 - 245346 .

3 0 0 0 1 0 4 2 9 1 .0 2 7 9 3 4 8 . 1 2 2 8 8 . 0 1 -7 59 73 0'83 0 5 0 1 0 6 1 6 4 . 5 7 8 08 0 5.9 5 2 8 8 . 6 3 - 7 7 4 1 4 6 .7310 0 108035 . 53 82261 .20 2119 . 24 - 788593 . 3~ 5 o m 9 9 m . 4 s s ~ 7~ .4 ~ 2 5 9 . s ~ -s o 3o 7 o. o3 2 0 0 1 1 1 76 7 .9 7 8 5 1 6 2 . 2 1 2 9 0 . 4 2 - 81 7 57 6 .43 2 5 0 1 1 3 6 2 8 . 5 5 8 66 0 7.0 8 2 9 1 . 0 0 4 1 3 2 11 1 .83 3 0 0 1 1 5 4 U . 8 8 80 4 7.5 9 2 9 1 . 5 6 - 8 4 6 6 7 5 . 93 3 5 0 1 1 7 3 3 6 . 1 8 8 94 8 3.2 8 2 9 2 . 1 2 - 8 6 1 2 68 . 1340 0 119182 .30 90913 .68 2 9 2 . 6 7 - 825887 . 83 4 5 0 1 2 1 0 2 2 . 6 6 9 23 3 8.3 2 2 9 3 . 2 1 - 8 9 0 5 3 4 . 735 00 12~56.28 ,93756.73 2 9 3 .2 3 i - ,905208.2

M e t h a n e , C H 4 9 h o = - 6 6 9 3 0 . 5 k J / I m m l

r , # i f ) m ( n a ( n ~ )5 0 9 0 6 .4 4 4 9 0 . ' / 2 1 3 4 .8 2 - 5 83 4 .7

100 2 0 1 1 . 6 2 1 1 8 0 . 1 9 1 4 9 . 9 9 - 12 9 87 .21 5 0 3 3 0 8 . 1 0 2 0 6 0 . 9 5 1 0 0 . 4 5 - ~ 5 9 . 5

200 4 7 8 8 . 6 0 3 1 2 5 . 7 4 1 6 0 . 9 1 - 29 0 00 .2250 6 4 4 6 . 0 8 4 3 6 7 . 5 0 1 7 6 . 3 3 - 3 76 3 5 .7

2# s 8+ 97 ,36 57+9, 7o +a.2, 73 - ~ 2 ~ + . . 43 00 8 2 2 3 . 6 4 5 ':T 7 9.3 5 1 8 2 . 9 8 - 4 6 6 2 1 . 13 50 1 0 2 6 4 . 6 2 2 3 5 4 . 6 2 1 89 .1 - 5 5 9 2 5 . 64 00 1 2 4 1 2 . 5 1 9 0 8 6 . 7 9 1 9 4 . 1 1 5 - 6 5 5 2 6 . $4 50 1 4 2 11 .~ 1 , 0 9 9 9 .5 7 , 2 0 0 . 2 6 - 7 5 4 0 6 . 05 00 - 1 7 1 5 3 .9 6 1 2 9 9 6 . 8 1 2 0 5 . 4 1 " 4 1 5 5 4 9 . 6550 1 9 / 3 5 . 4 4 1 5 1 6 2 . 5 7 2 1 0 . 3 3 - 9 5 94 5 . 26 00 2 2 4 4 9 . 6 5 1 7 4 6 1 . 0 2 2 1 5 . 0 5 - 1 06 5 8 2 .56 50 2 5 2 9 1 . 0 0 1 9 1 18 6 .7 0 2 1 9 . 6 1 - ! 1 7 4 5 2 . 37 00 2 8 2 5 4 . 0 ~ 2 2 4 3 4 . 0 4 2 2 4 . 0 0 -1 28 54 6.1 17 5 0 3 1 3 3 3 . 5 3 2 5 0 9 7 . 8 1 2 2 8 . 2 6 - 1 3 9 8 5 8 . 8

8 00 3 4 5 2 4 . 3 5 2 7 8 7 2 . 9 1 2 3 2 . 3 8 - 1 5 1 3 8 1 .78 50 3 7 8 2 1 . 5 7 3 0 7 5 4 . 4 1 2 3 6 . 3 9 - 1 6 3 1 0 9 .89 0 0 4 1 2 2 0 . 4 0 3 3 7 3 7 . 5 3 2 4 0 . 2 9 - 1 7 5 0 3 7 . 69 5 0 4 4 7 16 .2 3 3 6 8 1 7 . 6 5 2 4 4 . 0 8 - 18 7 16 0 .3

i 00 0 4 8 3 0 4 . 5 9 3 9 9 9 0 . 2 9 2 4 7 . 7 8 - 1 9 9 4 73 .31 0 5 0 5 1 9 8 1 . 1 2 4 3 2 5 1 . 1 6 2 5 1 . 3 8 - 2 il 9 /2 . 41 1 0 0 5 5 7 4 1 . 8 0 4 6 5 9 6 . 0 7 2 5 4 . 9 1 - 22 4 65 3 .91 1 5 0 5 9 5 8 2 . 4 6 5 0 0 2 1 . 0 2 2 5 1 1 . 3 4 - 2 3 75 1 4 .11 2 0 0 6 3 4 9 9 . 2 8 5 3 5 2 2 . 1 2 2 6 1 . 2 1 - 25 0 5 49 .71 2 5 0 6 2 4 8 8 . 5 2 5 7 0 9 5 . 6 5 2 6 5 . 0 0 - 2 63 7 5 7. 91 3 0 0 2 1 5 4 6 . 5 9 6 0 7 3 8 . 0 0 2 6 8 . 2 2 - 2 77 1 3 5. 71 3 5 0 2 5 6 7 0 . 0 2 6 4 4 4 5 . 7 1 2 2 1 . 3 2 - 2 9 0 6 8 0 . 71 4 0 0 7 9 8 5 5 . 4 9 6 8 2 1 5 . 4 2 2 7 4 . 4 6 - 30 4 39 0 .6

