Question

Using the equation of state $PV = nRT$, show that, at a given temperature, the density of a gas is proportional to its gas pressure P.

Answer 1

To answer this question, you must recall the ideal gas equation. Also it is important to know the mathematical definition of density. Density of a substance is the ratio between its mass and volume.
Formulae used:
$PV = nRT$
Where, $P$ is the pressure exerted by the gas on the walls of the container
$V$ is the total volume occupied by the gas or the volume of the container
$n$ is the number of moles of the gas present in the container
And, $T$ is the temperature at which the gas is present in the container
$\rho = \dfrac{m}{V}$
Where, $m$ is the mass of the gas
And, $V$ is the total volume occupied by the gas or the volume of the container

Complete step by step answer:
We know that the ideal gas equation is given as, $PV = nRT$
We know that the number of moles of a substance are given by the ratio between the given mass of the substance and its molecular/ atomic mass. So we can write the number of moles of the gas assuming its molecular mass to be M as, $n = \dfrac{m}{M}$.
Substituting this value in the ideal gas equation, we get,
$PV = \dfrac{m}{M}RT$ or $P = \dfrac{m}{V} \times \dfrac{{RT}}{M}$
We know that the density is given as
$\rho = \dfrac{m}{V}$.
So the ideal gas equation becomes
$P = \rho \dfrac{{RT}}{M}$
Rearranging the equation, we get
$\rho = \dfrac{{PM}}{R} \times \dfrac{1}{T}$
Hence, at constant pressure the whole term, $\dfrac{{PM}}{R}$attains a constant value and the density becomes inversely proportional to the temperature of the gas.

Note:
A gas is defined by four states namely volume, pressure, temperature and the number of moles. Individual relations between the states are given by the gas laws as follows:
Boyle’s Law: $P{\text{ }} \propto {\text{ }}\dfrac{1}{V}$
Charles’ Law: $V{\text{ }} \propto {\text{ }}\dfrac{1}{T}$
Gay- Lussac’s Law: $P{\text{ }} \propto {\text{ }}T$
Avogadro’s Law:$V{\text{ }} \propto {\text{ }}n$
Combining all these relations, we get a relation as $PV{\text{ }} \propto {\text{ nT}}$which satisfies all the gas laws. Adding an equality constant $R$, known as the gas constant, we get the Ideal gas equation.
$PV = nRT$