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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.
the gas is heated at constant volume, the energy supplied to raise
the temperature by dT is dQ1 = Cv'dT.
energy is completely absorbed by the gas. Therefore, change in internal
energy of the gas is given by
dv = Cv'dT
(\ dW= 0)
The energy supplied is dQ2 = Cp'dT
\ Cp'dT= du + dw
= change in internal energy
= work done by gas against external pressure.
For the same rise of temperature, change in internal energy remains the
\ Cp'dT= Cvn'dT+ dP
= force ´ displacement
= Pressure ´ Area ´ displacement
\ dW= P ´ A´ dx
But, A.dx = dv = increase in volume of the gas.
\ dW= PdV
\ Cp'dT= Cv'dT+ Pdv
For an ideal gas, PV= RT
(1 mole of gas)
\ P( V
+ dV) = R( T+
\ PdV= RdT
\ Cp'dT= Cv'dT+ RdT
= R Þ Mayer's relation for molar specific heats of gas
If R is in mechanical unit and Cp' and Cv'
are in heat units then,
relation for Molar specific heats (in heat units).
Mayer's Relation for Principal specific heats
\ MCp - MCv
Cp - Cv
In heat units,
Cp - Cv =
m : mass of gas
M: Molecular weight
\ P =
\ Cp - Cv