Let's discuss how enthalpy behaves when the temperature remains constant. Enthalpy (H) is a thermodynamic property defined as the sum of a system's internal energy (U) and the product of its pressure (p) and volume (V).

The relationship is given by the equation:

\( H = U + pV \)

Here's a breakdown of the components:

• U: Internal energy of the system.
• p: Pressure of the system.
• V: Volume of the system.

Enthalpy and Temperature Dependence

For an ideal gas, the internal energy (U) depends solely on its temperature. This is a fundamental concept derived from kinetic theory and the ideal gas model. If the temperature is constant, the internal energy (U) of an ideal gas remains constant.

Also, for an ideal gas, the relationship between pressure, volume, and temperature is described by the ideal gas law:

\( pV = nRT \)

where n is the number of moles and R is the ideal gas constant. If the temperature (T) and the amount of gas (n) are constant, then the product \( pV \) must also remain constant.

Now, let's look back at the enthalpy equation:

\( H = U + pV \)

If the temperature is constant, for an ideal gas:

• Internal energy (U) is constant.
• The product \( pV \) is constant (because \( pV = nRT \), and n and T are constant).

Therefore, if both U and \( pV \) are constant, their sum, which is enthalpy (H), must also be constant.

Impact of Pressure and Volume Changes

If the temperature remains constant for an ideal gas, pressure (p) and volume (V) can change, but they change in such a way that their product \( pV \) remains constant (Boyle's Law, a special case of the ideal gas law at constant temperature). For example, if pressure increases, volume decreases proportionally.

Since \( U \) depends only on temperature (and temperature is constant) and the product \( pV \) depends only on temperature (and temperature is constant) for an ideal gas, the enthalpy \( H \) is determined solely by the constant temperature and the amount of gas.

This means that for an ideal gas at constant temperature, enthalpy is unaffected by changes in pressure or volume individually because these changes occur in a way that keeps their product \( pV \) constant, and the internal energy \( U \) is also constant.

While real gases show slight deviations, exhibiting a small dependence of enthalpy on pressure even at constant temperature (related to the Joule-Thomson effect), the question and options suggest considering the behavior typical of an ideal gas or a scenario where temperature dependence is dominant.

Based on the principles for ideal gases, at constant temperature, the enthalpy is constant.