School of Chemical & Biological Engineering, Konkuk University

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-2

Lecture 11

• The 3rd law of thermodynamics

• The Helmholtz and Gibbs Energies

Ch. 3 The Second Law

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-3

• At T=0, all energy of thermal motion has been quenched, and

in a perfect crystal all the atoms or ions are in a regular, uniform

array.

• The localization of matter and the absence of thermal motion

suggest that such materials also have zero entropy.

• In the molecular interpretation of entropy, S=0 at T=0, since

there is only one way to arrange molecules and only one

microstate is accessible.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-4

• The entropy of a regular array of molecules is zero at T=0.

• The above expression summarized by the Nernst heat

theorem:

The entropy change accompanying any physical or

chemical transformation approaches zero as the

temperature approaches zero: S0 as T0, provided all

the substances involved are perfectly crystalline.

• If we arbitrarily ascribe the value zero to the entropies of

elements in their perfect crystalline form at T=0, all perfect

crystalline compounds also have zero entropy at T=0.

• The 3rd law of thermodynamics: The entropy of all perfect

crystalline substances is zero at T=0.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-5

• In most cases, there is only one accessible microstate (W=1)

at T=0, so the entropy is zero.

• However, in certain cases, W >1 even at T=0, and S >0 the

non-zero entropy at T=0 is called residual entropy.

• This is the case if there is no energy advantage in adopting a

particular orientation even at T=0.

• Ice has a residual entropy of 3.4 J/Kmol.

short O-H bond

long O-H bond

There is a degree of randomness in which two bonds are

short and which two are long.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-6

• The entropies reported on the basis that S(0)=0 for all perfect

crystalline substances are called Third-Law Entropies.

• When the substance is in its standard state (pure, 1 bar) at T,

the standard (Third-Law) entropy is denoted .

• The standard reaction entropy ( ) is defined as:

)(TS o

o

rS

o

m

o

m

o

r SSS ReactantsProducts

where is standard molar entropy of species J at the

temperature of interest.

omS

• Standard reaction entropies are likely to be positive if there is a

net formation of gas in a reaction, and negative if there is a net

consumption of gas.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-7

• For ions in solution, the standard entropy of the H+ ions in

water is taken as zero at all temperatures:

0)aq ,H( oS

• Because the entropies of ions in water are values relative to

the H+ ion in water, they may be either positive or negative.

• The positive entropy means that the ion has a higher molar

entropy than H+ in water.

• The negative entropy means that the ion has a lower molar

entropy than H+ in water.

• Small and highly charged ions induce local structure in the

surrounding water, and the disorder of the solution is decreased

more than in the case of large and singly charged ions.

molJ/K 57)aq ,Cl( o

mS molJ/K 128)aq ,M( 2 gS o

m

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-8

• Entropy is the basic concept for discussing the direction of

spontaneous change, but to use it we have to analyze

changes in both system and its surroundings.

• It’s very inconvenient to use the entropy.

• Therefore we will devise a simple method for taking the

contribution from the surroundings into account automatically.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-9

• Consider a system in thermal equilibrium with its surroundings

at a temperature T.

• When a change in the system occurs, there is a transfer of

energy as heat between the system and its surroundings.

• The changes that satisfies the Clausius inequality are

spontaneous.

T

dqdS 0

T

dqdS

• At constant volume, owing to the absence of non-expansion

work, dqV = dU 0

T

dUdS

This criterion for spontaneous change is expressed

solely in terms of the state functions of system.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-10

• For a constant-V process, at either constant internal energy

(dU=0) or constant entropy (dS=0), the criteria for spontaneity

are:

work)additional no V,(constant or dU TdST

dUdS

0or 0 ,, VSVU dUdS

The criterion for

spontaneity in a isolated

system

At const S and V, for a

spontaneous change, U should

be decreased to increase the

entropy in its surroundings.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-11

• At constant pressure and no additional work, dqp = dH

0T

dHdS

• For a constant-p process, at either constant enthalpy (dH=0)

or constant entropy (dS=0), the criteria for spontaneity are:

work)additional no p,(constant or dH TdST

dHdS

0or 0 ,, pSpH dHdS

At const p, for a

spontaneous change, S

should increase but H

should be constant to

ensure no change in Ssur.

