Question

A jar 'P' is filled with gas having pressure, volume and temperature P, V, T respectively. Another gas jar Q filled with a gas having pressure 2P, volume $\frac{V}{4}$ and temperature 2T. The ratio of the number of molecules in jar P to those in jar Q is

• A. 1:1
• B. 1:2
• C. 2:1
• D. 4:1

Hint:

Logic Tip: The proportionality $N \propto \frac{PV}{T}$ provides a rapid shortcut. For jar Q, $P$ is doubled ($\times 2$), $V$ is quartered ($\times \frac{1}{4}$), and $T$ is doubled ($\div 2$). So the modifier for $N_Q$ is $2 \times \frac{1}{4} \times \frac{1}{2} = \frac{1}{4}$. Therefore $N_P$ is 4 times larger than $N_Q$.

Answer 1

The Correct Option is D

Concept:
The ideal gas law can be expressed in terms of the number of molecules ($N$) rather than the number of moles ($n$). The formula is: $$PV = N k_B T$$ where $P$ is pressure, $V$ is volume, $N$ is the total number of molecules, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature.
Step 1: Write the equation for the number of molecules in jar P.
For jar P, the parameters are $P$, $V$, and $T$. $$N_P = \frac{PV}{k_B T}$$
Step 2: Write the equation for the number of molecules in jar Q.
For jar Q, the parameters are $2P$, $V/4$, and $2T$. Substitute these into the ideal gas equation: $$N_Q = \frac{(2P)\left(\frac{V}{4}\right)}{k_B (2T)}$$ Simplify the numerator: $$N_Q = \frac{\frac{2}{4}PV}{2 k_B T} = \frac{\frac{1}{2}PV}{2 k_B T}$$ $$N_Q = \frac{1}{4} \left(\frac{PV}{k_B T}\right)$$
Step 3: Calculate the required ratio.
We need the ratio $N_P : N_Q$. $$\frac{N_P}{N_Q} = \frac{\frac{PV}{k_B T{\frac{1}{4} \left(\frac{PV}{k_B T}\right)}$$ Cancel out the common term $\frac{PV}{k_B T}$: $$\frac{N_P}{N_Q} = \frac{1}{\frac{1}{4 = 4$$ The ratio is 4:1.