$4\mathrm{.}3$ Show that Eq$.(4\mathrm{.}143)$ can be written in the form

$\frac{{}_N}{N}=[\frac{k_{B}T}{-(del\frac{P}{d}el(hat(v)){)}_T(\frac{N}{N_A})hat(v{)}^2}{]}^{\frac{1}{2}}.$

where hat$(v)$ is the molar specific volume in $\frac{m^3}{k}$mol. $($PLEASE PROVIDE THE ANSWER WITH THE FINAL EXPRESSION IN FORM EXACTLY SHOWN AS ABOVE SINCE PREVIOUS FOUND ANSWER HAS DIFFERENT EXPRESSION$)$