Question

A vessel of volume $2V$ contains 4 moles of a gas at a certain temperature. Another vessel of the volume $3V$ contains 3 moles of the same gas at same temperature then find the ratio of pressure for both the vessels.
(A) 2:1
(B) 1:2
(C) 4:1
(D) 1:1

Answer 1

Hint: According to ideal gas law, the pressure is directly proportional to temperature while it varies inversely with volume. The ratio of pressure can be calculated using ideal gas law in which temperature is considered constant as per the given question conditions.

Formula used: $PV = nRT$

Complete Step by Step Solution:
Considering two vessels, A and B and applying ideal gas law under given conditions, the pressure for both the vessels can be represented as follows:

For vessel A:
The conditions given for the vessel A are
Volume $ = 2V$ and number of moles $n = 4$. Then according to ideal gas law, pressure can be represented as follows:
${P_A} = \frac{{4RT}}{{2V}}{\rm{ }}...{\rm{(1)}}$

For vessel B:
The conditions given for the vessel B are
Volume $ = 3V$ and number of moles $n = 3$. Then according to ideal gas law, pressure can be represented as follows:
${P_B} = \frac{{3RT}}{{3V}}{\rm{ }}...{\rm{(2)}}$

On dividing equation (1) and (2), the ratio of pressure in both vessels can be calculated as follows:
$\frac{{{P_A}}}{{{P_B}}} = \frac{{\frac{{4RT}}{{2V}}}}{{\frac{{3RT}}{{3V}}}}$
$ \Rightarrow \frac{{{P_A}}}{{{P_B}}} = \frac{{4RT \times 3V}}{{3RT \times 2V}}$
$ \Rightarrow \frac{{{P_A}}}{{{P_B}}} = \frac{2}{1}$
Hence, the ratio of pressure in the first vessel to second is $2:1$. So, option A is correct.

Note: It is important to note that the ideal gas law holds true only under ideal conditions. If the conditions are varying like high pressure and low temperature, other factors like molecular size and intermolecular forces come into consideration due to which the ideal gas law doesn’t work at such conditions.