dQ = dE + dW

where dQ is the heat energy input, dE is the increase in internal energy of the gas and dW is the work done by the gas.

But dE = C

This gives:

For an adiabatic change dQ = 0 and so C

Let the gas expand from V to V + dV at constant pressure and let the temperature fall from T to T- dT.

Then: C

But PV = nRT. Therefore for one mole: PV = RT and so C

But since C

Writing C

which when integrated gives:

From this the other versions of the adiabatic equation may be obtained using PV = RT for one mole of the gas.