3 This question is about modelling the relation between the pressure, $P$, volume, $V$, and temperature, $\theta$, of a fixed amount of gas in a container whose volume can be varied.

The amount of gas is measured in moles; 1 mole is a dimensionless constant representing a fixed number of molecules of gas. Gas temperatures are measured on the Kelvin scale; the unit for temperature is denoted by K . You may assume that temperature is a dimensionless quantity.

A gas in a container will always exert an outwards force on the walls of the container. The pressure of the gas is defined to be the magnitude of this force per unit area of the walls, with $P$ always positive.

An initial model of the relation is given by $P ^ { \alpha } V ^ { \beta } = n R \theta$, where $n$ is the number of moles of gas present and $R$ is a quantity called the Universal Gas Constant. The value of $R$, correct to 3 significant figures, is $8.31 \mathrm { JK } ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $[ P ] = \mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 2 }$ and $[ R ] = \mathrm { ML } ^ { 2 } \mathrm {~T} ^ { - 2 }$.
\item Hence show that $\alpha = 1$ and $\beta = 1$.

5 moles of gas are present in the container which initially has volume $0.03 \mathrm {~m} ^ { 3 }$ and which is maintained at a temperature of 300 K .
\item Find the pressure of the gas, as predicted by the model.

An improved model of the relation is given by $\left( P + \frac { a n ^ { 2 } } { V ^ { 2 } } \right) ( V - n b ) = n R \theta$, where $a$ and $b$ are constants.
\item Determine the dimensions of $b$ and $a$.

The values of $a$ and $b$ (in appropriate units) are measured as being 0.14 and $3.2 \times 10 ^ { - 5 }$ respectively.
\item Find the pressure of the gas as predicted by the improved model.

Suppose that the volume of the container is now reduced to $1.5 \times 10 ^ { - 4 } \mathrm {~m} ^ { 3 }$ while keeping the temperature at 300 K .
\item By considering the value of the pressure of the gas as predicted by the improved model, comment on the validity of this model in this situation.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2021 Q3 [17]}}