# Relation Between Specific Heat Capacities of Gas | Physics Grade XI

### Specific Heat Capacities of the Gas # Relation Between the Specific Heat Capacities of the Gas

Relation between the specific heat capacities of the gas
Let us consider n-mole of an ideal gas in a cylinder fitted with a weightless, frictionless and movable piston. Let P, V and T be the initial pressure, volume and temperature of the gas. Let the gas be heated at constant volume, and its temperature be increased by dT, then the amount of heat supplied to the system is given by
dQ = n Cv dT …….. (i)

When gas is heat at constant volume, it does not do any work so,
dQ = dU
dU = n Cv dT ………… (ii)

Similarly, if it was heated at constant pressure, and temperature be increased by dT, then
dQ = n Cp dT ……… (iii)

As the heat supplied is used to increase the internal energy as well as the work done against external force, then
dQ = dU + dW
dQ = dU + P dV ….... (iv)

from equation (iii) and (iv),
n Cp dT = dU + P dV

Since internal energy of a gas only depends upon temperature,
n Cp dT = n Cv dT + P dV  ……... (v)

for n-mole of gas we have,
PV = nRT …....... (vi)

Differentiating (vi) w.r.t T at constant pressure,
dPV/ dT = dnRT/dT
P dV/dT = nR dT/dT
P dV = nRdT …..... (vii)

Then equation (v) becomes,
n Cp dT = n Cv dT + nRdT
Cp = Cv + R
Cp – Cv = R …..... (viii)
Which is also known as Mayer’s formula

If M is the molar mass of the gas, then equation (viii) can be written as
C/ M – Cv/ M = R/M
Cp – Cv = r, where r is the gas constant per unit mass

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