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,
∴ CVdT = − PdV
For a system of 1 mole of an ideal gas, expanding adiabatically and reversibly from temp T1 to T2 and volume V1 to V2, we have
W = – CvT = −Cv(T2 − T1) = Cv (T1 − T2)
Where T1,T2 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 PVγ = K’
∴ P = K’/Vγ
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.