At const S and p, for a

spontaneous change, H should

be decreased to increase the

entropy in its surroundings.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-12

0TdSdU 0TdSdH

• Here two thermodynamic quantities, Helmholtz energy (A)

and Gibbs energy (G) are respectively defined as:

TSUA TSHG

where all the symbols in these definitions refer to the system.

• At constant T, the two properties of the system change as

below: TdSdUdA TdSdHdG

Clausius inequality at const V.

& no additional work

Clausius inequality at const p.

& no additional work

0, VTdA 0, pTdG

• “Concentrating on system” is successful !!!

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-13

0, VTdA

• At const T & V, if the Helmholtz energy of a system decreases,

the change is spontaneous.

• The criterion of equilibrium, when neither the forward nor

reverse process has a tendency to occur, is:

0, VTdA

• A false interpretation of 0, TdSdUdA VT

The tendency of a system to move to lower A is due to its

tendency to move towards states of lower U and higher S.

• That should be corrected: the tendency to lower A is solely a

tendency towards higher overall entropy (system + its

surroundings).

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-14

• The change in the Helmholtz energy is equal to the maximum

work accompanying a process: dAdw max

• So, A is sometimes called the “maximum work function” or the

“work function”.

• Proof: The Clausius inequality can be combined with the 1st

law as below:

TdSdUdw

TdSdUdwdwdUTdS

dwdqdUdqTdST

dqdS

max

dAdw max

where the maximum work is done by the system only when

the path is reversible because then the equality applies.

Because at const T, dA=dU-TdS,

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-15

• For a measurable change in isothermal process,

Aw max STUA with

• In the case of S<0, only some of U

(negative) is converted into work.

• Some of the energy must escape as heat

from the system in order to generate enough

entropy in the surroundings.

Uw max

• Nature demands a tax on the U when it is converted into work.

• Because A is that part of U which freely used to do work,

the A is also called Helmholtz free energy.

tax

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-16

• In the case of S>0, the maximum work

obtained from the system is greater than U

(negative).

• Since the entropy of the system increases,

we can afford to loss some entropy of the

surroundings.

• For a measurable change in isothermal process,

Aw max STUA

Uw max

• Heat may flow into the system from its surroundings as work

is done by the system.

• Nature is now providing a tax refund.

tax refund

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-17

0, pTdG

• At const T & p, if the Gibbs energy (often called free energy)

of a system decreases, the change is spontaneous.

• The Gibbs energy is more common in chemistry than the A

due to the limiting condition of the constant P.

• At constant T & p, chemical reactions are spontaneous in the

direction of decreasing Gibbs energy.

• If G decreases as the reaction proceeds, the forward reaction

is spontaneous.

• If G increases as the reaction proceeds, the reverse reaction

is spontaneous.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-18

• In the case of spontaneous (dG<0) endothermic (dH>0)

reactions, the entropy of the system should increase so much

that TdS can overcome the increase of the enthalpy (which is

related to the decrease of the entropy in its surroundings).

• Therefore the spontaneous endothermic reactions are driven

by the increase of entropy of the system.

0, TdSdHdG pT

pconst at T

dHdSsur

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-19

• The change in the Gibbs energy is equal to the maximum

non-expansion (or additional) work accompanying a process:

dGdwadd max,

• So, G is often called the “free energy”.

• This expression is particularly useful to assess the electrical

work in fuel cells or electrochemical cells.

• Proof: For a general change,

Gwadd max,

SdTTdSpVddwdqdG

SdTTdSdHdGpVddwdqdH

TSHGpVUH

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-20

TdSpVddwdqdG

• Proof: when the change is isothermal (dT=0),

when the change is reversible,

TdSdqdqdwdw revrev and

pVddwTdSpVddwTdSdG revrev

For a reversible isothermal process,

where the work consists of expansion work and some

additional work.

iswork expansion process, reversible afor

,revadddwpdVdw

-pdV

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-21

Vdpdw

VdppdVdwpdVdG

revadd

revadd

,

,

For a reversible isothermal process at constant P,

revadddwdG ,

Because the process is reversible, the additional work done

by the system must have its maximum value.

dGdwadd max,

Prof. Yo-Sep Min Physical Chemistry I, Spring 2008 Ch. 3-22

• Reading: page 100 ~ 109

• Problem set will be given in the next

class.

• The 1st Exam: Apr. 11 (Fri) 19:00

Room B566

Chapt. 1 & 2