Which one of the following is correct for the adsorption of a gas at a given temperature on a solid surface?

We begin by recalling that the adsorption of a gas on the surface of a solid is a spontaneous process at the temperature under consideration. For any spontaneous change, thermodynamics tells us that the Gibbs free-energy change is negative. The defining equation is first stated:

$$\Delta G \;=\; \Delta H \;-\; T\Delta S$$

Here, $$\Delta G$$ is the Gibbs free-energy change, $$\Delta H$$ is the enthalpy change, $$\Delta S$$ is the entropy change, and $$T$$ is the absolute temperature (always positive).

Because adsorption occurs on its own, we must have

$$\Delta G < 0.$$

Now, we examine how the gas molecules behave during adsorption. The gaseous molecules, which were previously free to move randomly in three dimensions, become attached to the solid surface and lose a large amount of translational freedom. This restriction of motion makes the system more ordered. Greater order implies a decrease in randomness, so we can write

$$\Delta S < 0.$$

Next, we consider the energy aspect. When the gas molecules adhere to the surface, new attractive forces (often van der Waals or chemical bonds) are established between the gas molecules and the solid. Formation of such interactions releases energy to the surroundings, meaning that the process is exothermic. Therefore, we have

$$\Delta H < 0.$$

Substituting these sign conventions into the Gibbs equation, we obtain

$$\Delta G = \underbrace{\Delta H}_{\;< 0\;} - T\;\underbrace{\Delta S}_{\;< 0\;}.$$

Since $$T\Delta S$$ is a product of a positive temperature and a negative entropy change, the term $$-T\Delta S$$ is positive. For $$\Delta G$$ to remain negative overall, the magnitude of the negative enthalpy term $$\Delta H$$ must exceed the magnitude of the positive term $$T|\Delta S|$$. Empirically, this is exactly what happens during adsorption, so the spontaneity criterion is satisfied.

Hence, both $$\Delta H$$ and $$\Delta S$$ are negative for the adsorption of a gas on a solid surface at a fixed temperature, matching:

Option D: $$\Delta H < 0, \; \Delta S < 0.$$

Hence, the correct answer is Option D.