,,, 1 4 5 0 . . . . ,84099.,80, ,, ,720414.07 .. .. 2 7 7 . 4 9 -,3111263.4i 5 00 8 83 99 .1 11 1 7 5 9 2 8 . 4 3 2 8 0 . 4 6 - 3 3 2 2 9 7 . 01 5 5 0 9 2 7 5 2 . 8 0 7 9 8 6 5 . 6 3 2 8 3 . 3 8 - 3 46 4 9 0. 01 6 0 0 9 /1 55 .7 2 8 3 8 5 2 . 8 4 2 8 6 . 2 5 - ~ . 9l S m 1 0 1 6 0 5 . 9 4 8 7 8 8 7 . 3 5 2 89 .0 6 - 3 7 5 3 4 8 . 31 7 0 0 1 0 6 1 0 0 . 8 9 9 1 9 6 6 . 5 8 2 9 1 . 8 3 - 3 9 0 0 1 1 . 41 7 5 0 1 1 0 6 3 8 .0 9 9 6 0 ~ .0 6 2 9 4 . 5 5 - 4 0 4 8 2 9 . 3I ~ I I 5 21 5 . 1 7 m 0 2 4 9 . 4 3 2 9 7 . 2 3 -459501.3185 0 11982 9 . 90 1044 48 . . 299 .87 , -, 434927 .0

1 9 0 0 1 2 4 4 8 0 .1 2 1 0 8 6 8 2 . 9 5 3 0 2 . 4 7 450206 . 2, , , 1950 ,129163.80 ! 12950291 3 0 5 . 0 3 - , ~J 638 . . I2 0 0 0 1 3 3 8 7 8 .9 9 1 1 7 2 5 0 . 3 9 3 0 7 . 5 5 - 4 81 2 2 5 .12 0 5 0 1 3 8 6 2 3 .8 6 1 2 1 5 7 9 . 5 5 3 1 0 . 0 4 4 9 69 6 5. 32 1 0 0 1 4 3 3 9 6 . 6 6 1 2 5 9 3 6 . 6 3 3 1 2. 50 - 5 1 2 8 6 0 .12 1 5 0 1 4 8 1 9 5 .2 4 1 3 0 3 2 0 . 0 0 3 1 4 . 9 3 - 5 2 89 1 0 .12 2 0 0 1 5 3 0 1 9 .5 4 1 3 4 7 2 8 . 0 8 3 1 7 . 3 3 - 5 4 5 1 1 6. 42 2 5 0 1 5 7 8 6 6 .5 7 1 3 9 1 5 9 . 4 0 3 1 9 . 7 1 - 5 6 14 8 0 .12 3 0 0 1 6 2 7 3 5 .4 6 1 4 3 6 1 2 . 5 7 3 2 2 . 0 6 - 5 7 8 0 02 . 62 3 5 0 1 6 7 6 2 4 . 8 9 1 4 8 0 8 6 . 2 8 3 2 4. 39 - 5 9 4 6 8 5 . $2 4 0 0 1 7 2 5 3 3 . 6 3 1 5 2 5 7 9 . 3 1 3 2 6. 69 - 6 1 1 5 3 0 . 6

_ 2 4 5 0 , ! 7 ,74 6 0 .5 4 1 5 7 0 9 0 .$ 0 , 3 2 8 .9 8 , ., .~2 8 53 9 .82 5 0 0 1 8 2 4 0 4 .5 3 1 6 1 6 1 8 . 7 8 3 3 1 . 2 5 - 6 4 5 21 5 . 42 5 5 0 1 8 7 3 64 . 61 1 6 6 1 6 3 . 1 4 3 3 3 . 5 0 - 6 6 30 5 9 .72 6 0 0 1 9 2 3 3 9 .8 2 1 7 0 7 2 2 . 6 4 3 3 5 . 7 4 - 6 8 0 52 5 . 52 6 5 0 1 9 / 32 9 . 3 1 1 7 5 2 9 6 .4 2 3 3 7 . 9 6 - 69 8 2 65 .52 7 0 0 2 0 2 3 3 2 . 2 2 1 7 9 8 8 3 . 6 6 3 4 0 . 1 2 - 2 1 6 13 2 . 8

2 7 5 0 2 0 7 3 4 2. 9 5 1 8 4 4 8 3 . 6 2 3 4 2 . 3 2 - 7 3 4 1 8 0 . 82 8 0 0 2 1 2 3 2 5 .6 5 1 8 9 0 9 5 . 6 1 3 4 4 . 5 2 - 7 5 24 1 2 .82 8 5 0 2 1 7 4 1 4 .7 5 1 9 3 7 1 9 . 0 0 3 4 6 . 7 5 - 7 7 0 83 2 . 72 9 0 0 2 2 2 4 64 . 62 1 9 8 3 5 3 . 2 0 3 4 8 . 9 3 - 7 8 9 4 4 4 . 42 9 5 0 2 2 2 5 2 4 . 8 8 2 0 2 ~ 7 .6 9 3 5 1 . 1 1 4 1 0 82 5 2 .

3 0 0 0 2 3 2 5 9 4 . 8 9 2 0 2 6 5 1 . 9 9 3 5 3 . ~ 4 2 72 6 0. 23 0 5 0 2 3 7 6 2 4 . 2 7 2 1 2 3 1 5 . 6 5 3 5 5. 46 4 4 6 4 7 3 . ~3 1 0 0 2 4 2 7 6 2 . 6 1 2 1 6 9 8 8 . 2 8 3 5 7 . 6 3 - 8 6 5 8 9 6 _ *3 1 5 0 2 4 7 8 5 9 .5 7 2 2 1 6 6 9 . 5 2 3 5 9 . 8 1 - 8 85 5 3 4. ~3 2 0 0 2 5 2 9 6 4 .8 2 2 2 6 3 5 9 . 0 6 3 6 1 . 9 9 - 9 05 3 9 3 .43 2 5 0 2 5 8 0 7 8 . 0 8 2 3 1 0 5 6 . 6 0 3 6 4. 12 - 9 2 5 4 7 8 .3 3 0 0 2 6 3 1 9 9 . 0 9 2 3 5 7 6 1 . 9 0 3 6 6. 36 . ,9 4 5 7 9 4 .73 3 5 0 2 6 8 3 2 7 .6 2 2 4 0 4 7 4 . ' / 2 3 6 8 . 5 6 - 9 6 6 3 49 . 43 4 0 0 2 7 3 4 5 6 . 9 5 2 4 5 1 8 8 . 3 3 3 7 0. 77 - 9 8 7 1 4 5 .13 4 5 0 2 7 8 6 0 6 .5 0 2 4 9 9 2 2 . 1 6 3 7 2 . 9 9 - 1 00 8 1 98 .33 5 0 0 2 8 3 7 5 6 . 5 0 2 5 ~ 5 6 .4 5 . 3 7 5 . 2 2 - 1 0 29 5 0 6 .1

