relation between two principal specific heats of a gas(mayer's formula)

 C_p - C_v = R

where C_p = specific heat at constant pressure

         C_v = specific heat at constant volume

          R = universal gas constant  

Consider one mole of a perfect gas enclosed in a perfectly conducting cylinder. Let the cylinder be fitted with a perfectly conducting, frictionless piston as shown in figure 19.1

Keeping the volume constant, let the temperature of the gas be increased from T to T + dT. If Cv is the specific heat of the gas at constant volume, then the heat supplied to increase the internal energy of the gas is CvdT.

Suppose that now we supply heat energy to the gas and allow the piston to move so as to keep the pressure constant. If the temperature changes from T to T + dT, then the part of the total energy supplied that is used in increasing the internal energy of the gas is equal to CydT. If the piston moves through a distance dx to keep the pressure constant, then the amount of external work done by the heat energy is

W = Force x Displacement

= Pressure x Surface area of the piston x Displacement = (P) (A) (dx) = P dV where dV = A dx is the increase in the volume of the gas.

Thus,

f Total heat supplied 1 = /Heat supplied at j + External work done i at constant pressure J Iconstant volume J

i.e., C dT = C dT + PdV ....................... (1)

' p V v 7

For a perfect gas,

PV = RT

.-. P (V + dV) = R (T+ d T) PV + PdV = RT + R dT

P dV = R dT, since PV = RT Hence eq (1) becomes

C dT = C dT + R dT

P v

or C = C + R

p ' V

or C - C = R

p V

This relation is known as Mayer's equation.

Note : For any mass m of the gas, cp - cy = r = nR where n = is the number of moles present in mass m.