Exercise 1

$P-T$ diagram of an ideal gas having three different densities ${\rho }_1, {\rho }_2, {\rho }_3$ (in three different cases) is shown in the figure. Which of the following is correct :

$$\begin{gathered} \mathrm{For} \mathrm{a} \mathrm{diatomic} \mathrm{gas}, \mathrm{if} {\mathrm{\gamma }}_1=\left(\frac{{\mathrm{C}}_{\mathrm{p}}}{{\mathrm{C}}_{\mathrm{v}}}\right) \mathrm{for} \mathrm{rigid} \mathrm{molecules} \mathrm{and} {\mathrm{\gamma }}_2=\left(\frac{{\displaystyle {\mathrm{C}}_{\mathrm{p}}}}{{\displaystyle {\mathrm{C}}_{\mathrm{v}}}}\right) \mathrm{for} \mathrm{another} \mathrm{diatomic} \mathrm{molecules}, \mathrm{but} \mathrm{also} \mathrm{having} \mathrm{vibrational} \mathrm{modes}. \mathrm{Then}, \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{options} \mathrm{is} \mathrm{correct}? \\ (\mathrm{Cp} \mathrm{and} \mathrm{Cv} \mathrm{are} \mathrm{specific} \mathrm{heats} \mathrm{of} \mathrm{the} \mathrm{gas} \mathrm{at} \mathrm{constant} \mathrm{pressure} \mathrm{and} \mathrm{volume}) \end{gathered}$$

$\mathrm{The} \mathrm{temperature} \mathrm{of} 1 \mathrm{mole} \mathrm{of} \mathrm{an} \mathrm{ideal} \mathrm{monoatomic} \mathrm{gas} \mathrm{is} \mathrm{increased} \mathrm{by} 50°\mathrm{C} \mathrm{at} \mathrm{constant} \mathrm{pressure}. \mathrm{The} \mathrm{total} \mathrm{heat} \mathrm{added} \mathrm{and} \mathrm{change} \mathrm{in} \mathrm{internal} \mathrm{energy} \mathrm{are} {\mathrm{E}}_1 \mathrm{and} {\mathrm{E}}_2, \mathrm{respectively}. \mathrm{If} \mathrm{X} == \frac{{\mathrm{E}}_1}{{\mathrm{E}}_2}=\frac{\mathrm{x}}{9} \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{x} \mathrm{is\_\_\_\_}.$

For a particular ideal gas which of the following graphs represents the variation of mean squared velocity of the gas molecules with temperature?

A container of fixed volume contains a gas at 27°C. To double the pressure of the gas, the temperature of gas should be raised to ______°C.

A poly-atomic molecule $(\mathrm{Cv} = 3\mathrm{R}, \mathrm{Cp} = 4\mathrm{R},$ where R is gas constant) goes from phase space point  $\mathrm{A}({\mathrm{P}}_{\mathrm{A}} = {10}^5 \mathrm{Pa}, {\mathrm{V}}_{\mathrm{A}} = 4 \times {10}^{-6} {\mathrm{m}}^3)$ to point $\mathrm{B} ({\mathrm{P}}_{\mathrm{B}} = 5 \times {10}^4 \mathrm{Pa}, {\mathrm{V}}_{\mathrm{B}} = 6 \times {10}^{-6} {\mathrm{m}}^3)$ to point $\mathrm{C} ({\mathrm{P}}_{\mathrm{C}} = {10}^4 \mathrm{Pa}, {\mathrm{V}}_{\mathrm{C}} = 8 \times {10}^{-6}{\mathrm{m}}^3)$. A to B is an adiabatic path and B to C is an isothermal path.

The net heat absorbed per unit mole by the system is:

$$\begin{gathered} \mathrm{The} \mathrm{ratio} \mathrm{of} \mathrm{vapour} \mathrm{densities} \mathrm{of} \mathrm{two} \mathrm{gases} \mathrm{at} \mathrm{the} \mathrm{same} \mathrm{temperature} \mathrm{is} \frac{4}{25}, \mathrm{then} \mathrm{the} \mathrm{ratio} \mathrm{of} \mathrm{r}.\mathrm{m}.\mathrm{s}. \\ \mathrm{velocities} \mathrm{will} \mathrm{be} : \end{gathered}$$

$\mathrm{The} \mathrm{kinetic} \mathrm{energy} \mathrm{of} \mathrm{translation} \mathrm{of} \mathrm{the} \mathrm{molecules} \mathrm{in} 50\mathrm{g} \mathrm{of} {\mathrm{CO}}_2 \mathrm{gas} \mathrm{at} 17°\mathrm{C} \mathrm{is} :$