Question

A container of volume $1\,{m^3}$ is divided into two equal compartments by a partition. One of these compartments contains an ideal gas at $300\,K$ . The other compartment is vacuum. The whole system is thermally isolated from its surroundings. The partition is removed and the gas expands to occupy the whole volume of the container. The temperature will be
A. $300\,K$
B. $250\,K$
C. $200\,K$
D. $100\,K$

Answer 1

Hint:There is no external force being applied on the system. Hence, this becomes a case of free expansion. If a gas expands freely, the work done in the process is $dW = 0$ . Also, if a system is thermally isolated, there will be no heat transfer. This means that $dQ = 0$ . We shall apply the first law of thermodynamics and substitute the values to get the change in the internal energy. We will then relate this change in the internal energy with the change in temperature to get the final answer.

Formula used: The first law of thermodynamics states that $dQ = dU + dW$ where $dW$ is the work done in a thermodynamic process, $dQ$ is the heat transferred in a thermodynamic process and $dU$ is the change in the internal energy.
Also, the internal energy is given $dU = mcdT$ where m is the mass of the gas, c is the specific heat and $dT$ is the change in the temperature.

Complete step by step solution:
When the partition is removed, the gas will expand freely since there is no external application of any force. Hence, $dW = 0$ where $dW$ is the work done in a thermodynamic process.
Also, the system is thermally isolated. Hence, there will be no heat transfer. This means that $dQ = 0$ where $dQ$ is the heat transferred in a thermodynamic process.
Using the first law of thermodynamics, we have
$dQ = dU + dW$
Now since $dW = 0$ and $dQ = 0$
$ \Rightarrow dU = 0$ where $dU$ is the change in the internal energy.
Since the derivative of a constant function is zero, we can say that the internal energy is constant.
The internal energy is given $dU = mcdT$ where m is the mass of the gas, c is the specific heat and $dT$ is the change in the temperature.
Since m and c are constant for a given gas and $dU = 0$we can say that $dT = 0$
Hence, the change in temperature is zero.
This means that the temperature remains constant.
Hence, option A is the correct answer.

Note:
In this question, the concept of free expansion could be applied. In cases where an external force is applied on the system, we calculate the force applied on a small length dx and then integrate it over the length of the container to get the work done. We can use the direct formulas as well but their application is limited to specific situations whereas going by the basics we need not worry about the applicability of the formulas.