Enthalpy and Heat Capacity

Most reactions happen in open vessels at constant atmospheric pressure. At constant pressure, heat absorbed by the system splits between raising internal energy and doing expansion work against the atmosphere, so a state function that captures "heat at constant pressure" directly is genuinely useful. That function is enthalpy:

where: $U$ is internal energy, $P$ is pressure, and $V$ is volume — all state functions, so $H$ is a state function too, depending only on the current state of the system, not on how it got there. For a process at constant pressure, $\Delta H = \Delta U + P\Delta V$, and it can be shown that this equals $q_p$, the heat exchanged at constant pressure. This is why $\Delta H$, not $\Delta U$, is the quantity tabulated for reaction enthalpies — it is exactly what a calorimeter open to the atmosphere measures. For reactions involving ideal gases, substituting $PV = n_gRT$ at constant temperature and pressure turns the volume term into a mole-number term:

where: $\Delta n_g$ is (moles of gaseous products) minus (moles of gaseous reactants) — solids and liquids are not counted — $R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}$ is the gas constant, and $T$ is the temperature in kelvin.

Heat capacity $C$ is the heat needed to raise a system's temperature by 1 K:

Two heat capacities matter because the "1 K rise" can happen at constant volume or constant pressure, and the heat needed is different in each case:

• $C_v$ (constant volume): $C_v = \left(\dfrac{\partial U}{\partial T}\right)_V$, so $q_v = \Delta U = C_v\Delta T$ — all the heat goes into internal energy, none into work, because the volume cannot change.
• $C_p$ (constant pressure): $C_p = \left(\dfrac{\partial H}{\partial T}\right)_P$, so $q_p = \Delta H = C_p\Delta T$ — some heat also does expansion work as the gas is free to push outward.

Because constant-pressure heating must additionally supply expansion work, $C_p$ is always larger than $C_v$ for a gas; the two are equal only for near-incompressible solids and liquids where volume barely changes with temperature. Molar heat capacity is per mole of substance; specific heat capacity is per unit mass — always check which one a problem is quoting before you substitute.

Problem: For $\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)$ at $298\ \text{K}$, $\Delta U = -91.8\ \text{kJ mol}^{-1}$. Find $\Delta H$. Separately, an ideal monatomic gas has $C_v = 12.5\ \text{J K}^{-1}\text{mol}^{-1}$; find $C_p$ and the heat needed to warm $2.0\ \text{mol}$ of it by $15\ \text{K}$ at constant pressure.

Step 1. Count gas moles on each side. Products: $2$ mol $\text{NH}_3$. Reactants: $1$ mol $\text{N}_2 + 3$ mol $\text{H}_2 = 4$ mol. So $\Delta n_g = 2 - 4 = -2$.

Step 2. Apply $\Delta H = \Delta U + \Delta n_g RT$ with $R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}$, $T = 298\ \text{K}$: $\Delta n_g RT = (-2)(8.314)(298) = -4955\ \text{J mol}^{-1} = -4.955\ \text{kJ mol}^{-1}$.

Step 3. Add to $\Delta U$: $\Delta H = -91.8 + (-4.955) = -96.8\ \text{kJ mol}^{-1}$. The gas contracts (4 mol gas becomes 2 mol), so the surroundings do work on the shrinking system, and the reaction releases slightly more heat at constant pressure than the bare $\Delta U$ suggests.

Step 4. Now the heat capacity part. Using $C_p = C_v + R$: $C_p = 12.5 + 8.314 = 20.8\ \text{J K}^{-1}\text{mol}^{-1}$. Then $q_p = nC_p\Delta T = (2.0)(20.8)(15) = 624\ \text{J}$.

Final answer: $\Delta H = -96.8\ \text{kJ mol}^{-1}$ for the ammonia synthesis, and $624\ \text{J}$ of heat is needed to warm $2.0\ \text{mol}$ of the gas by $15\ \text{K}$ at constant pressure (with $C_p = 20.8\ \text{J K}^{-1}\text{mol}^{-1}$).

• Computing $\Delta n_g$ from all species, including solids and liquids — only gaseous species count; a pure solid or liquid product contributes zero.
• Flipping the sign of $\Delta n_g$ — it is always (products) minus (reactants), so a reaction that produces fewer moles of gas gives a negative $\Delta n_g$ and $\Delta H < \Delta U$.
• Assuming $C_p = C_v$ in general — this only holds approximately for solids and liquids with negligible expansion; for an ideal gas $C_p - C_v = R$ exactly.
• Mixing up heat capacity (extensive, scales with amount) with molar or specific heat capacity (intensive, per mole or per gram) — always confirm the units before plugging into $q = nC\Delta T$ or $q = mc\Delta T$.

Ask the tutor: Stuck on when to use $C_v$ versus $C_p$, or how to compute $\Delta n_g$ for a tricky reaction — ask the AI tutor to walk through a fresh example with you.