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 volume 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:
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:
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_{P} = (dU/dT)_{P} + P(dV/dT)_{P}
or, C_{P} = C_{V} + {P×(nR/P)}
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_{P} and C_{V}. Since for C_{P}, some mechanical work is required as additional energy to absorbed for definite piston from volume V₁ to V₂. Thus , C_{P}  C_{V} = Mechanical Work = PdV = P(V₂V₁) = PV₂PV₁ = R(T+1)  RT = R
Again, C_{P} = C_{V}+R for 1 mole ideal gas.
Now let us find the expression of C_{V} from the point of view of Kinetic Theory.
C_{V} = Energy required to increase transnational kinetic energy of 1 mole gas for rise of 1° temperature + energy required to increase inter molecular energy of 1 mole gas for rise temperature of 1°.
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.
Monoatomic Gases:
C_{V} = (dU/dT)v = (3/2)R and C_{P} = (5/2)R.
Thus Î³ = C_{P}/C_{V} = 5/3 ≃ 1.667
Polyatomic Gases:
Linear → C_{V} = (dU/dT)v = (3/2)R + R +(3N  5)R
Nonlinear → C_{V} = (dU/dT)v = (3/2)R + (3/2)R +(3N  6)R
Where N is the number of particles.
Molar Heat Capacities for Diatomic Molecules:
For diatomic molecule N = 2.
Thus C_{V} = (3/2)R + R + R = (7/2)R and C_{P} = (9/2)R.
∴ ⋎ = C_{P}/C_{V} = 9/7 ≃ 1.286
Molar Heat Capacities for Triatomic Molecules:
For triatomic molecule N = 3.Thus C_{V}(Linear) =