3.2 Internal energy and enthalpy

Internal Energy vs Enthalpy

Defining Internal Energy and Enthalpy

Internal energy ($U$) is the total kinetic and potential energy of all particles within a system. This includes molecular translations, rotations, vibrations, and intermolecular interactions. It's an extensive property, meaning it scales with the amount of matter: 2 moles of water have twice the internal energy of 1 mole at the same conditions.

Enthalpy ($H$) is defined as:

$H = U + PV$

Enthalpy packages together the internal energy and the pressure-volume "work content" of a system. Why bother defining a new quantity? Because most chemistry happens in open beakers and flasks at constant pressure, and enthalpy turns out to be the natural way to track heat flow under those conditions.

Comparing Internal Energy and Enthalpy

Changes in internal energy follow directly from the First Law:

$\Delta U = q + w$

• $q$ is heat transferred into or out of the system
• $w$ is work done on or by the system

At constant pressure, the change in enthalpy simplifies to:

$\Delta H = q_p$

This is what makes enthalpy so useful in chemistry: at constant pressure, $\Delta H$ is the heat exchanged with the surroundings. No need to separately calculate work.

Both $U$ and $H$ are state functions. Their values depend only on the current state of the system (temperature, pressure, composition), not on how the system got there. This property is what makes Hess's law and thermochemical cycles possible.

First Law of Thermodynamics Applications

Calculating Changes in Internal Energy

The First Law:

$\Delta U = q + w$

Sign conventions (IUPAC convention used in most physical chemistry texts):

• $q > 0$: heat absorbed by the system
• $q < 0$: heat released by the system
• $w > 0$: work done on the system
• $w < 0$: work done by the system

For processes involving only pressure-volume work against a constant external pressure:

$w = -P_{ext}\Delta V$

This applies whenever a gas expands or compresses against an external pressure, like a gas pushing a piston outward. Expansion ($\Delta V > 0$) means the system does work on the surroundings, so $w < 0$.

Calculating Changes in Enthalpy

At constant pressure:

$\Delta H = q_p$

You can also relate $\Delta H$ to $\Delta U$:

$\Delta H = \Delta U + P\Delta V$

For reactions involving only solids and liquids, $P\Delta V$ is negligibly small, so $\Delta H \approx \Delta U$. The distinction matters most when gases are produced or consumed, since gases occupy much more volume.

Example: For the combustion of methane at constant pressure:

$CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(l)$

The left side has 3 moles of gas; the right side has 1 mole of gas. The decrease in gas moles means $P\Delta V$ is negative, so $\Delta H$ and $\Delta U$ differ slightly. For most purposes in this course, though, you'll work directly with $\Delta H = q_p$.

Heat, Work, and Energy Changes

Adiabatic and Isothermal Processes

Adiabatic process ($q = 0$): No heat enters or leaves the system (think of a perfectly insulated container).

$\Delta U = w$

All energy change comes from work. Rapid compression heats the gas; rapid expansion cools it.

Isothermal process ($\Delta T = 0$): Temperature stays constant. For an ideal gas, internal energy depends only on temperature, so:

$\Delta U = 0 \quad \Rightarrow \quad q = -w$

Whatever work the gas does on the surroundings is exactly compensated by heat flowing in from a thermal reservoir (or vice versa). A classic example is the slow, reversible expansion of an ideal gas in contact with a heat bath.

Isobaric and Isochoric Processes

Isobaric process (constant pressure):

$\Delta H = q_p$ $\Delta U = q_p + w = q_p - P\Delta V$

Heating a liquid in an open container is a straightforward example. The enthalpy change directly equals the heat you supply.

Isochoric process (constant volume, $\Delta V = 0$):

Since no expansion or compression occurs, $w = 0$:

$\Delta U = q_v$

This is why bomb calorimeters (rigid, constant-volume devices) measure $\Delta U$ directly. Note that at constant volume, $\Delta H = \Delta U + V\Delta P$, which is not simply equal to $\Delta U$ unless the pressure also stays constant. For reactions involving gases, $\Delta H$ and $\Delta U$ at constant volume are related by:

$\Delta H = \Delta U + \Delta n_{gas}RT$

where $\Delta n_{gas}$ is the change in moles of gas. This equation is very useful for converting bomb calorimetry data to enthalpy changes.

