An adiabatic change by definition, is one which does not allow any transfer of heat, i.e., q = 0 , it follows from the 1st law,

ΔU = − W

dU = − dW

Let only mechanical work of expansion or contraction is involved, dW = PdV. Moreover,

dU= C_{V}dT

∴ C_{V}dT = − PdV

For a system of 1 mole of an ideal gas, expanding adiabatically and reversibly from temp T_{1} to T_{2} and volume V_{1} to V_{2}, we have

## Adiabatic work:

W = – C_{v}T = −C_{v}(T_{2} − T_{1}) = C_{v} (T_{1} − T_{2})

Where T_{1},T_{2} are initial and final temperatures.

For 1 mole of gas T = PV/R

Hence adiabatic work

Slope of PV curve in adiabatic & isothermal expansion.

For isothermal expansion of the gas, PV = K

The slope of the PV curve will be obtained from

For the adiabatic expansion of the gas PV^{γ} = K’

∴ P = K’/V^{γ}

In the both the changes the slope is negative, since γ , is greater than 1 , the slope in the adiabatic P − V curve will be large than that in the isothermal one.