Using first law of thermodynamics, derive the equation for heat capacity at constant volume and constant pressure

The first law of thermodynamics is given by the equation: \[ dQ = dU + dW \] where: - \( dQ \) is the heat added to the system, - \( dU \) is the change in internal energy of the system, - \( dW \) is the work done by the system. For a closed system, we will derive the equations for heat capacity at constant volume (\(C_V\)) and constant pressure (\(C_P\)). ### Heat Capacity at Constant Volume (\(C_V\)) At constant volume, the work done by the system is given by: \[ dW = P dV \] However, since the volume is constant (\(dV = 0\)), the work done is zero: \[ dW = 0 \] Substituting this into the first law, we have: \[ dQ = dU + 0 \implies dQ = dU \] The heat capacity at constant volume is defined as the amount of heat absorbed or released per unit change in temperature, at constant volume: \[ C_V = \frac{dQ}{dT} \] Since \(dQ = dU\) in this scenario, we can write: \[ C_V = \frac{dU}{dT} \] This expression gives the heat capacity at constant volume in terms of the change in internal energy with respect to temperature. ### Heat Capacity at Constant Pressure (\(C_P\)) At constant pressure, the work done by the system is given by: \[ dW = P dV \] If we assume an infinitesimal process, the work done can’t be ignored. In this case, the first law becomes: \[ dQ = dU + P dV \] The heat capacity at constant pressure is defined as: \[ C_P = \frac{dQ}{dT} \] From the first law, we can rearrange \(dQ\): \[ dQ = dU + P dV \] Using the definition of enthalpy (\(H\)), which is defined as: \[ H = U + PV \] The differential change in enthalpy is: \[ dH = dU + P dV + V dP \] Since we are considering a process at constant pressure (\(dP = 0\)), this simplifies to: \[ dH = dU + P dV \] From this, we get \(dQ\) as: \[ dQ = dH \] Now substituting this back into our expression for heat capacity at constant pressure, we have: \[ C_P = \frac{dQ}{dT} = \frac{dH}{dT} \] So we can summarize: - The heat capacity at constant volume is given by \(C_V = \left( \frac{dU}{dT} \right)_{V}\). - The heat capacity at constant pressure is given by \(C_P = \left( \frac{dH}{dT} \right)_{P}\). These relations allow us to understand how heat transfer varies with temperature in systems at constant volume and constant pressure.