Question

Two different masses m and 3m of an ideal gas are heated separately in a vessel of constant volume, the pressure P and absolute temperature T, graphs for these two cases are shown in the figure as A and B. The ratio of slopes of curves B to A is -

• A. 3 : 1
• B. 1 : 3
• C. 9 : 1
• D. 1 : 9

Answer 1

Answer: (A)

3 : 1

Answer 2

For an ideal gas at constant volume:

$$P = \frac{nRT}{V} \quad \Rightarrow \quad \frac{dP}{dT} = \frac{nR}{V}$$

For the sample with mass $m$, let the number of moles be $n$. For the sample with mass $3m$, the number of moles is $3n$. Therefore, the slopes of the $P$ vs. $T$ curves are proportional to $n$ (and $3n$ respectively).

Thus, the ratio of the slopes (curve B corresponding to mass $3m$ to curve A corresponding to mass $m$) is:

$$\frac{3n}{n} = 3:1$$

Brief Explanation:
At constant volume, $\frac{dP}{dT} \propto n$. For masses $m$ and $3m$, the mole ratio is $1:3$, hence the slope ratio is $3:1$.