The equation of state for a gas is given by $$PV = nRT + \alpha V$$, where n is the number of moles and $$\alpha$$ is a positive constant. The initial temperature and pressure of one mole of the gas contained in a cylinder are T$$_0$$ and P$$_0$$ respectively. The work done by the gas when its temperature doubles isobarically will be:

$$PV = nRT + \alpha V$$

$$PV = RT + \alpha V$$ ($$n = 1$$)

$$PV - \alpha V = RT$$

$$V(P - \alpha) = RT \implies V = \frac{RT}{P - \alpha}$$

$$V_{\text{initial}} = \frac{RT_0}{P_0 - \alpha}$$

$$V_{\text{final}} = \frac{R(2T_0)}{P_0 - \alpha} = \frac{2RT_0}{P_0 - \alpha}$$

$$W = P_0 \cdot \Delta V = P_0 (V_{\text{final}} - V_{\text{initial}})$$

$$W = P_0 \left( \frac{2RT_0}{P_0 - \alpha} - \frac{RT_0}{P_0 - \alpha} \right)$$

$$W = P_0 \left( \frac{RT_0}{P_0 - \alpha} \right)$$

$$W = \frac{P_0 T_0 R}{P_0 - \alpha}$$