CONSTANT PRESSURE AND VOLUME SPECIFIC HEATS CLASS NOTES FOR ENGINEERING

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 1oC. 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 1oC. 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
Hydro-chloric 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 mono-atomic, diatomic and poly-atomic gases

There can be three values of C p and three values of C v depending upon whether the gas is mono-atomic, diatomic or poly-atomic. As per Equi-partition 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 mono-atomic gas has only three degrees of translation motion. A diatomic gas has three degrees of translation motion and two degrees of rotational motion A poly-atomic gas has three degrees of translation motion as well as three degrees of rotational motion.

Specific heats

Gas

Mono-atomic

Diatomic

Poly-atomic

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 200 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 H-bonds. 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.