Chapter 2

Part 3 of 4

Dr.Khaled S. Al-zahrani

Boyle’s Law

Born:25 January 1627

Died:31 December 1691

(aged 64)

London

𝒊𝒇 𝑻 = 𝑪 𝑷 ∝𝟏

𝑽

Experiment

Dr.Khaled S. Al-zahrani

Charles's Law

Born: Nov. 1746

Died: April 1823

(aged 76)

Paris

𝒊𝒇 𝑷 = 𝑪 𝑽 ∝ 𝑻

Experiment

Dr.Khaled S. Al-zahrani

Avogadro's Law

Born: Aug. 1776

Died: April 1856

(aged 79)

Italy

𝒊𝒇 𝑷&𝑻 = 𝑪 𝑽 ∝ 𝒏

The volume of the gas is

proportional to the number of gas

particles.

Boyle’s Law Charles’s Law

𝑃 ∝1

𝑉 @ 𝑇 = 𝑐

𝑃𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑃𝑉

𝑇𝑛= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

V ∝ 𝑇 @ 𝑃 = 𝑐

𝑉

𝑇= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑃𝑉

𝑇𝑛= 𝑅

𝑃𝑉 = 𝑛𝑇𝑅

Avogadro’s Law

v ∝ 𝑛 @ 𝑃& 𝑇 = 𝑐

𝑉

𝑛= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑖𝑓 𝑛 =𝑚

𝑀

𝑡ℎ𝑒𝑛:𝑚 = 𝑛𝑀

n: Number of kg mole

R: specific gas constant

287.04J/ kg K

M: Molecular weight

kg/kg mole

Characteristic gas equation

Dr.Khaled S. Al-zahrani

Ideal Gas The gas which obeys all the gas laws and the characteristic

gas equation at all pressure and temperature

𝑪𝑷 specific heat @ constant pressure

𝑪𝑽 Specific heat @ constant volume

𝛾 =𝐶𝑃𝐶𝑉

Specific ratio

𝑅 = 𝐶𝑃 − 𝐶𝑉 Gas constant

𝐶𝑃 =𝛾 𝑅

(𝛾 − 1)

𝐶𝑉 = 𝑅

(𝛾 − 1)

𝚫𝐇 = 𝑚 𝐶𝑝 𝑇2 − 𝑇1 ∆𝐔 = 𝑚 𝐶𝑣 (𝑇2 − 𝑇1)

Dr.Khaled S. Al-zahrani

Gas Process

V

P

PVn = C General (polytropic) Process

Value of n PVn = C Process law Process

name

Isobaric

Isochoric

Isothermal

Adiabatic

n = 0

n = 1

n = 𝛾

n = ∞

PV0 = C P = C Isobaric

PV1 = C PV = C Isothermal

PV𝜸 = C PV𝛾 = C

Adiabatic

PV∞= C V = C Isochoric

Dr.Khaled S. Al-zahrani

Isobaric Process

V

P

V1 V2

P1 = P2

1 2 Work Done:

𝑊 = 𝑃𝑑𝑉

2

1

= 𝑃 𝑑𝑉

2

1

= 𝑃 𝑉2 − 𝑉1 𝑖𝑓 𝑃𝑉 = 𝑚𝑅𝑇

= 𝑚𝑅 𝑇2 − 𝑇1

= 𝑅 𝑇2 − 𝑇1 𝑝𝑒𝑟 𝑈𝑛𝑖𝑡 𝑚𝑎𝑠𝑠

Internal Energy:

Δ𝑈 = 𝑈2 − 𝑈1

= 𝑚𝐶𝑉 𝑇2 − 𝑇1

Heat Supplied:

𝛿𝑄 = 𝑑𝑈 + 𝛿𝑊 = 𝑑𝑈 + 𝑃𝑑𝑉 = 𝑑𝑈 + 𝑑(𝑃𝑉) = 𝑑 (𝑈 + 𝑃𝑉) = 𝑑𝐻

= 𝑚𝐶𝑃 𝑇2 − 𝑇1

P = C

Dr.Khaled S. Al-zahrani

Isochoric Process

V

P

P1

P2

V1 = V2

1

2

Work Done:

𝛿𝑄 = 𝑑𝑈 + 𝛿𝑊 = 𝑑𝑈 + 𝑍𝑒𝑟𝑜

𝑊 = 𝑃𝑑𝑉

2

1

= 𝑃 𝑑𝑉

2

1

= 𝑃 𝑉2 − 𝑉1 = 𝑃 𝑍𝑒𝑟𝑜

Internal Energy:

𝑑𝑈 = 𝛿𝑄

Heat Supplied:

𝛿𝑄 = 𝑑𝑈 + 𝛿𝑊 = 𝑑𝑈 + 𝑍𝑒𝑟𝑜

𝛿𝑄 = 𝑑𝑈

= 𝑍𝑒𝑟𝑜

= 𝑚𝐶𝑉 𝑇2 − 𝑇1

V = C

Dr.Khaled S. Al-zahrani

Isothermal Process

Work Done:

