Consider the reaction: $$\text{N}_2 + 3\text{H}_2 \longrightarrow 2\text{NH}_3$$ carried out at constant temperature and pressure. If $$\Delta H$$ and $$\Delta U$$ are the enthalpy and internal energy changes for the reaction, which of the following expressions is true?

For a chemical reaction carried out at constant temperature and pressure, the relation between the enthalpy change $$\Delta H$$ and the internal energy change $$\Delta U$$ is

$$\Delta H = \Delta U + \Delta n_g\,RT$$

where $$\Delta n_g$$ is the change in the number of moles of gaseous species (moles of gaseous products $$-$$ moles of gaseous reactants), $$R$$ is the gas constant and $$T$$ is the absolute temperature.

For the Haber reaction
$$\text{N}_2(g) + 3\text{H}_2(g) \longrightarrow 2\text{NH}_3(g)$$

moles of gaseous reactants $$= 1 + 3 = 4$$
moles of gaseous products $$= 2$$

Therefore, $$\Delta n_g = 2 - 4 = -2$$.

Substituting into the formula:

$$\Delta H = \Delta U + (-2)RT = \Delta U - 2RT$$

The term $$2RT$$ is positive, so $$\Delta H$$ is smaller than $$\Delta U$$.

Hence, $$\Delta H \lt \Delta U$$.

Option C which is: $$\Delta H \lt \Delta U$$