The figure shows a plot of $PV/T$ versus $P$ for $1.00 \times 10^{-3} \; kg$ of oxygen gas at two different temperatures.
$(a)$ What does the dotted plot signify?
$(b)$ Which is true: $T_{1} > T_{2}$ or $T_{1} < T_{2}$?
$(c)$ What is the value of $PV/T$ where the curves meet on the $y$-axis?
$(d)$ If we obtained similar plots for $1.00 \times 10^{-3} \; kg$ of hydrogen,would we get the same value of $PV/T$ at the point where the curves meet on the $y$-axis? If not,what mass of hydrogen yields the same value of $PV/T$ (for the low-pressure,high-temperature region of the plot)?
(Molecular mass of $H_{2} = 2.02 \; u$,of $O_{2} = 32.0 \; u$,$R = 8.31 \; J \; mol^{-1} K^{-1}$.)

(A) The dotted plot in the graph signifies the ideal behavior of the gas,i.e.,the ratio $PV/T$ is equal to $\mu R$ (where $\mu$ is the number of moles and $R$ is the universal gas constant),which is a constant quantity independent of the pressure of the gas.
$(b)$ The dotted plot represents an ideal gas. $A$ real gas approaches the behavior of an ideal gas as its temperature increases. Since the curve at temperature $T_{1}$ is closer to the dotted plot than the curve at temperature $T_{2}$,it follows that $T_{1} > T_{2}$.
$(c)$ The value of the ratio $PV/T$ where the curves meet on the $y$-axis is $\mu R$. For oxygen,the number of moles $\mu = \frac{\text{mass}}{\text{molecular mass}} = \frac{1.00 \times 10^{-3} \; kg}{32.0 \times 10^{-3} \; kg/mol} = \frac{1}{32} \; mol$. Thus,$PV/T = \mu R = \frac{1}{32} \times 8.31 \approx 0.26 \; J K^{-1}$.
$(d)$ For hydrogen,the molecular mass is $2.02 \; u$. Since $\mu$ depends on the molecular mass,the value of $PV/T = \mu R$ will be different for the same mass of hydrogen. To get the same value of $PV/T = 0.26 \; J K^{-1}$,we need $\mu = \frac{PV/T}{R} = \frac{0.26}{8.31} \approx 0.0313 \; mol$. The required mass of hydrogen is $m = \mu \times M = 0.0313 \; mol \times 2.02 \times 10^{-3} \; kg/mol \approx 6.32 \times 10^{-5} \; kg$.