Match the thermodynamic processes taking place in a system with the correct conditions. In the table: \(\Delta Q\) is the heat supplied \(\Delta W\) is the work done, and \(\Delta U\) is the change in internal energy of the system.

(I) – (2), (II) – (4), (III) – (1), (IV) – (3)

The correct answer is option 3. (I) – (2), (II) – (4), (III) – (1), (IV) – (3).

Let us delve into each thermodynamic process and their associated conditions:

Thermodynamic Processes:

I. Adiabatic Process: An adiabatic process occurs when no heat exchange (\(\Delta Q = 0\)) takes place between the system and its surroundings. The system is thermally isolated, meaning there is no heat transfer into or out of the system. Any change in internal energy (\(\Delta U\)) during an adiabatic process is solely due to work done (\(\Delta W\)). Example: Compression or expansion of a gas in a perfectly insulated cylinder.

II. Isothermal Process: An isothermal process occurs at a constant temperature. For an ideal gas, the internal energy change (\(\Delta U\)) is zero (\(\Delta U = 0\)) because the temperature remains constant. Therefore, any heat (\(\Delta Q\)) added to the system is completely converted into work (\(\Delta W\)). Example: Heating or cooling a gas in thermal equilibrium with its surroundings.

III. Isochoric (Isovolumetric) Process: An isochoric process occurs at constant volume. The work done (\(\Delta W\)) during an isochoric process is zero (\(\Delta W = 0\)) because there is no change in volume. Any heat added (\(\Delta Q\)) results solely in an increase in the internal energy of the system (\(\Delta U\)). Example: Heating a sealed container without allowing the piston to move.

IV. Isobaric Process: An isobaric process occurs at constant pressure. The internal energy change (\(\Delta U\)) is not zero (\(\Delta U \neq 0\)) because the temperature and volume can change while keeping pressure constant. Both heat (\(\Delta Q\)) and work (\(\Delta W\)) are involved in changing the internal energy of the system. Example: Heating or cooling a gas in a container with a movable piston where pressure is kept constant.

The correct matching is: 3. (I) – (2), (II) – (4), (III) – (1), (IV) – (3)

This means:

Adiabatic process (I) corresponds to condition (2): \(\Delta Q = 0\).

Isothermal process (II) corresponds to condition (4): \(\Delta U = 0\).

Isochoric process (III) corresponds to condition (1): \(\Delta W = 0\).

Isobaric process (IV) corresponds to condition (3): \(\Delta U \neq 0, \, \Delta W \neq 0, \, \Delta Q \neq 0\).

These matches correctly describe the key thermodynamic characteristics and conditions associated with each type of process.