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T a b l e s o f u ( T ) a n d h (T ) a g a i n s t T 1 71

9 .3 .1 T A B L E S O F M E A N S P E C I F I C H E A T

Som et imes d a t a a re g iven in t e rms of mean spec i f i c hea t ra ther t han the ac tua l spe c i f i c hea t

a t a par t i cu l a r tempe ra ture . Th i s approach w i l l now b e descr ibed .

Con sider t he change o f en tha lpy , h , be tween t em pera tures T~ and T2. Thi s i s

I r~ cp (T ) d T = h(T2) - h(TO (9 .43)2 - h~ ffi r,

No w, th i s can be wr i t ten

ep(T , - 7'1) = h( T2) - h ( T~) (9 .44)

where ~p i s the m ean spec i f i c hea t be tween T~ and T2. No rm al ly ~p i s g iven a t a par t i cu l a r

t empera ture T , and i t i s t hen def ined as t he mean be tween the t empera ture T and a

reference t empera ture T=f , i .e .

h ( T ) - h (T ~f )(6p)r = (9.45)

T - T r a

I t i s a l so poss ib le to wr i t e t he m ean spec i fi c hea t as a po ly nom ia l func t ion o f tem pera ture ,

in which case

( ~p) = a + bT + c T" + . . . (9 .46)

w he re a , b a n d c are t abula t ed coeff i c ien t s .

I f i t i s requireA to ca l cu la t e t he en tha lpy d i f fe rence be tw een two t empera tures T~ an d T2,

then

h 2 - h i = (ep)r~(T2- Tre f)- (Cp)T,(T I- Tref) (9.47 )

H av i ng ca l cu l a t ed t he change i n en t ha l py , h2 - h~ , t hen t he change i n i n t e rna l ene rgy ,u , - u~, m ay be eva l ua ted a s

u 2 - u~ ffi h 2 - R T 2 - ( h~ - RT~) ffi h ~ - h~ - R ( T , - T~) ( 9 . 4 8 a )

oN

To ffi 0 T2 Temp era ture ,

Fig. 9 .1 M ean specific heat at constant pressure

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172 Thermodynamic properties of ideal gases and ideal gas mixtures

or , in molar t e rms , by

Um.,+- Um .~- hm.,+- 9~T 2- (hm .~- 9~T~) - hm .2- hm .~- 9~ (T2 - T~) (9.4 8b)

I f t h e ch an g e o f in t e r n a l en e r g y , u 2 - u ~, i s k n o w n , t h en th e ch an g e o f en t h a l p y , h , - h ~,

m ay b e ca l cu l a t ed i n a s i m i l a r w ay . T h e u s e o f m ean s p ec i f i c h ea t v a l u e s en ab l e s m o r e

accu r a t e ev a l u a t i o n o f t em p e r a t u r e ch an g es b ecau s e t h e m ean s p ec i f i c h ea t s en ab l e t h een e r g y eq u a t i o n t o b e s o l v ed w i t h an a l l o w an ce f o r t h e v a r i a t i o n o f s p ec i f i c h ea t s . T h e

m e an s p ec i f ic h ea t s ap p r o ach i s d ep i c t ed i n F i g 9 . 1 .

9 . 4 M i x t u r e s o f i d e a l g a s e s

M an y p r o b l em s en co u n t e r ed i n en g i n ee r i n g i n v o l v e m i x t u r e s o f g a se s - a i r i t s e l f i s a

m i x t u r e o f m a n y g a s e s a l t h o u g h i t c a n b e c o n s i d e r e d t o b e o x y g e n a n d a t m o s p h e r i c

n i t r o g en ( w h i ch can b e a s s u m ed t o co n t a i n t h e n i t r o g en i n t h e a i r an d t h e o t h e r ' i n e r t '

g a s e s s u ch a s a r g o n an d ca r b o n d i o x i d e ) . I f th e g a s e s i n a m i x t u r e a r e a t a t em p e r a t u r e w e l l

ab o v e t h e i r c ri ti c a l tem p e r a t u r e an d a p r e s s u r e b e l o w t h e c r i ti c a l p re s s u r e , t h ey ac t a s i d ea l

g a s e s . E x p r e s s i o n s r e l a t i n g t h e p r o p e r t i e s o f m i x t u r e s o f i d ea l g a s e s w i l l n o w b ef o rm u l a t ed . T h e s e e x p r e ss i o n s a r e a d i re c t c o n s e q u e n c e o f th e G i b b s - D a l t o n l a w s .

9.4.1 DALTON PRINCIP LE

Dal ton s t a ted tha t

an y g a s is a s a v acu u m t o an y g a s m i x ed w i t h it .

T h i s st a te m e n t w a s e x p a n d e d a n d c l a r if ie d b y G i b b s t o g i v e th e G i b b s - D a l t o n l a w s .

9.4.2

(a)

(b )

(c )

GIBBS-DALTON LAW

A g as m i x t u r e a s a w h o l e o b ey s t h e eq u a t i o n o f s t a te ( eq n ( 9 . 2 ))

pVffi ng~T

wh ere n i s the total a m o u n t o f s u b s t a n c e i n t h e m i x t u re .