Enthalpy as a State Function

Path Independence and Hess's Law

Because enthalpy is a state function, $\Delta H$ between two states is the same regardless of the route taken. This is the foundation of Hess's law: the total enthalpy change for a reaction equals the sum of the enthalpy changes for any sequence of steps that connect the same reactants to the same products.

In practice, this means you can combine reactions whose $\Delta H$ values are known to find $\Delta H$ for a reaction that's hard to measure directly.

Steps for applying Hess's law:

1. Write the target reaction clearly.
2. Identify known reactions whose enthalpies are given.
3. Manipulate the known reactions (reverse them, multiply by coefficients) so they sum to the target reaction. Reversing a reaction flips the sign of $\Delta H$; multiplying by a factor scales $\Delta H$ by that factor.
4. Add the adjusted $\Delta H$ values to get $\Delta H$ for the target reaction.

Thermochemical Cycles

The state function property also enables thermochemical cycles, where you arrange a series of steps into a closed loop. Since enthalpy is path-independent, the sum of all $\Delta H$ values around the cycle must equal zero:

$\sum \Delta H = 0$

The Born-Haber cycle is a classic example. It breaks the formation of an ionic compound (like NaCl) into individual steps: sublimation, ionization, dissociation, electron affinity, and lattice formation. If you know all the enthalpy changes except one, you can solve for the unknown, which is typically the lattice energy since it can't be measured directly.

Enthalpy Changes in Reactions and Transitions

Standard Enthalpies of Formation and Reaction

The standard enthalpy of formation ($\Delta H^\circ_f$) is the enthalpy change when one mole of a compound forms from its elements in their standard states (1 bar pressure, usually 298 K).

• By definition, $\Delta H^\circ_f = 0$ for any element in its standard state (e.g., $O_2(g)$, $C(graphite)$).
• Example: $\Delta H^\circ_f$ for $H_2O(l)$ is $-285.8 \text{ kJ/mol}$.

The standard enthalpy of reaction is calculated from formation enthalpies:

$\Delta H^\circ_{rxn} = \sum n \cdot \Delta H^\circ_f(\text{products}) - \sum n \cdot \Delta H^\circ_f(\text{reactants})$

where $n$ is the stoichiometric coefficient of each species. This works because of Hess's law: you're conceptually decomposing all reactants into elements and then reassembling them into products.

Phase Transitions and Heat Capacities

Phase transitions occur at constant temperature and pressure, so the heat involved equals the enthalpy change:

• Enthalpy of fusion ($\Delta H_{fus}$): heat absorbed on melting. For ice, $\Delta H_{fus} = 6.01 \text{ kJ/mol}$.
• Enthalpy of vaporization ($\Delta H_{vap}$): heat absorbed on boiling. For water, $\Delta H_{vap} = 40.7 \text{ kJ/mol}$ at 100°C.

These are always positive for the forward transition (solid $\rightarrow$ liquid, liquid $\rightarrow$ gas) and negative for the reverse.

Heat capacity ($C$) tells you how much heat is needed to raise the temperature of a substance by one degree. Two common forms:

• Specific heat capacity ($c$): per unit mass (J/(g·K))
• Molar heat capacity ($C_m$): per mole (J/(mol·K))

At constant pressure, the enthalpy change for a temperature change is:

$\Delta H = nC_p\Delta T$

where $C_p$ is the molar heat capacity at constant pressure. At constant volume, the analogous expression gives $\Delta U$:

$\Delta U = nC_v\Delta T$

For an ideal gas, these two heat capacities are related by:

$C_p - C_v = R$

where $R = 8.314 \text{ J/(mol·K)}$. This difference arises because heating at constant pressure requires extra energy to do expansion work against the surroundings.