Question

At a given temperature, the density of an ideal gas is proportional to (P = pressure of ideal gas)

• A. \( \frac{1}{P} \)
• B. \( P \)
• C. \( P^2 \)
• D. \( \sqrt{P} \)

Hint:

Start with the ideal gas law \( PV = nRT \). Express the number of moles \( n \) in terms of mass \( m \) and molar mass \( M \). Then, relate mass and volume to density \( \rho = m/V \). By rearranging the ideal gas law, you can find the proportionality between density and pressure at constant temperature.

Answer 1

The Correct Option is B

The ideal gas law is given by: $$ PV = nRT $$ where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.
We want to find the relationship between the density \( \rho \) of the gas and its pressure \( P \) at a constant temperature \( T \).
Density \( \rho \) is defined as mass per unit volume: $$ \rho = \frac{m}{V} $$ The number of moles \( n \) can be expressed in terms of the mass \( m \) of the gas and its molar mass \( M \): $$ n = \frac{m}{M} $$ Substitute this expression for \( n \) into the ideal gas law: $$ PV = \frac{m}{M} RT $$ Rearrange the equation to solve for \( \frac{m}{V} \), which is the density \( \rho \): $$ \frac{m}{V} = \frac{PM}{RT} $$ $$ \rho = \frac{PM}{RT} $$ At a given temperature \( T \), and for a specific ideal gas (which has a constant molar mass \( M \)), the terms \( \frac{M}{RT} \) are constant.
Let \( k = \frac{M}{RT} \).
Then the equation becomes: $$ \rho = kP $$ This shows that the density \( \rho \) of an ideal gas at a given temperature is directly proportional to its pressure \( P \).
Therefore, \( \rho \propto P \).