Q.Derive the relationship between ΔH and ΔU for an ideal gas. Explain each term involved in the equation.

The key idea is that enthalpy $H$ is defined as $H = U + PV$, so the change in enthalpy $\Delta H$ differs from the change in internal energy $\Delta U$ by the change in the $PV$ product.

For an ideal gas, $PV = nRT$. At constant temperature, $PV$ changes only if the number of moles of gas changes during the reaction. The difference between $\Delta H$ and $\Delta U$ is therefore:

At constant temperature $T$, $R$ is constant, so $\Delta (nRT) = RT \Delta n_g$, where $\Delta n_g$ is the change in the number of moles of gaseous species (moles of gaseous products minus moles of gaseous reactants). Thus:

• $\Delta H$: enthalpy change (heat exchanged at constant pressure). …

The relationship between $\Delta H$ and $\Delta U$ is one of the most fundamental ideas in thermochemistry. It tells you how much of the heat absorbed or released in a reaction goes into changing the internal energy of the system versus doing pressure-volume work on the surroundings. For an ideal gas, this relationship takes a particularly clean and useful form.

Let’s start with the definition of enthalpy. Enthalpy $H$ is defined as:

where $U$ is the internal energy, $P$ is the pressure, and $V$ is the volume. This definition is not arbitrary — it’s constructed so that at constant pressure, the heat exchanged equals the change in enthalpy. For a change from an initial state to a final state, the change in enthalpy is:

Now, for an ideal gas, we have the equation of state $PV = nRT$. So $PV$ is not a constant — it depends on the number of moles $n$ and the temperature $T$. If the temperature is constant (which is often the case in calorimetry, or when we compare $\Delta H$ and $\Delta U$ at the same temperature), then:

Here, $\Delta n$ is the change in the number of moles of gas during the reaction. But careful: this $\Delta n$ is specifically the change in gaseous moles only — solids and liquids contribute negligibly to $PV$ compared to gases.

So the relationship becomes:

where $\Delta n_g = n_{\text{gaseous products}} - n_{\text{gaseous reactants}}$.

Let’s break down each term:

1. $\Delta H$ — Enthalpy change. This is the heat absorbed or released at constant pressure. In bomb calorimetry, we measure $\Delta U$ directly (constant volume), but most reactions in open containers occur at constant pressure, so $\Delta H$ is what we usually want.

2. $\Delta U$ — Internal energy change. This accounts for changes in bond energies, molecular motions, and interactions. At constant volume, no work is done, so $\Delta U = q_V$ (heat at constant volume).

3. $RT \Delta n_g$ — The work term. This is the $P\Delta V$ work done by (or on) the system as the number of gas moles changes. If $\Delta n_g > 0$ (more gas moles produced), the system expands against the atmosphere, doing work, so $\Delta H > \Delta U$ — some of the heat goes into work. If $\Delta n_g < 0$, the surroundings do work on the system, so $\Delta H < \Delta U$.