T h e t o t a l p r e s s u r e ex e r t ed b y a m i x t u r e i s t h e s u m o f t h e p r e s s u r e s ex e r t ed b y t h e

i n d i v i d u a l co m p o n en t s a s e ach o ccu p i e s t h e whole volume o f th e m i x t u r e a t t h e s a m e

tempera ture .

T h e i n t e r n a l en e r g y , en t h a l p y an d en t r o p y o f t h e m i x t u r e a r e , r e s p ec t i v e l y , eq u a l t o

t h e s u m s o f t h e i n te r n a l e n e r g y , e n t h a l p y a n d e n t r o p y o f th e v a r io u s c o m p o n e n t s a s

each o ccu p i e s t h e whole volume a t th e t em p e r a t u r e o f t h e m i x t u re .

9.4.3 MIXTURE RELAT IONS HIPS

T h e t o t a l m as s , m , o f a m i x t u r e can b e r e l a t ed t o t h e m as s o f co n s ti t u en ts b y

m = ma + mb + m~ + . . . . ~ ~

i = l

From the idea l gas l aw

mpV = mRT = - - mwRT = ng~T

m w

(9 .49)

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M i x t u r e s o f i d e a l g a s e s 173

an d h en ce f o r t h e i n d i v i d u a l co n s t i t u en t s

p V o = m , f l o T = n , , g tT

P V b = m b R b T - - n b g ~ T ( 9 . 5 0 )

p V ~ = m c R f f = n ~ 9~ T , e tc

I n eq n ( 9 . 5 0 ) , V o ffi v o l u m e o f co n s t i t u en t a a t th e p r e s s u re an d t em p e r a t u r e o f t h e m i x t u r e ,

and Vb and V~ are s i m i l a r v o l u m es f o r co n s t i t u en t s b an d c . B u t t h e t o t a l v o l u m e , V , i s

g i v e n b y

V = V ~ + Vb + V~ + . . . . ~ V ~ (9 .51)i f f i l

an d t h e r e fo r e

n = no + nb + n~ + .- . ffi ~ , n~ (9.52 )

i - I

L e t

n~ = Xo ffi m ol e f rac t ion of a in the m ixtu re (9.53 )n

S i m i l a r l y , nb /n = Xb , and n~ / n ffi x~ , e tc , w i th n~/n = x~. T h er e f o r e

Xo + Xb + X~ + - . - = ~ X~ = 1 ( 9 . 5 4 )

i - 1

I t i s p o s s i b l e t o d ev e l o p t h e t e r m f o r t h e p a r t i a l p r e s s u r e o f e a c h c o n s t i t u e n t f r o m

s t a te m e n t ( b ) o f t h e G i b b s - D a l t o n l aw s . T h e n , f o r c o n s t it u e n t a o c c u p y i n g t h e t o ta l

v o l u m e o f th e m i x t u r e a t t h e p r e s s u r e an d t em p e r a t u r e o f th e m i x t u r e ,

p o V = n o g ~T

B u t , f o r t h e m i x t u r e a s a w h o l e ( eq n ( 9 . 2 ) )

p V = n g ~T

H en ce , b y d i v i d i n g eq n ( 9 . 5 0 ) b y eq n ( 9 . 2 )

g i v i n g

p~ n~- - = - - = X o ( 9 . 5 5 )

p n

p o = x o p , an d i n g en e r a l p ~ = x~p

and , a l so

( 9 . 5 6 )

E pi = .r ip = p xi = P ( 9 . 5 7 )i f f il i = 1 i f f i l

O f t en m i x t u r e s a r e an a l y s ed o n a v o l u m e t r i c b a s i s , an d f r o m t h e v o l u m e t r i c r e s u l t s i t i s

p o s s i b l e t o o b t a i n t h e p a r t i a l p r e s s u r e s an d m o l e f r ac t i o n s o f t h e co m p o n en t s . N o r m a l l y

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174 Thermodynam ic prop ertie s o f ideal gases and ideal gas mixtures

v o l u m e t r i c a n a l y s e s a r e p e r f o r m e d a t c o n s t a n t p r e s s u r e a n d t e m p e r a t u r e . T h e n , c o n s i d e r i n g

t h e i th c o m p o n e n t ,

n i ~ Tv , = ( 9 . 5 8 )

Pa n d t h e t o ta l v o l u m e o f t h e m i x t u r e , V , i s

ng~TV =

P

H e n c e , f r o m e q n s ( 9 . 5 8 ) a n d ( 9 .5 9 )

( 9 . 5 9 )

Vt nt- - = - - = x t ( 9 .6 0 )V n

I f t h e g as c o m p o s i t i o n i s g i v e n i n v o l u m e p e r c e n t a g e o f th e m i x t u r e , t h e m o l e f r a c t i o n i s

v~(%)x~ = 9 ( 9 . 6 1 )

1 0 0

and t he pa r t i a l p re s sure i s

Vt(%)Pi = - - - - P (9 . 62)

1 0 0

T h e t e r m s r el a ti n g t o t h e e n e r g y o f a m i x t u r e c a n b e e v a l u a te d f r o m s t a t e m e n t ( c ) o f th e

G i b b s - D a l t o n l a w s . I f e m = m o l a r i n t e r n a l e n e r g y , a n d h m ---m olar e n t h a l p y , t h e n f o r t h e

m i x t u r e

and

E = nero = ~ , niem.i ( 9 . 6 3 )i=1

H = nh= = ~ nih =.i ( 9 . 64 )i - I

I f t h e m o l a r in t e rn a l e n e r g y a n d m o l a r e n t h a lp y of the mixture a r e r e q u i r e d t h e n

e m = - - n i e m . i = X i e m . i ( 9 . 6 5 )n i--I i- I

and

h m - - - n i h m . l - - x i h m , i ( 9 . 6 6 )

n i -1 i l l

N e g l e c t i n g m o t i o n , g r a v i t y , e l e c t r i c i t y , m a g n e t i s m a n d c a p i l l a r y e f f e c t s , t h e n e = = u m , a n d

h e n c e

- -- - ~ Xi Um , i (9.67)

