Problem $2$: A long rigid cylindrical container $($a tube$)$ of diameter $D=5cm$ and length, $L=100cm$ contains $0\mathrm{.}1$ kg of gas the pressure, p of which is related to its temperature, T and specific volume, $v$ by the following function:

$p(v,T)=\frac{RT}{v-b}-\frac{a}{v(v+b)\sqrt[\phantom{2}]{T}}($I$)$

where $R,$a and $b$ are constants.

a$)$ What are the base dimensions and base SI units of the constants $R,$a and $b?[6]$

b$)$ Show that $\frac{J}{k}g=\frac{m^2}{s^2}.$

c$)$ Calculate the value of a and $b$ in base SI units if

$a=0\mathrm{.}4278\frac{R^2{T}_c^{25}}{p_c}$ and $b=0\mathrm{.}0867\frac{R{T}_c}{p_c}$

where $p_c=38$atm and $T_c=135K$ are the so$-$called critical pressure and temperature,

respectively and the value of $R$ in base SI units is $189.$

d$)$ Convert the constants $R,a,$ and $b$ to US customary units.

e$)$ Using $($I$)$ calculate the gas temperature if the pressure is $21$ atm $.$

f$)$ More of the same gas is added to the rigid container until the pressure and temperature

are $25$ atm and $250$ K $,$ respectively. Using equation, $($I$)$ determine the exact amount of gas mass added in grams. $($Exact means that we CANNOT use an iterative procedure or use graphical methods$/$root$-$finder programs via our calculator because both are approximations. To find the exact amount we have to find exact expressions for the roots of the polynomial represented by equation $($I$).)$

g$)$ For the situation depicted by part f$),$ evaluate $\frac{dp}{dT}{|}_T=300x$ in SI units.

h$)$ Develop an expression for ${\displaystyle \underset{v_1}{\overset{2}{}}}pdv$ in terms of $u_1,v_2$ and T for an isothermal process. Then evaluate the integral's value if $v_1=1\mathrm{.}6{10}^{-3}\frac{m^3}{k}g,v_2=2{10}^{-3}\frac{m^3}{k}g$ and $T=300K.$