Q.Figure 12.8 shows plot of $PV/T$ versus $P$ for $1.00\times10^{-3}$ kg of oxygen gas at two different temperatures.

Concept: Real gas vs. ideal gas behaviour — for an ideal gas $PV/T = \mu R$ is a constant, independent of $P$, so a horizontal line on a $PV/T$ vs $P$ graph represents ideal behaviour; a real gas approaches this only in the limit $P \to 0$.

(a) The dotted plot. The dotted horizontal line represents the ideal-gas prediction $PV/T = \mu R = \text{constant}$, unaffected by pressure. The solid curves (real oxygen) deviate from this constant value as pressure increases, showing oxygen departing from ideal behaviour at higher pressure.

(b) Comparing $T_1$ and $T_2$. A real gas behaves more like an ideal gas at higher temperature (molecules are farther apart on average relative to their size, so intermolecular interactions matter less). The curve that stays closer to the dotted ideal-gas line is therefore at the higher temperature. Reading the graph, that curve corresponds to $T_1$, so $T_1 > T_2$.

(c) Value of $PV/T$ where the curves meet the $y$-axis. As $P \to 0$, real-gas behaviour merges with ideal-gas behaviour, so both curves converge to the same value $\mu R$. For the given mass $1.00\times10^{-3}\ \text{kg} = 1\ \text{g}$ of O$_2$ (molar mass $32.0$ u, so $n = 1/32\ \text{mol}$):

(d) Same mass of hydrogen. $PV/T = nR = (\mu/M)R$ depends on the number of moles present, which depends on molar mass $M$. Since $M_{\text{H}_2} = 2.02\ \text{u} \ne M_{\text{O}_2} = 32.0\ \text{u}$, the same mass ($1.00\times10^{-3}$ kg) of H$_2$ would NOT give the same $PV/T$ — being far lighter per mole, it has many more moles for the same mass, giving a larger $PV/T$.

To match the same value $0.26\ \text{J K}^{-1}$ with hydrogen, solve $\mu_{\text{H}_2} R / M_{\text{H}_2} = 0.26\ \text{J K}^{-1}$ for $\mu_{\text{H}_2}$:

Concept — why $PV/T$ is the natural quantity to plot

The ideal-gas equation is $PV=\mu R T$, so

where $\mu$ is the number of moles and $R=8.31\ \text{J mol}^{-1}\text{K}^{-1}$. For a fixed mass of gas $\mu$ is fixed, so an ideal gas would give a value of $PV/T$ that is completely independent of $P$ and of $T$ — a horizontal straight line. A real gas obeys $PV=\mu RT$ only approximately (best at low pressure and high temperature), so its $PV/T$ deviates from the constant, dipping and rising as $P$ grows.

(a) Meaning of the dotted straight line

The dotted horizontal line has $PV/T$ constant for all $P$. That is exactly ideal-gas behaviour:

So the dotted line represents the ideal-gas value $\mu R$; the solid curves show how real oxygen departs from it.

(b) Ordering of $T_1$ and $T_2$

A real gas behaves more like an ideal gas at higher temperature, i.e. its $PV/T$ stays closer to the constant $\mu R$ line. The $T_1$ curve lies closer to the dotted line than the $T_2$ curve, so

(The $T_2$ curve dips further below $\mu R$, showing stronger non-ideality, which happens at the lower temperature.)

(c) Value of $PV/T$ where the curves meet the axis

At the axis the pressure is vanishingly small; there every gas is ideal, so both curves converge to $PV/T=\mu R$. For $m=1.00\times10^{-3}\ \text{kg}=1\ \text{g}$ of oxygen with molar mass $M=32.0\ \text{g mol}^{-1}$:

(d) The same experiment with hydrogen

The intercept value is $PV/T=\mu R$, which depends only on the number of moles, not on the identity of the gas. Taking $1.00\times10^{-3}\ \text{kg}=1\ \text{g}$ of hydrogen ($M=2.02\ \text{g mol}^{-1}$) would give

many more moles than $0.03125\ \text{mol}$, so $PV/T$ would be much larger — not the same value.

To reproduce the oxygen value we need the same number of moles, $\mu=0.03125\ \text{mol}$. The required mass of hydrogen is

One physical idea ties all four parts together: $PV/T=\mu R$ is exactly constant only for an ideal gas, so the dotted line marks the ideal-gas limit, and how close a real curve sits to that line at a given pressure measures how nearly ideal the gas is behaving there. Real gases behave more ideally at lower pressure and higher temperature (weaker intermolecular interactions, more free volume), which is precisely why the curve lying closer to the dotted line is the higher-temperature one — $T_1>T_2$ can be read straight off the graph shape, no calculation needed. Part (d)'s key insight is that $PV/T=\mu R$ tracks the number of moles, not mass or gas identity — so equal masses of different gases never give equal $PV/T$ unless the mass is deliberately chosen to give equal moles, which is why the matching mass scales by the ratio of molar masses.