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The de f in i t ion o f en tha lpy fo r an idea l gas i s

h , ffi u , + f l iT

Thus . fo r the mix tu re

h , = Y~(x i (u , + ~RT)t )ffi ~, XjUm.t + Y . x i~ Rr ffi ~, XtUm.a+ 9~ T Y~ x~

= ~ x~u.,t+ ~ T

Entropy o f mix tures 175

(9 .68)

(9 .69)

9 .4 . 4 S P E C I F I C H E A T S O F M I X T U R E S

S t a t e m e n t ( c ) o f t h e G i b b s - D a l t o n l a w a n d t h e a b o v e e x p r e s s i o n s s h o w t h a t

u, ffi Y~ Xl Um.t

and

h , ffi ~ x~h ..t

By def in i t ion , the spec if ic hea ts a re

d u d hc~ffi dT... and cpf fi dT "

-- E Xi(Cv,m)/ (9.70)

T h u s

d r l iand , s imi la r ly

C~,m = E x,(C p..), (9 .71)

9 . 5 E n t r opy o f m ix t ur es

Co n s i d e r a m i x t u re o f tw o i d e al g a s e s , a a n d b . T h e e n t r o p y o f t h e m i x t ur e i s e q u a l t o t h e

s u m o f th e e nt r o p ie s w h i c h e a c h c o m p o n e n t o f t he m i x t u r e w o u l d h a v e i f it a lone occupied

the whole vo lume o f the mix ture a t the same temperature . T h i s c o n c ep t ca n p r e s e n t s o m e

di f f icu l ty because i t means tha t when gases a re sepa ra te bu t a l l a t the same tempera tu re

a n d p r e s s u r e t h e y h a v e a l o w e r e n t r o p y t h a n w h e n t h e y a r e m i x e d t o g e t h e r i n a v o l u m e

e q u a l t o th e s u m s o f t he i r p r e v i o u s v o l u m e s ; t h e s i t u a t io n c a n b e e n v i s a g e d f r o m F i g 9 .2 .

T h i s c a n b e a n a l y s ed b y c o n s i d e r i n g t h e m i x i n g p r o c e s s i n t h e f o l l o w i n g w a y :

( i ) G a s e s a a n d b , a t t h e s a m e p r e s s u r e , a r e c o n t a i n e d i n a c o n t r o l v o l u m e b u t p r e v e n t e d

f r o m m i x i n g b y a n i m p e r m e a b l e m e m b r a n e .

( i i) T h e m e m b r a n e i s t h e n r e m o v e d , a n d t he c o m p o n e n t s m i x d u e t o d i f fu s i o n , i . e . d u e to

t h e c o n c e n t r a t i o n g r a d i e n t t h e r e w i l l b e a n e t m i g r a t i o n o f m o l e c u l e s f r o m t h e i r

o r i g i n a l v o l u m e s . T h i s m e a n s t h a t t h e p r o b a b i l i t y o f f i n d i n g a p a r t i c u l a r m o l e c u l e a t

any par t icu lar po in t in the vo lume i s dec reased , and the sys tem i s in a le ss o rde red

s ta te . Th is change in p robab i l i ty can be re la ted to an inc rease in en t ropy by s ta t i s t i ca l

the rmodynamics . Hence , qua l i ta t ive ly , i t can be expec teA tha t mix ing wi l l g ive r i se to

an increase in entropy.

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1 76 Therm odynam ic proper t ies o f ideal gases and ideal gas m ix tures

s ' \ 9 \ I ns - " - ~ , , , , n

\, G a s ' a' G a s

! p . P b

T , , T

Fig. 9.2 Tw o gases a t the sam e pressure , p , conta ined in an insula ted conta iner and separa ted by a

memb rane; po = Pb =P

C o n s i d e r i n g t h e m i x i n g p r o c e s s f r o m a m a c r o s c o p i c v i e w p o i n t , w h e n t h e m e m b r a n e i s

b r o k e n t h e p r e s s u r e i s u n a f f e c t e d but the par t ia l pressures o f the indiv idual components

are decreased.

T h e e x p r e s s i o n f o r t h e e n t r o p y o f a g a s is ( e q n ( 9 .2 0 ) )

Sm -- sin(T ) - 9~ In p + $0.m190

w h e r e Sm(T) i s a f u n c t i o n o f T a l o n e . H e n c e , c o n s i d e r i n g t h e p r e s s u r e t e r m , w h i c h i s

ac t ua l l y t he par t ia l pressure o f a c o m p o n e n t , a d e c r e a s e i n t h e p a r t i a l p r e s s u r e w i l l c a u s e

a n in c r e a s e i n t h e e n t r o p y o f t h e g a s . E q u a t i o n ( 9 . 2 0 ) c a n b e s i m p l i f i e d b y w r i t i n g t h e

pre ssu re ra t i o a s P r = P / P o , g i v i n g

Sm -- Sm (T ) - ~1~ In p , + 5'0,m

N o w c o n s i d e r t h e c h a n g e f r o m a n a n a l y t i c a l v i e w p o i n t . C o n s i d e r t h e t w o g a s e s a a n d b .

T h e t o t a l a m o u n t o f s u b s t a n c e i n t h e g a s e s i s

n r - no + n b ( 9 .7 2 )

T h e e n t r o p i e s o f th e g a s e s b e f o r e m i x i n g a r e

am . - - , naSm~ na[Sm~ ) - ~ I n Pr~+ S O , m ~

a n d

( 9 . 7 3 )

Smb---- nbS mb - n b [$ m b (T )- ~ In Prb + S0,mb] ( 9 . 7 4 )

g i v i n g th e s u m o f th e e n t r o p i e s b e f o r e m i x i n g ( i. e. s e p a r a t e d ) a s

(SmT)sep m (?lTSmT)sep m nT[XaSma(Y) + XbSmb(T + Xa$O,ma']" XbSO,mb'- ~ ( X a l n pra'l" X b In P r b ) ]

( 9 . 7 5 )

S i n c e t h e g a s e s a r e s e p a r a t e d a n d b o t h a r e a t t h e p r e s s u r e , P r , t h e n

(Smr)~p = nr[XoSmo(T) + XbSm b(T) + XaSO.mo + Xb $O .mb- ~ In p ]

H e n c e , t h e m o l a r e n t r o p y , w h e n t h e g a s e s a r e s e p a r a te , i s

(SmT)lep--'-- XaSm o(T + XbSmb(T) + XaSO,m~+ Xb$O,mb- ~ In p

m E XiSm 4" E Xi$O.m, "- ~ In p

( 9 . 7 6 )

( 9 . 7 7 )

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Entropy o f mixtures 17 7

The en t ropy o f the gases a f t e r m ix ing i s

(S m ) , ~ = nTSm = ~., nSm = naSm,, + nbS~ (9.78)

whe re Smo and Smb are the m olar entropies i f each c om pon ent occup ied the wh ole v olum e.

