${P_0}V = NRT$

${T_f} = 2{T_0}$

$W = \int {PdV} $

$ = \int {\left( {\frac{{nRT}}{V} + \alpha } \right)dV} $

$PV = nRT + \alpha V$

$\int {PdV = \int\limits_{{T_0}}^{2{T_0}} {nRdT + \int\limits_{{V_1}}^{{V_1}} {\alpha dV} } } $

$ = nR{T_0} + \alpha \,{V_i}$

$ = nR{T_0} + \alpha \left( {\frac{{nR{T_0}}}{{{P_0}}}} \right)$

$ = nR{T_0} \left( {1 + \frac{\alpha }{{{P_0}}}} \right)$

$PV = nRT + \alpha V$

$\int {PdV = \int {nRdT + \int {\alpha dV} } } $

$W = nR{T_0} + \alpha \left[ {\frac{{nR{T_0}}}{{{P_0} - \alpha }}} \right]$

$W = nR{T_0}\left[ {1 + \frac{\alpha }{{{P_0} - \alpha }}} \right]$

$ = n{R_0}{T_0}\left[ {\frac{{{P_0}}}{{{P_0} - \alpha }}} \right]$

$ = \frac{{nR{T_0}{P_0}}}{{{P_0} - \alpha }}$