If A is the area of piston then force is given by
F = P × A
Where P = pressure of the gas
dW = F × dx
= P × Adx
dW = P × dV
dV = Adx
Small increase in volume of the gas.
Total work done by the gas in adiabatic expansion from volume V1 to V2
W=\(\int_{v_1}^{v_2}PdV\)
The equation of adiabatic change is
PV = K, a constant
\(\gamma=\frac{c_p}{c_v}\)
W=\(\int_{v_1}^{v_2}KV-^\gamma.dV\)
=K\([\frac{V_1-\gamma}{1-\gamma}]_{v_1}^{v_2}\)
W=\(\frac{K}{1-\gamma}[V_{2}^{1-\gamma}-V_{1}^{1-\gamma}]\)
W=\(\frac{1}{1-\gamma}[KV_{2}^{1-\gamma}-KV_{1}^{1-\gamma}]\)
Equation of adiabatic change
P2V\(_{2}^{\gamma}=p_1V_{1}^{\gamma}=K\)
W=\(\frac{1}{1-\gamma}[P_2V{_2}^{1-\gamma}-P_1V_{1}^{\gamma}V_{1}^{1-\gamma}]\)
W=\(\frac{1}{1-\gamma}[P_2V{_2}-P_1V_{1}]\)
W=\(\frac{1}{1-\gamma}[T_2-T_1]\)
