#$3.$ In the above problem, we assumed that the volume was independent of $p($i$.$e$.,$ we assumed that $=0).$ In this problem, we remove this assumption by including a nonzero $($but you may assume that $$ is independent of $p).$

$($a$)$ Derive an improved expression for $G$ upon an isothermal change in $p.$ The expression should contain only $G,,$ initial and final pressures $p_0$ and $p,$ and the initial volume $V_0.$ Hint: start with the definition of $,$ and integrate it under the assumption of constant $.$ This should give you an expression for the volume as a function of pressure, which you can use in your expression for $G.$

$($b$)$ Show that in the limit of small $,$ this expression becomes equal to the expression in Problem #$2.$

$($c$)$ Use your equation in part $($a$)$ to repeat the calculation of Problem #$2.$ For water, $=4\mathrm{.}5{10}^{-5}at{m}^{-1}.$ What is the $\%$ error when we ignore the compressibility $($i$.$e$.,$ how does your answer in prob. $3($c$)$ compare to your answer in prob. $2?)$

Info from previous question $100$cm$^3$ of liquid water at $25$ C is isothermally compressed from $1$ atm to $100$ atm