Thus, subst i tu t ing for Sm fro m eqn (9 .20a) gives

(n rS mr)mix f f i n a S m o ( T ) + r l b S m b ( T ) -- 9~(n o In p~o + n b In P rb ) + n.So,m,, + rlb$O,mb (9.79)

(S=T)~ = XoSmo(T) + XbSmb(T) -- 9~(Xa ln pra + Xb ln prb) + X a$O,ma + X b$ O. m b (9.80)

wh ere p,~ = P JP o , and Pi = par t ia l pressure of const i tuen t i .

Equat ion (9 .80) may be wri t ten

($m.r)mix -- ~. X iS m i( T) + ~-~ XiS O,m i- ~ ~-~ Xi In Pri (9.81)

Now, consider the f inal term in eqn (9 .81) , based on two const i tuents to s impli fy the

mathemat ic s

Xo In p,o + Xb In P 'b = Xo ln(XoPr) + Xb l n ( X b P r )= X a In x a + X a In Pr + Xb 111 X b + x b In P r

-- ~., X i In x i + I n P r E Xi

1~--. X xi ln xi + l n g --- ln g -- Z xi l n -

xi

H e n c e

($mT)mix ffi ~" XiSm'(T) + ~" Xi$O'm' + ~ ( v~ xi In 1---x~ npr)

Conside r ing the terms on the f ight-hand s ide of eqn (9 .83) :

1st and 2nd terms

3rd term

(9.82)

(9.83)

4th term

The change o f en tropy due to mix ing i s g iven by

A S = (SmT)mix -- (Sm T )sep (9 .8 4)

which is the dif ference between eqns (9 .83) and (9 .77) . This gives

1A S = 9 ~ ,Y _. ,x~ In - - (9.85)

xi

The nu me rical value of AS m ust be posi t ive bec ause x~ is a lw ay s less than uni ty .

Equat ion (9 .85) shows that there is an entropy increase due to mixing, and this is caused

by the reduc t ion in the o rde r o f the molecu les . B efore m ix ing i t i s poss ib le to go in to one

s ide o f the con ta ine r and guaran tee t ak ing a pa r t icu la r m olecu le , because on ly molec u les o f

a are in the lef t -hand conta iner , and only molecules of b are in the f ight-hand conta iner .

Af te r mix ing i t i s no t poss ib le to know whe the r the molecu le ob ta ined wi l l be o f a o r b :

the order of the system has been reduced.

sum m at ion o f en trop ies be fo re mix ing , (Smv),,p

change o f en t ropy due to mix ing ; th i s i s due to a change o f partial

pressures w h e n mi x e d

pressure term

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17 8 Thermodynamic properties of ideal gases and ideal ga s mixtures

W hat happens i f both a and b in Fig 9.2 are the same gas?

Superf ic ia l ly i t might be imagined that there wil l s t i l l be an increase of entropy on mixing.

However , s ince a and b a re now the same the re i s no c hang e in partial pressure due tomixing. This means tha t the mola r f r ac t ion i s una l te red and

A S = 0 ( 9 . 8 6 )

9.6 Concluding rem arks

A deta i led s tudy of ideal gases and ideal gas mixtures has been under taken in preparat ion

for la ter chapters . Equat ions have been developed for a l l proper t ies , and enthalpy

coef f i c ien t s have been in t roduced fo r n ine commonly encoun te red gases . Tab les o f gas

propert i es have been p resen ted fo r m os t gases occur r ing in co m bus t ion ca lcu la t ions .

Equa t ions fo r gas mix tu res have been deve loped and the e f fec t s o f mix ing on en t ropyand Gibbs ene rgy have been shown.

PROBLEMS

Assum e tha t a i r cons i s t s o f 79% N2 and 21% 02 by volume.

A c losed vesse l o f 0 .1 m 3 capac i ty con ta ins a m ix tu re o f m e thane (CIL ) and a i r , thea i r be ing 20% in excess o f tha t r equ i red fo r chemica l ly co r rec t combus t ion . The

pressure and temperature in the vessel before combust ion are , respect ively , 3 bar and100~ Determine:

(a ) the ind iv idua l pa r t ia l p ressures and the w e igh t s o f me thane , n i t rogen andoxygen p resen t be fo re combus t ion ;

(b ) the ind iv idua l pa r t ia l p ressures o f the burn t p rodu c t s , o n the a ssumpt ion tha tthese are coo led to 100~ wi thou t change o f vo lum e and tha t a l l the vapourproduced by com bus t ion i s condensed .

[ (a ) 0 .2415 , 2 .179 , 0 .579 ba r ; 0 .01868 , 0 .2951 , 0 .08969 kg

Co) 2 .179, 0 .241 5, 0 .0966 b ar]

An eng ine runs on a ri ch mix tu re o f me thy l and e thy l a lcoho l and a i r. A t a p ressure o f

1 ba r and 10~ the fue l i s com ple te ly vapor ised . Ca lcu la te the a i r - fu e l r a t io by

vo lume under these cond i t ions , and the pe rcen tage o f e thy l a lcoho l in the fue l by

weight . I f the to ta l pressure of the exhaust gas is 1 bar , ca lcula te the dew point of the

wate r vapour in the exhaus t and the pe rcen tage by vo lum e o f carbon mon oxide in the

dry exhaus t gas , a s suming a l l the hydrog en in the fue l fo rm s w a te r vapour .

