$2)$ The van der Waal$$s equation of state

$2$

$2$

V

n a

V nb

nRT P $$

$$

$$

is difficult to solve for V when

p$,$ T$,$ n are given. It is possible, however, to derive an approximation for V$,$ which is$,$

a$)$ Derive an expression for

T P

V

$$

$$

$$

$$

$$

$$

$$

$$

for the approximate volume above in terms of the

constants$($a and b$)$ and the variables n$,$ p$,$ T$.$

b$)$ Evaluate

T P

V

$$

$$

$$

$$

$$

$$

$$

$$

in $($a$)$ for n $=0\mathrm{.}1000$ moles N$2,$ T $=276$ K$,$ p$=105\mathrm{.}1$ atm,

a $=1\mathrm{.}39$ L$2$ atm mol$-2$

$,$ and b $=3\mathrm{.}92$ x $10-2$ L mol$-1$

$.$

c$)$ Using the exact van der Waal$$s equation evaluate

T V

P

$$

$$

$$

$$

$$

$$

$$

$$

algebraically and

numerically for the conditions in $($b$).$

d$)$ Similarly to part $($c$)$ evaluate

V T

P

$$

$$

$$

$$

$$

$$

$$

$$

algebraically and numerically.

e$)$ Using the identity

T

V

P

V

P

T

P

T

V

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

compare the exact numerical value of

T P

V

$$

$$

$$

$$

$$

$$

$$

$$

with the approximation in $($b$)$ and determine the percent error in the approximate

derivative.The van der Waal's equation of state $P=\frac{nRT}{V-nb}-\frac{n^{2}a}{V^2}$ is difficult to solve for $V$ when

$p,T,n$ are given. It is possible, however, to derive an approximation for $V,$ which is$,$

$V$~~$\frac{nRT}{p(1-\frac{bp}{RT})}-\frac{na}{RT}$

a$)$ Derive an expression for $(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ for the approximate volume above in terms of the

constants and $b$ and the variables $n,p,T.$

b$)$ Evaluate $(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$ in $($a$)$ for $n=0\mathrm{.}1000$ moles $N2,T=276K,p=105\mathrm{.}1$atm,

$a=1\mathrm{.}39L^{2}atmmo{l}^{-2},$ and $b=3\mathrm{.}92{10}^{-2}Lmo{l}^{-1}.$

c$)$ Using the exact van der Waal's equation evaluate $(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V$ algebraically and

numerically for the conditions in $($b$).$

d$)$ Similarly to part $($c$)$ evaluate $(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{V}}{)}_T$ algebraically and numerically.

e$)$ Using the identity $(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P=-\frac{(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V}{(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{V}}{)}_T}$ compare the exact numerical value of $(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P$

with the approximation in $($b$)$ and determine the percent error in the approximate

derivative.