Question

A thermally insulated container is divided into two parts by a screen. In one part the pressure and temperature are $P$ and $T$ for an ideal gas filled. In the second part it is vacuum. If now a small hole is created in the screen, then the temperature of the gas will
A. decrease
B. increase
C. remains same
D. none of these

Answer 1

Hint:
This problem is based on thermodynamics; we know that pressure and temperature vary with the given conditions of the system and surroundings. As the gas is expanding from a small hole against a vacuum, therefore, the process will be free expansion i.e., $\Delta W = 0$. Hence, use the first law of thermodynamics to state the answer for the given problem.


Formula used:
First Law of the thermodynamics used in this problem is defined as: -
$\Delta U + \Delta W = \Delta Q$
and, for ideal gas internal energy is the function of temperature i.e., $U = f\left( T \right)$



Complete step by step solution:

Let us illustrate the given problem by a diagram given as follows: -

It is given that the container is thermally insulated which means $\Delta Q = 0$.
Additionally, the process of the gas expanding against a vacuum (surroundings) is known as free expansion, and for this process, work done by gas will be: -
$\Delta W = \int {PdV} = 0$
Now we know that the First Law of Thermodynamics is the application of conservation of energy and according to the first law: -
$\Delta U + \Delta W\, = \Delta Q\,\,\,\,\,\,\,\,\,\,\,...\,(1)$
where $\Delta U = $ Change in the Internal Energy of the system
$\Delta Q = $ Heat Added to the system
$\Delta W = $ Work done
From the equation $(1)$, we get
$0 = \Delta U + 0 \Rightarrow \Delta U = 0$
i.e., ${U_{final}} = {U_{initial}}$
or, $U{\text{ }} = constant$
Since the given gas is ideal and for an ideal gas, the internal energy is the function of temperature.
i.e., $U = f\left( T \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\,(2)$.
As, $U = constant$
Therefore, from the equation $(2)$
$T = constant$
Thus, in the given scenario, the temperature of the gas will remain constant.
Hence, (C) is the correct option.




Note:
 In this problem, to determine the effect on the temperature of an ideal gas when it is freely expanding against a vacuum from a small hole (in a container that is thermally insulated), use $\Delta Q = 0$ and $\Delta W = 0$ in the first law of thermodynamics and then apply the condition of an ideal gas $U = f\left( T \right)$ to state the final answer.