Vapour p ressures a t 10~ m ethy l a lcoho l (CH 3OH ) , 0 .0745 bar ; and e thy l a lcoho l

(C2H5OH), 0.310 bar.

[ 6 3 ~ 4 . 15 % ]

An eng ine work ing on the cons tan t vo lume (Ot to ) cyc le has a compress ion ra t io t~ f

6.5 to 1 , and the compression fol lows the law pVL3ffi C, the in i t ia l pressure and

tempera tu re be ing 1 ba r and 40~ The spec i f ic hea t s a t cons tan t p ressure and

c o n s ta n t v o lu m e t h r o u g h o u t c o m p r e s s i o n a n d c o m b u s t i o n a re 0 . 9 6 + 0 . 0 0 0 0 2 T

kJ /k g K and 0 .67 + 0 .00002 T kJ /kg K respec t ive ly , w here T is in K .

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Problems 179

Find(a )

(b )(e )

the change in en t ropy dur ing com press ion ;

the hea t r e jec ted pe r u n i t mass dur ing com press ion ;

the hea t r e jec ted pe r un i t mass dur ing combus t ion i f the max imum pressure i s

43 ba r and the ene rgy l ibe ra ted by the combu s t ion i s 2150 kJ /kg o f a i r.

[ ( a ) -0.1621 kJ/kg K; ( b ) - 6 7 . 8 k J / k g ; ( e ) - 1 0 9 0 . 6 k J / k g ]

4 A comp ression- igni t ion engine runs on a fuel of the fol low ing analysis by weight :

ca rbon 84% , hydrog en 16%. I f the p ressure a t the end o f com bus t ion i s 55 ba r , the

vo lume ra t io o f expans ion i s 15 :1 , and the p ressure and t empera tu re a t the end o f

expansion are 1 .75 bar and 600"C respect ively , ca lcula te

(a) the var iable specif ic heat a t constant volu m e for the products of com bust ion; and

Ca) the change in entrop y dur ing the exp ansion s t roke per km ol .

The expans ion fo l lows the l aw pW = C an d there is 60% excess a i r. The specif ic

hea t s a t cons tan t vo lume in kJ /km ol K be tw een 6000C and 24000C a re:

0 2 3 2 + 0 . 0 0 2 5 T ; N 2 2 9 + 0 . 0 0 2 5 TH20 34 + 0 .08 T; CO2 50 + 0 .0067 T

The water vapo ur ( I-120) m ay be co nsidered to act as a perfect gas .

[ ( a ) 22.1 +0 . 01 8 4 "[ '; C a) - 1 1 . 4 2 k J / k m o l K ]

w The exhaus t gases o f a com press io n- ign i t io n eng ine a re to be used to d r ive an

exhaus t gas tu rbocharge r . Es t ima te the mean p ressure ra t io o f expans ion and thei sen t rop ic en tha lpy d rop pe r kmol o f gas in the tu rb ine i f the mean exhaus t

t empera tu re i s 6000C and the i sen t rop ie t empera tu re d rop i s 1000C. The com pos i t ion

of the exhaus t gas by vo lum e i s CO2, 8%; 1 -120, 9 .1%; 02 , 7 .5%; N2 , 75 .4 % . The

spec if ic hea t s a t cons tan t vo lume in kJ /k m ol K a re:0 2 3 2 + 0 . 0 0 2 5 T ; N 2 2 9 + 0 . 0 0 2 5 T

H20 34 + 0 .08 T CO2 50 + 0 .0067T

The water vapo ur (1-120) m ay be co nsidered to act as a perfe ct gas .

[ 1 . 9 4 6 ; - 4 5 4 7 . 1 ]

(a) An am ount of substance equal to 2 km ol of an ideal gas at tem per am ~ T and

pressure p is conta ined in a compartment . In an adjacent eompamnent is an amount of

substance equal to 1 km ol o f an ideal gas a t tem pem tme 2T and pressure p . The g ase s m ix

adiabatically but do no t react chemically wh en the parti t ion separating the com partm ents is

withdrawn. S how th at, as a result of the mix ing process, the entropy increases by

9~ (ln 274 + ~c-l~C In 3 ~ )

provided that the gases are dif ferent and that ~c, the ra t io of specif ic heats , i s the same

for bo th gases and rem ains a cons tan t in the t em pera tu re rang e T to 2T .

(b ) W hat would be the en t ropy change i f the gases be ing mixed were o f t he samespecies?

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1 80 Thermodynam ic propert ies of ideal gases and ideal gas m ixtures

7 T h e ex h au s t g a s f r o m a t w o - s t ro k e cy c l e co m p r es s i o n - i g n i t i o n en g i n e i s ex h au s t ed at

an e l ev a t ed p r e s s u r e i n t o a l a r g e ch am b er . T h e g a s f r o m t h e ch am b er i s s u b s eq u en t l y

ex p an d ed i n a t u r b i n e . I f t h e m ean t em p e r a t u r e i n t h e ch am b er i s 8 1 1 K an d t h e

p r e s s u r e r a t i o o f ex p an s i o n i n t h e t u r b i n e i s 4 " 1 , c a l cu l a t e t h e i s en t r o p i e en t h a l p y

d r o p i n t h e t u r b i n e p e r u n i t m as s o f g a s.

[ 2 5 6 . 6 8 ]

T h e f o l l o w i n g d a t a r e f e r t o an an a l y s i s o f a d u a l co m b u s t i o n cy c l e w i t h a g a s h av i n g

s p ec i fi c h ea t s v a r y i n g l i n ea r l y w i t h t em p e r a t u r e .

