1.State the First Law of Thermodynamics and write its sign conventions for heat supplied and work done.
Sol: $\Delta Q = \Delta U + \Delta W$. Heat absorbed by system is positive; work done BY system is positive.
2.Derive the relation $C_p - C_v = R$ (Mayer's relation) for an ideal gas.
Sol: $dQ_v = dU = C_v dT$. $dQ_p = dU + P dV = C_p dT \implies C_v dT + R dT = C_p dT \implies C_p - C_v = R$.
3.Write expression for work done in an Isothermal process ($W = nRT \ln(V_2/V_1)$).
Sol: $W = \int_{V_1}^{V_2} P dV = \int_{V_1}^{V_2} \frac{nRT}{V} dV = nRT \ln\left(\frac{V_2}{V_1}\right) = 2.303 nRT \log_{10}\left(\frac{V_2}{V_1}\right)$.
4.Write expression for work done in an Adiabatic process ($W = \frac{nR(T_1 - T_2)}{\gamma - 1}$).
Sol: $Q=0 \implies W = -\Delta U = -n C_v (T_2 - T_1) = n \left(\frac{R}{\gamma-1}\right) (T_1 - T_2) = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1}$.
