Question

An ideal gas occupies a volume \(V\) at pressure \(P\) and absolute temperature \(T\). The mass of each molecule is \(m\). If \(K_B\) is the Boltzmann constant, then the density of the gas is given by

• A. \(\dfrac{K_B T}{P m}\)
• B. \(\dfrac{3K_B T}{2P m}\)
• C. \(\dfrac{P m}{2K_B T}\)
• D. \(\dfrac{P m}{K_B T}\)

Hint:

Gas density increases with pressure and decreases with temperature.

Answer 1

The Correct Option is D

Step 1: Ideal gas equation in molecular form.
\[ PV = Nk_B T \]
Step 2: Express number density.
\[ \frac{N}{V} = \frac{P}{k_B T} \]
Step 3: Write expression for density.
Density is mass per unit volume:
\[ \rho = \frac{Nm}{V} \]
Step 4: Substitute number density.
\[ \rho = m \frac{P}{k_B T} \]
Step 5: Conclusion.
The density of the gas is \[ \rho = \frac{P m}{K_B T} \]