T h e p r e s s u r e an d t em p e r a t u r e o f t h e g a s a t t h e en d o f co m p r es s i o n a r e 3 1 b a r an d

2 2 7 ~ r e s p ec t i v e l y ; t h e m ax i m u m p re s s u re ach i ev ed d u r i n g t h e cy c l e i s 6 2 b a r , w h i l e

t h e m a x i m u m t e m p e r a t u r e a c h ie v e d i s 1 7 0 0~ T h e t e m p e r a t u r e a t t h e e n d o f t h e

ex p an s i o n s t ro k e i s 1 2 4 0 ~ T h e i n c rea s e s i n en t r o p y d u r i n g co n s t an t v o l u m e an d

c o n s t a n t p r e s s u r e c o m b u s t i o n a r e 0 . 8 8 2 a n d 1 . 4 5 0 k J / k g K r e s p e c t i v e l y . A s s u m i n g

t h a t t h e f l u i d b eh a v es a s an i d ea l g a s o f m o l ecu l a r w e i g h t m , ffi 3 0 . 5 , c a l cu l a t e t h e

eq u a t i o n s f o r t h e s p ec i f i c h ea t s , an d a l s o t h e ex p an s i o n r a t i o i f t h a t p r o ces s i s

i s en t rop ie .

[ 0 .6 7 4 7 + 0 . 0 0 0 8 2 9 T ; 0 . 9 4 7 3 5 + 0 . 0 0 0 8 2 9 T ; 7 . 8 1 4 ]

9 D i s t i n g u i s h b e t w een an i d ea l an d a p e r f ec t g a s an d s h o w t h a t i n b o t h c a s e s t h e

s p ec i fi c en t r o p y , s , i s g i v en b y

s f f i s o + - g t l nro T

T w o s t r eam s o f p e r f ec t g a s e s , A an d B , m i x ad i ab a t i c a l l y a t co n s t an t p r e s s u r e an d

w i t h o u t ch em i ca l ch an g e t o f o r m a t h i r d s t r eam . T h e m o l a r s p ec i f i c h ea t a t co n s t an tp r e s s u r e , C p.m , o f t h e g a s i n s t r e am A i s eq u a l t o th a t i n s t r e am B . S t r ea m A f l o w s a t

M k m o l / s an d i s a t a t em p e r a t u r e 7"1, w h i l e s t r e am B f l o w s a t 1 k m o l / s an d i s a t a

t em p e r a t u r e nT~. A s s u m i n g t h a t t h e g a s e s A an d B a r e d i f f e r en t , s h o w t h a t t h e r a t e o f

en t ropy increase i s

% l n M + I

H o w i s t h e ab o v e ex p r e s s i o n m o d i f i ed i f t h e g a s e s A a n d B a r e th e s am e?

F o r t h e ca s e n = 1 , ev a l u a t e t h e ra t e o f en t r o p y i n c r ea s e

( a )

( b )

w h en d i f f e r en t g a s e s m i x , an d

w h en t h e g a s i n each s t r eam i s t h e s am e .

1 0 A j e t en g i n e b u m s a w eak m i x t u r e o f o c t an e ( C sH ~ s) an d a i r , w i t h an eq u i v a l en ce

r a t io , ~ = 2 . T h e p r o d u c t s o f co m b u s t i o n , i n w h i ch d i s s o c i a t i o n m ay b e n eg l ec t ed ,

en t e r th e n o zz l e w i t h n eg l i g i b l e v e l o c i ty a t a tem p e r a t u r e o f 1 0 0 0 K . T h e g a s e s ,

w h i c h m a y b e c o n s i d e r e d t o b e id e a l, l e a v e th e n o z z l e a t t h e a t m o s p h e r i c p r e s su r e o f

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Problems 181

1 .0 1 3 b a r w i t h a n e x i t v e lo c i ty o f 5 0 0 m / s . T h e n o z z l e m a y b e c o n s i d e r e d t o b e

ad i ab a t ic a n d f r i c ti o n l e s s .

D e t e r m i n e :

( a )

(b )(c )(d )

t h e s p ec i f i c h ea t a t co n s t an t p r e s su r e , C p, o f t h e p r o d u c t s a s a f u n c t i o n o ft e m p e r a t u r e ;

t h e m o l ecu l a r w e i g h t , m w , o f t h e p ro d u c t s ;

t h e t em p e r a t u r e o f t h e p r o d u c t s a t t h e n o zz l e ex i t;

t h e p r e s s u r e o f t h e p r o d u c t s a t t h e n o zz l e i n l e t.

S p ec i f ic h ea t a t co n s t an t p r e s s u r e , C p , m , i n J / k m o l K , w i t h T i n K :

CO 2

H2o

0 2

N2

Cp , m ~-- 21 x 103 + 34.0 T

C p, m -----33 x 103 + 8.3 T

C p , m = 28 x 103 + 6.4 T

Cp,m = 29 X 103 + 3 .4 T[0 .9986 • 103 + 0 .21 05T ; 28 .71 ; 895 .7 ; 1 .591 bar ]

11 T h e p r o d u c t s o f c o m b u s t i o n o f a j e t e n g in e h a v e a m o l e c u l a r w e i g h t, m w , o f 3 0 a n d a

m o l a r s p ec i f i c h ea t a t co n s t an t p r e s s u r e g i v en b y Cp.m fi 3 . 3 X 1 0 4 + 1 5 T J / k m o l K

w h e r e T i s t h e g a s t e m p e r a t u r e i n K e lv i n . W h e n t h e j e t p i p e s t a g n a ti o n t e m p e r a t u r e i s

1 2 0 0 K t h e g a s e s l e av e t h e n o zz l e a t a r e l a t i v e s p eed o f 6 0 0 m / s . E v a l u a t e t h e s t a t i c

t em p e r a t u r e o f t h e g a s a t t h e n o zz l e ex i t an d e s t i m a t e t h e t o t a l to s t a ti c p r e s s u r e r a t io

a c ro s s th e n o z z l e . A s s u m e t h a t t h e p ro d u c t s o f c o m b u s t io n b e h a v e a s a n i d e a l g a s a n d

t h a t t h e f l o w i s i sen t r o p i c .

I n a f r i c t i o n a l n o z z l e p r o d u c i n g t h e m e a n o u t l e t s p e e d f r o m t h e s a m e i n l e t gas

t e m p e r a t u r e, w h a t w o u l d b e t h e e f f e c t o n( a ) t h e m ea n o u t l e t s t a ti c t em p e r a t u r e , an d

(b) the to ta l to s t a t i c p res su re r a t io?

[ 1 0 9 2 ; 1 . 7 3 2 ; ( a ) u n a f f ec t ed ; ( b ) i n c r ea s ed ]