Q$1.$ a$)$ show that the van der Waals equation can be given by the equation $P=\frac{RT}{((\frac{?}{\text{b}\text{a}\text{r}}(V))-b)}-\frac{a}{\frac{?}{\text{b}\text{a}\text{r}}(V{)}^2},$ where $\frac{?}{\text{b}\text{a}\text{r}}(V)=$ molar volume?

b$)$ A gas that behaves ideally has a density of $1\mathrm{.}92gd{m}^{-3}$ at $150$kPa and $298K.$ What is the molar mass of the sample?

Q$2.$ a$)$ Prove that: $\frac{P{V}_c}{T_c}=\frac{3}{8}R?$ Find the critical temperature and critical pressure of the gas?

Q$3.($a$)$ Starting from the main KMT equation show that $\sqrt[\phantom{2}]{\frac{?}{\text{b}\text{a}\text{r}}(u{)}^2}=H_m==\sqrt[\phantom{2}]{\frac{3RT}{{}_m}}?$

$($b$)$ It takes gas A $2\mathrm{.}3$ times as long to effuse through an orifice as the same amount of nitrogen. What is the molar mass of gas $A?$

Q$4.($a$)$ What is the total kinetic energy of $0\mathrm{.}50$mol of an ideal monatomic gas confined to $8\mathrm{.}0d{m}^3$ at $200$kPa $?$

$($b$)$ Calculate the pressure of nitrogen $($molar mass $=28g*mo{l}^{-1})$ at an altitude of $100km$ assuming that sea$-$level pressure is exactly $1$ atm and that the temperature of $298\mathrm{.}15K$ is constant?

Q$5.$ The critical constants of methane are $P_c=45\mathrm{.}6$atm,$V_c=98\mathrm{.}7c{m}^{3}mo{l}^{-1},$ and $T_c=190\mathrm{.}6K.$ Calculate the van der Waals parameters of the gas and estimate the radius of the molecules.

Q$6.$ A sample consisting of $1\mathrm{.}0$mol of perfect gas molecules with $C_V=20\mathrm{.}8J{K}^{-1}$ is initially at $3\mathrm{.}25$ atm and $310K.$ It undergoes reversible adiabatic expansion until its pressure reaches $2\mathrm{.}50$atm. Calculate the final volume and temperature and the work done.

Q$7.$ A sample consisting of $1$mol of perfect gas atoms $($for which $\frac{?}{\text{b}\text{a}\text{r}}(C{)}_V=\frac{3}{2}R)$ is taken through the cycle shown in the following figure $($a$)$ Determine the temperature at the points $1,2,$ and $3.($b$)$ Calculate q$,$ w$,U,$ and $H$ for each step and for the overall cycle. If a numerical answer cannot be obtained from the information given, then write in$,,+-0,$ or $?$ as appropriate.