Q.In an adiabatic process, no transfer of heat takes place between system and surroundings. Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following.

The key idea here is the First Law of Thermodynamics, which relates the change in internal energy to heat and work.

1. Adiabatic condition: By definition, an adiabatic process involves no heat transfer, so $q = 0$.
2. Free expansion: This implies the gas expands against zero external pressure ($P_{ext} = 0$). Therefore, the work done by the system is $w = -P_{ext} \Delta V = 0$.
3. First Law of Thermodynamics: The change in internal energy is given by $\Delta U = q + w$. Substituting $q = 0$ and $w = 0$, we get $\Delta U = 0$. …

The problem asks us to identify the correct thermodynamic parameters ($q$, $\Delta T$, $w$) for the free expansion of an ideal gas under adiabatic conditions. To solve this, we need to understand the definitions of an adiabatic process and free expansion, and then apply the First Law of Thermodynamics.

Concept and Intuition

The First Law of Thermodynamics is a statement of energy conservation. It relates the change in internal energy ($\Delta U$) of a system to the heat ($q$) added to the system and the work ($w$) done on the system:

Here's how each term behaves under the given conditions:

• Adiabatic condition: This means the system is perfectly insulated, preventing any heat transfer with the surroundings.
• Free expansion: This refers to the expansion of a gas into a vacuum. Since there is no external pressure to push against, the gas does no work on the surroundings, and the surroundings do no work on the gas.
• Ideal gas: For an ideal gas, the internal energy ($U$) depends solely on its temperature ($T$). This means if the internal energy does not change ($\Delta U = 0$), then the temperature must also remain constant ($\Delta T = 0$).

Let's apply these concepts step-by-step.

Step-by-Step Solution

1. Analyze the adiabatic condition: The problem states that the process is adiabatic. By definition, an adiabatic process is one where no heat transfer occurs between the system and its surroundings. Therefore, the heat exchanged, $q$, is zero.

2. Analyze the free expansion condition: The gas undergoes free expansion. Free expansion occurs when a gas expands into a vacuum. In such a scenario, there is no external pressure opposing the expansion. Work done by or on the gas is given by $w = -P_{ext} \Delta V$, where $P_{ext}$ is the external pressure. Since the gas expands into a vacuum, the external pressure is zero ($P_{ext} = 0$). Therefore, the work done, $w$, is zero.

> [!WARNING]
> It's a common mistake to confuse free expansion with reversible expansion. In a reversible expansion, work is done against a non-zero external pressure. In free expansion, the external pressure is zero.

3. Apply the First Law of Thermodynamics:

Now we use the First Law of Thermodynamics, $\Delta U = q + w$.

Substitute the values of $q$ and $w$ we found:

$$ \Delta U = 0 $$ …