𝑊 = 𝑃𝑑𝑉

2

1

PV = C P = 𝐶

𝑉

= 𝐶

𝑉𝑑𝑉

2

1

= 𝐶 1

𝑉𝑑𝑉

2

1

= 𝐶 𝑙𝑛𝑉2𝑉1

C =𝑃1𝑉1 OR =𝑃2𝑉2

= 𝑃1𝑉1 ln𝑉2𝑉1

= 𝑃2𝑉2 ln𝑉2𝑉1

OR

V

P

P1

P2

1

2

V1 V2

PV = T = C

Dr.Khaled S. Al-zahrani

Isothermal Process

V

P

P1

P2

1

2

V1 V2

PV = T = C

Internal Energy:

Δ𝑈 = 𝑚𝐶𝑉 𝑇2 − 𝑇1

= 𝑚𝐶𝑉 𝑍𝑒𝑟𝑜

= 𝑍𝑒𝑟𝑜

Heat Supplied:

𝛿𝑄 = 𝑑𝑈 + 𝛿𝑊 = 𝑍𝑒𝑟𝑜 + 𝛿𝑊 = 𝛿𝑊

= 𝑃1𝑉1 ln𝑉2𝑉1

= 𝑃2𝑉2 ln𝑉2𝑉1

OR

Dr.Khaled S. Al-zahrani

Polytropic Process:

Work Done:

𝑊 = 𝑃𝑑𝑉

2

1

PVn = C P = 𝐶

Vn = 𝐶𝑉−𝑛

= 𝐶𝑉−𝑛 𝑑𝑉

2

1

= 𝐶 𝑉−𝑛𝑑𝑉

2

1

=𝐶

1 − 𝑛(𝑉2

1−𝑛 − 𝑉11−𝑛)

=𝑃2 𝑉2

𝑛 𝑉21−𝑛 − 𝑛 − 𝑃1 𝑉1

𝑛 𝑉11−𝑛

1 − 𝑛

C = 𝑃2 𝑉2𝑛 or 𝑃1 𝑉1

𝑛

=𝑃2𝑉2 − 𝑃1𝑉1

1 − 𝑛

=𝑃1𝑉1 − 𝑃2𝑉2

𝑛 − 1

OR 𝑖𝑓 𝑃𝑉 = 𝑚𝑅𝑇

= 𝑚𝑅 𝑇2 − 𝑇1

1 − 𝑛

= 𝑅 𝑇2 − 𝑇1

1 − 𝑛 𝑈𝑛𝑖𝑡 𝑚𝑎𝑠𝑠

Dr.Khaled S. Al-zahrani

Adiabatic Process:

PVn = C P𝑉𝛾 = C 𝑛 = 𝛾

Dr.Khaled S. Al-zahrani

Ideal Gas Process

Equations

Internal energy

∆U

Work

(𝑾 )

Heat

(𝑸 )

Equation of state

(𝑷𝑽 = 𝒎𝑹𝑻)

Isothermal Process

𝑇 = 𝐶 Zero 𝑃1𝑉1 𝑙𝑛

𝑉2 𝑉1

W 𝑃1𝑉1 = 𝑃2𝑉2

Isochoric Process

𝑉 = 𝐶 ∆U Zero ∆U

𝑃1𝑇1

=𝑃2𝑇2

Adiabatic Process

𝑃𝑣𝛾 = 𝐶

𝑆 = 𝐶

-W

𝑃1𝑉1 − 𝑃2𝑉2 γ − 1

𝑃2𝑉2 − 𝑃1𝑉1 1 − 𝛾

Zero 𝑃1𝑉1

𝛾= 𝑃2𝑉2

𝛾

𝑇1𝑇2

=𝑉2𝑉1

𝛾−1

=𝑃1𝑃2

𝛾−1𝛾

Isobaric Process

𝑃 = 𝐶 ∆U 𝑃 𝑉2 − 𝑉1

𝑅 (𝑇2 − 𝑇1)

𝚫𝐇 𝑉1𝑇1

=𝑉2𝑇2

Polytropic process

𝑃𝑣𝑛 = 𝐶 ∆U

𝑃1𝑉1 − 𝑃2𝑉2 𝑛 − 1

𝑅 (𝑇1 − 𝑇2)

𝑛 − 1

W+ Δ𝑈

𝛾 − 𝑛

𝛾 − 1 𝑅

1 − 𝑛 (𝑇2 − 𝑇1)

𝑃1𝑉1 𝑛= 𝑃2𝑉2

𝑛

𝑇1𝑇2

=𝑉2𝑉1

𝑛−1

=𝑃1𝑃2

𝑛−1𝑛

Dr.Khaled S. Al-zahrani

Dr.Khaled S. Al-zahrani

Information manual: pages 84 86

Questions: 12 20