Question

A perfect gas of a given mass is heated first in a small vessel and then in a large vessel , such that their volume remains unchanged. The $P - T$ curve are
(A) Parabolic with same curvature
(B) Parabolic with different curvature
(C) Linear with same slopes
(D) Linear with different slopes

Answer 1

Hint : To answer this question, we have to use the ideal gas equation to get the relation between the pressure and the temperature. Then, on applying the two conditions given in the question, we can get the final answer.

Formula Used: The formula which is used in solving this question is given by
$\Rightarrow PV = nRT$ , where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles and $T$ is the temperature.

Complete step by step answer
We know that the ideal gas, or the perfect gas equation is given by
$\Rightarrow PV = nRT$
Dividing both the sides by the volume $V$ , we have
$\Rightarrow P = \dfrac{{nR}}{V}T$
According to the question, the volume of the gas is unchanged in both the cases of heating. Also the mass of the gas is fixed, which means that the number of moles, $n$ is also a constant. And $R$ is a universal gas constant, as we know. So the coefficient of the temperature in the above equation is a constant. Therefore there exists a linear relation between the pressure and the temperature in both the cases. Comparing the above equation with the equation of the line $y = mx + c$ , we get the slope of the $P - T$ curve as
$\Rightarrow m = \dfrac{{nR}}{V}$ ……………………….(1)
Now, it is given in the question that the gas is first heated in a small vessel and then in a large vessel. So the volume of the first vessel is less than that of the second vessel. We know that when a gas is filled inside a vessel, then the gas occupies all the volume of that vessel. So, the volume of the gas in the first case will be less than that in the second case. From (1) we can see that the slope of the $P - T$ curve is inversely proportional to the volume of the gas. So the slope will be more in the first case than that in the second case.
Thus, the $P - T$ curves are linear with different slopes.
Hence, the correct answer is option D.

Note
There must be no confusion regarding the volume term which appears in the ideal gas equation. It is the volume occupied by the gas, not that of the container. The volume occupied by the gas will be equal to that of the container only if the container is closed.