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Mayer's Relation between Specific Heats of a Gas

Consider one mole of a perfect gas which is enclosed in a cylinder fitted with an air-tight piston. Let P,V,T be the initial values of pressure, volume and temperature of a gas.

When the gas is heated at constant volume, the energy supplied to raise the temperature by dT is dQ_{1} = C_{v}'dT.

This energy is completely absorbed by the gas. Therefore, change in internal energy of the gas is given by

dv = C_{v}'dT

(\ dW= 0)

The energy supplied is dQ_{2} = C_{p}'dT \ C_{p}'dT= du + dw

Where, du

= change in internal energy

dw

= work done by gas against external pressure.

For the same rise of temperature, change in internal energy remains the same. \ du=C'_{v} dT \ C_{p}'dT= Cv_{n}'dT+ dP

Work done

= force ´ displacement

= Pressure ´ Area ´ displacement

\ dW= P ´ A´ dx

But, A.dx = dv = increase in volume of the gas. \ dW= PdV \ C_{p}'dT= C_{v}'dT+ Pdv

For an ideal gas, PV= RT

(1 mole of gas)

\ P( V + dV) = R( T+ dT) \ PdV= RdT \ C_{p}'dT= C_{v}'dT+ RdT \ C_{p}' - C_{v}' = R Þ Mayer's relation for molar specific heats of gas If R is in mechanical unit and C_{p}' and C_{v}' are in heat units then,

C_{p}' - C_{v}' =

R

Þ Mayer's relation for Molar specific heats (in heat units).

J

Mayer's Relation for Principal specific heats

C_{v} =

C_{v}'

M

C_{p} =

C_{p}'

\ MC_{p} - MC_{v} = R

C_{p} - C_{v} =

In heat units, C_{p} - C_{v} =

MJ

Further, PV= nRT=

m

RT

m : mass of gas M: Molecular weight

\ P

V

=

P

r

\ P =

RTr

\

rT

\ C_{p} - C_{v} =

rTJ