4.3 Enthalpy and heat capacity

Enthalpy and heat capacity are foundational to energy balances in chemical engineering. Enthalpy tracks the total heat content of a system at constant pressure, while heat capacity tells you how much energy is needed to change a substance's temperature. Together, they let you calculate how energy flows through chemical processes.

These concepts show up everywhere: designing heat exchangers, sizing reactors, running energy balances on distillation columns. To get accurate results, you need to understand not just the definitions but how heat capacity changes with temperature and how phase changes factor into enthalpy calculations.

Enthalpy and Internal Energy

Definition and Relationship

Enthalpy ($H$) is defined as the sum of a system's internal energy ($U$) and the pressure-volume product:

$H = U + PV$

Why does this combination matter? In most chemical engineering processes, you're working at constant pressure (open systems, reactors at atmospheric pressure, etc.). Under those conditions, the change in enthalpy equals the heat transferred to or from the system:

$\Delta H = Q_p$

This is what makes enthalpy so useful. Instead of tracking both internal energy and work separately, enthalpy wraps them into one quantity that directly equals the measurable heat flow at constant pressure.

Enthalpy is a state function, meaning its value depends only on the current state of the system (temperature, pressure, composition), not on how the system got there. This is a big deal for calculations because it means you can pick any convenient path between two states and get the same $\Delta H$.

Pressure-Volume Work

Pressure-volume work is the work a system does when it expands or contracts against an external pressure. At constant pressure:

$W = -P\Delta V$

The negative sign follows the convention that work done by the system on the surroundings (expansion, $\Delta V > 0$) is negative from the system's perspective. Unlike enthalpy, pressure-volume work is path-dependent, so it can differ depending on how the process is carried out.

Calculating Enthalpy Changes

Heat Capacity and Specific Heat

Heat capacity ($C$) is the amount of heat needed to raise the temperature of a substance by one degree (Celsius or Kelvin). You'll encounter it in three forms:

• Specific heat capacity ($c$): heat capacity per unit mass, $c = C/m$. Units are typically J/(g·K) or kJ/(kg·K).
• Molar heat capacity ($C_m$): heat capacity per mole, $C_m = C/n$. Units are typically J/(mol·K).
• $C_p$ vs. $C_v$: $C_p$ is measured at constant pressure; $C_v$ at constant volume. For most chemical engineering applications (open systems, flow processes), you'll use $C_p$.

To calculate the enthalpy change for heating or cooling a substance (no phase change), use:

$\Delta H = C_p \times \Delta T$

This version assumes $C_p$ is roughly constant over the temperature range. For large temperature changes, you'll need the integral form covered below.

Phase Changes and Latent Heat

During a phase change, temperature stays constant even though heat is being added or removed. The enthalpy change for a phase transition equals the latent heat:

$\Delta H = \pm L$

Two key types:

• Latent heat of fusion ($L_f$): energy to melt a solid into a liquid at the melting point. For water, this is about 334 J/g.
• Latent heat of vaporization ($L_v$): energy to vaporize a liquid into a gas at the boiling point. For water, this is about 2,260 J/g.

The sign convention: positive $\Delta H$ for melting and vaporization (endothermic, system absorbs heat), negative for freezing and condensation (exothermic, system releases heat).

When you're calculating the total enthalpy change for a process that involves both temperature changes and phase changes, you need to handle each segment separately. For example, heating ice from $-10°C$ to steam at $120°C$ requires five separate calculations: heating ice, melting, heating liquid water, vaporizing, and heating steam.

Significance of Heat Capacity

Energy Balance Calculations

Heat capacity is central to nearly every energy balance you'll write. A few specific applications:

• Heat exchangers: The heat capacities of the hot and cold fluids determine the outlet temperatures and the required heat transfer area. A fluid with a high $C_p$ can carry more thermal energy per degree of temperature change.
• Mixing streams: When combining streams at different temperatures, you use each stream's mass flow rate and heat capacity to find the final mixture temperature.
• Equipment sizing: Accurate $C_p$ values feed directly into the design equations for heat exchangers, reactors, and distillation columns.

Process Design and Optimization

Beyond individual equipment, heat capacity data drives larger design decisions:

• Heat integration techniques like pinch analysis use heat capacity flow rates ($\dot{m} \times C_p$) to identify where you can recover energy between process streams and reduce utility costs.
• Reactor design depends on $C_p$ to predict temperature profiles inside the reactor and determine how much cooling or heating is needed to maintain safe, productive operating conditions.
• Distillation energy requirements are tied to the heat capacities and latent heats of the components being separated, since you're repeatedly vaporizing and condensing material.

Temperature Dependence of Heat Capacity

Empirical Models

Heat capacity is not truly constant. It varies with temperature, especially for gases and liquids over wide temperature ranges. Engineers account for this using empirical polynomial expressions:

$C_p(T) = a + bT + cT^2 + dT^3$

Here, $a$, $b$, $c$, and $d$ are constants specific to each substance, found in reference tables (like Perry's Chemical Engineers' Handbook or appendix tables in your textbook). Another common form is the Shomate equation, which uses a slightly different polynomial structure and is widely used in thermodynamic databases like NIST.

For ideal gases, the heat capacity depends on both temperature and molecular structure. Monatomic gases (like He or Ar) have nearly constant $C_p$, while polyatomic molecules (like $CO_2$ or $CH_4$) show stronger temperature dependence because more vibrational modes become active at higher temperatures.

Impact on Enthalpy Calculations

When the temperature range is small (say, 10-20°C), using a constant average $C_p$ is usually fine. But for large temperature changes, you need to integrate:

$\Delta H = \int_{T_1}^{T_2} C_p(T) \, dT$

Here's how to carry out this calculation:

1. Look up the polynomial coefficients ($a, b, c, d$) for your substance.
2. Substitute the polynomial expression for $C_p(T)$ into the integral.
3. Integrate term by term: $\Delta H = aT + \frac{b}{2}T^2 + \frac{c}{3}T^3 + \frac{d}{4}T^4 \Big|_{T_1}^{T_2}$
4. Evaluate at $T_2$ and $T_1$, then subtract.

Skipping this integration and using a single constant $C_p$ value can introduce significant error. For example, in a gas-phase process where the temperature rises from 300 K to 1,000 K, the heat capacity of many gases increases substantially. Using the value at 300 K would underestimate the actual enthalpy change, throwing off your energy balance and potentially leading to undersized equipment.