A fluid undergoes a process where its pressure $(P)$, temperature $(T)$ and volume $(V)$ changes from $\left(P_{1}, T_{1}, V_{1}\right)$ to $\left(P_{2}, T_{2}, V_{2}\right)$. During the process, volume expansivity ( $\beta$ ) and isothermal compressibility ( $\kappa_{T}$ ) remains constant. Given that $\beta=\dfrac{1}{V}\left(\dfrac{\partial V}{\partial T}\right)_{P}$ and $\kappa_{T}=-\dfrac{1}{V}\left(\dfrac{\partial V}{\partial P}\right)_{T}$, the ratio $\left(\dfrac{V_{2}}{V_{1}}\right)$ is
- $\dfrac{\beta\left(T_{2}-T_{1}\right)}{\kappa_{T}\left(P_{2}-P_{1}\right)}$
- $\left[\beta\left(T_{2}-T_{1}\right)\right]\left[\kappa_{T}\left(P_{2}-P_{1}\right)\right]$
- $\dfrac{\exp \left[\beta\left(T_{2}-T_{1}\right)\right]}{\exp \left[\kappa_{T}\left(P_{2}-P_{1}\right)\right]}$
- $\beta\left(T_{2}-T_{1}\right) - \kappa_{T}\left(P_{2}-P_{1}\right)$
