CONSTANT PRESSURE AND VOLUME
SPECIFIC HEATS CLASS NOTES FOR
ENGINEERING
Constant pressure specific heat
Specific heat at constant pressure,’ C p’ is quantity of heat required for raising the temperature of 1 kg of a substance (solid, liquid or gas) at constant pressure by 1^{o}C. Its units are kJ/kg ^{o }C.
Specific heat at constant Volume
C v is quantity of heat required for raising the temperature of 1 kg of a substance (solid, liquid or gas) at constant volume by 1^{o}C. Its units are kJ/kg ^{o }C.
Two specific heats C v and C p of Solids are equal because all solids have fixed volume and has no effect of pressure.
C p and C v of one solid have one fixed value and different for different solids..
For example, C v = C p = of copper= 0.386 kJ/kg K
TABLE: Specific heats of solids
Substance

C p, kJ/kg K

C v, kJ/kg K

Copper

0.386

0.386

Aluminum

0.900

0.900

Brass

0.380

0.380

Iron

0.450

0.450

Specific heats of liquids
C p and C v of one liquid have one fixed value because all liquids have fixed volume and has no effect of pressure.
Specific heats of water C p = C v= 4.186 kJ/kg K
Liquid substance

C p, kJ/kg K

C v, kJ/kg K

Water

4.186

4.186

Milk

3.930

3.930

Hydrochloric Acid

3.140

3.140

Mercury

0.140

0.140

Specific heats of gases
A gas has a different value of C p and C v because volume is affected by pressure.
C p of air = 1.005 kJ/kg K
C v of air = 0.714 kJ/kg K
VALUES ARE AT 20 C.
Gas

C p, kJ/kg K

C v, kJ/kg K

Air

1.005

0.714

Nitrogen

1.04

0.743

Hydrogen

14.32

10.16

Oxygen

0.919

0.659

Helium

5.19

3.12

Different values of C p and C v for monoatomic, diatomic and polyatomic gases
There can be three values of C p and three values of C v depending upon whether the gas is monoatomic, diatomic or polyatomic. As per Equipartition theorem, energy is equally distributed for each degree of freedom by the amount RT/2 (J/ mol). R is universal gas constant. T is absolute temperature. A monoatomic gas has only three degrees of translation motion. A diatomic gas has three degrees of translation motion and two degrees of rotational motion A polyatomic gas has three degrees of translation motion as well as three degrees of rotational motion.
Specific heats
Gas

Monoatomic

Diatomic

Polyatomic

Degrees of freedom= Trans + Rotational

3+0=3

3+2=5

3+3=6

C v

3RT/2

5RT/2

6RT/2

C p

5RT/2

7RT/2

8RT/2

NOTE:
Vibration modes have been neglected since these are found only at very high temperature which is not found in day to day applications.
Values have been tabulated as per 20^{0} C or 293 K.
Why liquids has the highest value of specific heat than its corresponding solid or gas?
In liquids, the molecules are neither very closely packed (so that heat can pass by just pure vibration as in solids) nor very far apart (so heat can pass by just flying around and banging other molecules as in gases). The molecules in a liquid are bonded by van der Waals forces and some bonds like Hbonds. Thus some energy is used in overcoming van der Waals forces and also in breaking the bonds. Thus the specific heat of a liquid is highest as compared to its solid or its gaseous form. For example
Ice specific heat = 2.108 kJ/kg K
Water specific heat = 4.186 kJ/kg K
Specific heat of water vapor = 1.996 kJ/kg K
Relations between Cp and Cv of a gas
(a) Cp – Cv =R in heat units i.e kJ/kg K
(b) Ratio of Cp/Cv = γ
γ is used as an index of compression or expansion in an isentropic process and is also used in finding the speed of sound in a gas. The value of γ for air is fixed and is 1.4.
In a polytropic process, Ratio of Cp/Cv = n
The value of n for air varies between 1.0 and 1.4.