Question

In a given process on an ideal gas $dW = 0$ and $dQ < 0$, then for the gas which one of the following is true:
A. The temperature will decrease
B. The volume will increase
C. The pressure will remain constant
D. The temperature will decrease

Answer 1

This question is related to the first law of thermodynamics. The first law states that the energy of the whole universe remains constant, it is only exchanged between system and surroundings. To solve these types of questions use the formula of the first law of thermodynamics.

Formula used:
Formula of first law of thermodynamics:
$dQ = dU + dW.....(1)$
Where:
$dU = Change{\text{ in internal energy of the system}}{\text{.}} \\\ \Rightarrow {\text{dW = Work done by the system}}{\text{.}} \\\$
$dU = n{C_v}\Delta T......(2)$
Where:
$dU = Change{\text{ in internal energy of the system}}{\text{.}} \\\ \Rightarrow n = total{\text{ no}}{\text{. of moles}}{\text{.}} \\\ \Rightarrow {{\text{C}}_v} = Heat{\text{ capacity at constant volume}}{\text{.}} \\\ \Rightarrow \Delta T = Change{\text{ in temperature}}{\text{.}}$

Complete step by step answer:
Here in the question it is mentioned that work done by the gas $dW = 0$ and $dQ < 0$ .
Therefore from equation (1):
$dQ = dU + 0$.................(Since $dW = 0$ )
As $dQ < 0$ the internal energy $dU < 0$
From equation (2):
$n{C_v}\Delta T < 0$
Since $n$ and ${C_v}$ both are constants, therefore $\Delta T < 0$
Now, $\Delta T = {T_f} - {T_i}$
$\Delta T < 0 \\\ \Rightarrow {T_f} - {T_i} < 0 \\\ \Rightarrow {T_f} < {T_i}$
As the final temperature is less than the initial temperature, the temperature decreases.

So option A is correct.

Additional Information: The first law of thermodynamics is based on the principle that thermodynamic processes are governed by the principle of conservation of energy. It also confirms that heat is a form of energy. For an ideal gas, the internal energy of a system depends only on the temperature.

Note: Whenever energy in the form of heat is provided to an engine it does some work and the rest of the energy is used by it to increase its internal energy. Students usually get confused regarding the sign convention in thermodynamics. We must remember that in physics we take the work done by the system as positive whereas in chemistry we take the work done on the system as positive. There is a difference only in our assumption.