Heat capacity of a substance is defined as the amount of heat required to rise of the temperature by one degree. Heat capacity per gram of substance is called specific heat and per mole called molar heat capacity.

Thus, Molar heat capacity = Molar mass × Specific heat

C

_{P}= M × c_{p}
C

_{V}= M × c_{v}
Where C

_{P}and C_{V}are the molar heat capacities at constant pressure and constant volume respectively. c_{p}and c_{v}are their specific heats.

*Problem*- The specific heat at constant pressure and constant volumes are 0.125 and 0.075 Cal gm⁻¹K⁻¹ respectively. Calculate the molecular weight and atomicity of the gas. Name the gas if possible.

*Answer*For gases, there are two heat capacities at constant volume and constant pressure.

These are Represented as,

C

_{V}= (dq/dT)_{V}= (dU/dT)_{V}
C

_{P}= (dq/dT)_{P}= (dU/dT)_{P}+ P(dV/dT)_{P}###
*Difference in Heat Capacities of an Ideal Gas*

*Difference in Heat Capacities of an Ideal Gas*

If the gas is assumed to be ideal, then,

PV = nRT and (dU/dT)

_{V}= (dU/dT)_{P}
Again, P(dV/dT)

_{P}= nR
Thus for an ideal gas, PV = nRT; or, P(dV/dT)

_{P}= nR.
Again,

**C**= (dU/dT)_{P}_{P}+ P(dV/dT)_{P}
or,

**C**=_{P}**C**+ {P×(nR/P)}_{V}
or,

**C**_{P}**=****C**_{V}**+nR**
For 1 mole ideal gas

**C**_{P}**= Cv +R**Molar Heat Capacity Of Gases |

From the above two descriptions, it is clear that

**C**and_{P}**C**. Since for_{V}**C**, some mechanical work is required as additional energy to absorbed for definite piston from volume_{P}**V₁**to**V₂**. Thus ,**C**-_{P}**C**= Mechanical Work = PdV = P(V₂-V₁) = PV₂-PV₁ = R(T+1) - RT = R_{V}
Again,

**C**=_{P}**C**+R for 1 mole ideal gas._{V}
Now let us find the expression of

**C**from the point of view of Kinetic Theory._{V}**C**= Energy required to increase the translational kinetic energy of 1-mole gas for a rise of 1° temperature + energy required to increase the intermolecular energy of 1-mole gas for the rising temperature of 1°.

_{V}
Increase of transnational K.E. = (3/2)R(T+1) - (3/2)R = (3/2) R for 1 mole gas for 1° rise in temperature.

####
__Mono-atomic Gases:__

__Mono-atomic Gases:__

**C**= (dU/dT)v = (3/2)R and

_{V}**C**= (5/2)R.

_{P}
Thus Î³ =

**C**/_{P}**C**= 5/3 ≃ 1.667_{V}####
__Poly-atomic Gases:__

__Poly-atomic Gases:__

Linear →

**C**= (dU/dT)v = (3/2)R + R +(3N - 5)R_{V}
Non-linear →

**C**= (dU/dT)v = (3/2)R + (3/2)R +(3N - 6)R_{V}
Where N is the number of particles.

####
__Molar Heat Capacities for Diatomic Molecules:__

__Molar Heat Capacities for Diatomic Molecules:__

For diatomic molecule N = 2.

Thus

**C**= (3/2)R + R + R = (7/2)R and_{V}**C**= (9/2)R._{P}
∴ ⋎ =

**C**/_{P}**C**= 9/7 ≃ 1.286_{V}####
__Molar Heat Capacities for Tri-atomic Molecules:__

For tri-atomic molecule N = 3.__Molar Heat Capacities for Tri-atomic Molecules:__

Thus

**C**(Linear) =

_{V}