Question

If the internal energy of an ideal gas decreases by the same amount as the work done by the system, the process is:
A) Cyclic
B) Isothermal
C) Adiabatic
D) Isolated

Answer 1

Hint: The process is said to be adiabatic if no heat enters and leaves the system during any stage of the process. The first law of thermodynamics relates the internal energy $\text{ }\!\!\Delta\!\!\text{ U}$ with the heat of system q and work done w. When the internal energy becomes the same as that of the internal energy, no heat is allowed to enter or leave the system.

Complete step by step answer:
Let the system absorb the heat ‘q’ from the surrounding and perform the work which tends to lower the energy of the system because the performance of the work requires the expenditure of energy. Hence, the change of internal energy $\text{ }\!\!\Delta\!\!\text{ U}$ can be given by
$\text{ }\!\!\Delta\!\!\text{ U = q + w}$ (1)
Where $\text{ }\!\!\Delta\!\!\text{ U}$the internal energy, q is the heat transferred from the surrounding to the system and w is the work done.
This equation (1) is called the first law of thermodynamics.

Let a condition in which the internal energy of the system decreases to reach the value of work. That is the internal energy is equal to the work done by the system. That is, at a certain condition the internal energy and work done becomes equal.
$\text{ }\!\!\Delta\!\!\text{ U = w}$ (2)
Then by substituting the equation (2) in (1) we get,
$\begin{align}
 & \text{ }\!\!\Delta\!\!\text{ U = q + w} \\
 & \Rightarrow \text{ w = q + w} \\
 & \Rightarrow \text{ q = w}-\text{w} \\
 & \therefore \text{ q = 0} \\
\end{align}$

Thus, when the internal energy of an ideal gas decreases by the same amount as the work done by the system, then the (heat) $\text{q = 0}$.
We know that a process in which no heat enters or leaves the system is called the adiabatic process. Since, no heat is allowed to enter or leave the system, the above process is adiabatic.
So, the correct answer is “Option C”.

Note: There are other thermodynamic processes:
1) Isothermal process: This is a process in which the temperature of the system remains constant. During the isothermal process, the enthalpy of reaction is zero, $\text{ }\!\!\Delta\!\!\text{ H = 0}$
2) Isobaric process: A process is said to be isobaric if the pressure of the system remains constant.