2.2 Internal energy and enthalpy

Internal Energy and Enthalpy

Internal energy and enthalpy are two ways of tracking energy in a thermodynamic system. Internal energy captures everything happening inside the system, while enthalpy builds on that by accounting for the energy tied up in maintaining the system's pressure and volume. The distinction matters because which one you use depends on the constraints of your process: constant volume or constant pressure.

Internal Energy and Enthalpy

Internal energy and enthalpy concepts

Internal energy ($U$) is the total energy contained within a thermodynamic system. It includes the kinetic energy of molecular motion, the potential energy from intermolecular forces, and any other microscopic energy contributions. Because $U$ is a state function, it depends only on the current state variables (temperature, pressure, volume), not on how the system got there.

The change in internal energy follows directly from the First Law:

$\Delta U = Q + W$

where $Q$ is heat added to the system and $W$ is work done on the system. (Be careful with sign conventions here. Some textbooks define $W$ as work done by the system, which flips the sign.)

Enthalpy ($H$) is a state function defined as:

$H = U + PV$

where $P$ is pressure and $V$ is volume. Enthalpy is especially useful at constant pressure because, under that constraint, the change in enthalpy equals the heat transferred:

$\Delta H = Q_p$

This is why enthalpy shows up constantly in chemistry and engineering: most lab and real-world processes happen at atmospheric (constant) pressure.

Internal energy vs enthalpy

The core distinction comes down to what's held constant:

• Constant volume (isochoric) processes → track $\Delta U$. No expansion or compression work occurs, so all heat goes into changing internal energy. Example: heating a gas in a sealed, rigid container.
• Constant pressure (isobaric) processes → track $\Delta H$. Some energy goes into doing expansion work against the surroundings, and enthalpy captures the full heat exchange. Examples: chemical reactions open to the atmosphere, phase changes, heating a gas in a piston at constant pressure.

For an isobaric process, $\Delta H$ equals the heat transferred, but $\Delta U$ is smaller because part of the heat does $PV$ work on the surroundings. The relationship is:

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

Calculations for thermodynamic processes

Each type of idealized process has its own set of simplifications. Here's how $\Delta U$ and $\Delta H$ work out in each case.

Isochoric process (constant volume):

$\Delta U = Q_v = nC_v\Delta T$

• $n$ = number of moles, $C_v$ = molar heat capacity at constant volume, $\Delta T$ = temperature change
• Since volume doesn't change, no pressure-volume work is done
• For an ideal gas, $\Delta H = nC_p\Delta T$ still applies (enthalpy is a state function, so you can always compute it from the temperature change regardless of the process path)

Isobaric process (constant pressure):

$\Delta H = Q_p = nC_p\Delta T$

• $C_p$ = molar heat capacity at constant pressure
• To find the internal energy change: $\Delta U = \Delta H - P\Delta V$
• For an ideal gas, $P\Delta V = nR\Delta T$, so $\Delta U = nC_p\Delta T - nR\Delta T = nC_v\Delta T$

Isothermal process (constant temperature):

For an ideal gas, internal energy depends only on temperature, so:

$\Delta U = 0 \quad \text{and} \quad \Delta H = 0$

Since $\Delta U = 0$, the First Law gives $Q = -W$ (using the sign convention where $W$ is work done on the system). All heat absorbed is converted into work done by the system. For a reversible isothermal expansion:

$Q = nRT\ln\left(\frac{V_2}{V_1}\right)$

Adiabatic process (no heat exchange):

$Q = 0$, so $\Delta U = W$

• The temperature changes, so $\Delta U = nC_v\Delta T$ (nonzero)
• Similarly, $\Delta H = nC_p\Delta T$ (also nonzero)

Enthalpy changes and heat capacity

Heat capacity measures how much heat a substance needs to raise its temperature by one degree. It comes in two common forms:

• Molar heat capacity: heat per mole per degree (units: J/(mol·K))
• Specific heat capacity: heat per unit mass per degree (units: J/(kg·K) or J/(g·°C))

At constant pressure, the enthalpy change connects directly to heat capacity:

$\Delta H = nC_p\Delta T \quad \text{(using moles)}$ $\Delta H = mC\Delta T \quad \text{(using mass, where } C \text{ is specific heat capacity)}$

Why is $C_p$ always greater than $C_v$ for an ideal gas? At constant pressure, some of the added heat goes into expanding the gas against external pressure rather than raising the temperature. You need to supply more heat per degree of temperature rise. The exact relationship is:

$C_p - C_v = R$

where $R = 8.314 \text{ J/(mol·K)}$ is the universal gas constant. This is known as Mayer's relation.

Phase changes at constant pressure involve enthalpy changes equal to the latent heat:

$\Delta H = \pm nL \quad \text{(molar)} \qquad \Delta H = \pm mL \quad \text{(mass-based)}$

• $L$ is the molar or specific latent heat of the transition
• Positive for endothermic processes (melting, vaporization): the system absorbs heat
• Negative for exothermic processes (freezing, condensation): the system releases heat

During a phase change, temperature stays constant even though heat is being transferred. All the energy goes into breaking or forming intermolecular bonds rather than increasing molecular kinetic energy.