MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Kinetic Theory of Gases Page 3

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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 dQ1 = Cv'dT.

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

 dv = Cv'dT (\ dW= 0)

• Starting with the same initial conditions, the gas is heated at constant pressure so that the temperature rises by dT. In this case, the gas expands.

The energy supplied is dQ2 = Cp'dT
\ Cp'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
\
Cp'dT= Cvn'dT+ dP

• For a gas enclosed in the cylinder with a piston of cross-sectional area A, dx is the displacement of the piston during the expansion of the gas. The expansion is due to heating of the gas under constant pressure.

 Work done = 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+ dT)
\ PdV= RdT
\ Cp'dT= Cv'dT+ RdT
\ Cp' - Cv' = 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,

 Cp' - Cv' = R Þ Mayer's relation for Molar specific heats (in heat units). J

Mayer's Relation for Principal specific heats

 Cv = Cv' M

 Cp = Cp' M

\ MCp - MCv = R

 Cp - Cv = R M

 In heat units, Cp - Cv = R MJ

 Further, PV= nRT= m RT M

m : mass of gas
M: Molecular weight

 \ P V = RT m M

 P = RT r M

 \ P = RTr M

 \ R = P M rT

 \ Cp - Cv = P rTJ

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