What is an adiabatic process? Derive an expression for work done in an adiabatic process.

(N/A) An adiabatic process is a thermodynamic process in which there is no exchange of heat between the system and its surroundings,i.e.,$\Delta Q = 0$.
From the first law of thermodynamics:
$\Delta Q = \Delta U + W$
Since $\Delta Q = 0$,we have $W = -\Delta U$.
For an ideal gas undergoing an adiabatic process,the relation between pressure $P$ and volume $V$ is given by $PV^{\gamma} = K$ (constant),where $\gamma = \frac{C_P}{C_V}$ is the adiabatic index.
The work done $W$ during the expansion from state $(P_1, V_1)$ to $(P_2, V_2)$ is:
$W = \int_{V_1}^{V_2} P \, dV$
Since $P = K V^{-\gamma}$,we substitute this into the integral:
$W = \int_{V_1}^{V_2} K V^{-\gamma} \, dV = K \left[ \frac{V^{-\gamma+1}}{-\gamma+1} \right]_{V_1}^{V_2}$
$W = \frac{K}{1-\gamma} (V_2^{1-\gamma} - V_1^{1-\gamma})$
Since $K = P_1 V_1^{\gamma} = P_2 V_2^{\gamma}$,we get:
$W = \frac{1}{1-\gamma} (P_2 V_2^{\gamma} V_2^{1-\gamma} - P_1 V_1^{\gamma} V_1^{1-\gamma})$
$W = \frac{P_2 V_2 - P_1 V_1}{1-\gamma} = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1}$
Using the ideal gas law $PV = nRT$,the expression can also be written as:
$W = \frac{nR(T_1 - T_2)}{\gamma - 1}$