4.4 Heat of reaction and heat of formation

Heat of Reaction and Formation

Definition and Significance

Heat of reaction ($\Delta H_{rxn}$) is the enthalpy change that occurs during a chemical reaction at constant pressure. It represents the difference in enthalpy between the products and reactants. If the products hold less enthalpy than the reactants, energy is released; if they hold more, energy is absorbed.

Heat of formation ($\Delta H_f^\circ$) is the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states (25°C and 1 atm). For example, the standard heat of formation of water describes the enthalpy change when $H_2(g)$ and $\frac{1}{2}O_2(g)$ combine to form one mole of $H_2O(l)$. That value is $-285.8 \text{ kJ/mol}$, meaning the reaction releases energy.

By convention, the standard heat of formation of any element in its most stable form (like $O_2(g)$ or $C(s, \text{graphite})$) is defined as zero. This gives you a consistent reference point for all calculations.

Role in Energy Balance Calculations

These values are the foundation of the energy balance around any reactive system. Specifically, they let you:

• Determine whether a reactor needs heating or cooling, and how much
• Size heat exchange equipment (heat exchangers, furnaces, cooling systems) based on the magnitude of energy released or absorbed
• Use tabulated formation data to calculate the heat of reaction for processes that would be difficult or dangerous to measure directly

Without reliable $\Delta H_f^\circ$ data, you can't close an energy balance on a reactive system.

Calculating Heat of Reaction

Hess's Law

Hess's law states that the total enthalpy change for a reaction is independent of the pathway taken. It depends only on the initial and final states. This means you can combine known reactions to find the enthalpy change of a reaction you haven't measured.

The reason this works is that enthalpy is a state function. It doesn't matter how you get from reactants to products; the total $\Delta H$ is the same.

The most common application in this course is calculating $\Delta H_{rxn}^\circ$ from standard heats of formation:

$\Delta H_{rxn}^\circ = \sum \nu_i \Delta H_{f,\text{products}}^\circ - \sum \nu_i \Delta H_{f,\text{reactants}}^\circ$

where $\nu_i$ represents the stoichiometric coefficients.

Steps to calculate heat of reaction using Hess's law:

1. Write the balanced chemical equation for the reaction.
2. Look up the standard heats of formation ($\Delta H_f^\circ$) for every product and reactant. Remember: elements in their standard states have $\Delta H_f^\circ = 0$.
3. Multiply each $\Delta H_f^\circ$ by its stoichiometric coefficient.
4. Sum the values for all products, then sum the values for all reactants.
5. Subtract the reactant total from the product total. A negative result means exothermic; positive means endothermic.

Quick example: For the combustion of methane, $CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(l)$:

• Products: $\Delta H_f^\circ[CO_2(g)] = -393.5$ and $2 \times \Delta H_f^\circ[H_2O(l)] = 2(-285.8) = -571.6$
• Reactants: $\Delta H_f^\circ[CH_4(g)] = -74.8$ and $2 \times \Delta H_f^\circ[O_2(g)] = 0$
• $\Delta H_{rxn}^\circ = (-393.5 - 571.6) - (-74.8) = -890.3 \text{ kJ/mol}$

Make sure all your heats of formation are in the same units (typically kJ/mol) before you start adding them up.

Standard Heats of Formation

Standard heats of formation are tabulated at 25°C and 1 atm. You'll find them in references like the NIST Chemistry WebBook or the CRC Handbook of Chemistry and Physics.

A few things to watch for when using these tables:

• Check the phase of the compound. $\Delta H_f^\circ$ for $H_2O(l)$ is $-285.8 \text{ kJ/mol}$, while $\Delta H_f^\circ$ for $H_2O(g)$ is $-241.8 \text{ kJ/mol}$. Picking the wrong one throws off your answer by the enthalpy of vaporization.
• The values are per mole of compound formed, so you must multiply by the stoichiometric coefficient in your balanced equation.
• If a compound isn't in your table, you may need to combine multiple formation reactions using Hess's law to get the value you need.

Enthalpy Change at Different Temperatures

Kirchhoff's Law

Standard heats of formation are tabulated at 25°C, but real processes rarely run at 25°C. Kirchhoff's law lets you adjust the heat of reaction to any temperature by accounting for the difference in heat capacities between products and reactants.

The general form is:

$\Delta H_{rxn}(T) = \Delta H_{rxn}(T_{ref}) + \int_{T_{ref}}^{T} \Delta C_p \, dT$

where $\Delta C_p = \sum \nu_i C_{p,\text{products}} - \sum \nu_i C_{p,\text{reactants}}$.

Think of it this way: you're taking a hypothetical path where you first cool everything to 25°C, react at 25°C (where you know $\Delta H_{rxn}^\circ$), and then heat the products to the actual temperature. The integral accounts for those heating/cooling steps.

Steps to apply Kirchhoff's law:

1. Calculate $\Delta H_{rxn}^\circ$ at the reference temperature (usually 25°C = 298.15 K) using Hess's law.
2. Look up or calculate $C_p$ for each reactant and product. These are often given as polynomial functions of temperature: $C_p = a + bT + cT^2 + dT^3$.
3. Compute $\Delta C_p$ by taking the stoichiometric-weighted sum of product heat capacities minus the stoichiometric-weighted sum of reactant heat capacities.
4. Integrate $\Delta C_p$ from $T_{ref}$ to your target temperature $T$.
5. Add the integral result to $\Delta H_{rxn}^\circ$ to get the heat of reaction at the new temperature.

Application of Kirchhoff's Law

This correction matters most when there's a large temperature difference between 25°C and your operating conditions, or when $\Delta C_p$ is significant. For reactions where products and reactants have similar heat capacities, the correction may be small.

Keep two assumptions in mind:

• The method assumes no phase changes occur between $T_{ref}$ and $T$. If a reactant or product changes phase in that range, you need to include the enthalpy of that phase transition ($\Delta H_{vap}$, $\Delta H_{fus}$, etc.) as a separate term.
• If you're using a constant $\Delta C_p$ (not a polynomial), the integral simplifies to $\Delta C_p \cdot (T - T_{ref})$. This is a reasonable approximation over narrow temperature ranges but can introduce noticeable error over wide ones. For a 500 K span, always use the polynomial form.

Impact of Heat of Reaction on Energy Requirements

Exothermic and Endothermic Reactions

• Exothermic reactions ($\Delta H_{rxn} < 0$) release heat. The reactor may need cooling to maintain a safe operating temperature, or that heat can be recovered and used elsewhere in the process.
• Endothermic reactions ($\Delta H_{rxn} > 0$) absorb heat. You'll need to supply energy (via a furnace, steam, etc.) to keep the reaction going at the desired temperature.

The magnitude of $\Delta H_{rxn}$ directly affects equipment sizing. A highly exothermic reaction like combustion of methane ($\Delta H_{rxn}^\circ = -890.3 \text{ kJ/mol}$) demands substantial cooling capacity, while a mildly endothermic reaction may only need a modest heat input. This is where your energy balance connects to real engineering decisions about reactor design.

Process Integration and Optimization

In a real plant, you rarely look at one reactor in isolation. Heat integration (sometimes called pinch analysis at a more advanced level) is the practice of matching heat sources with heat sinks across an entire process. The idea is straightforward: use the heat released by exothermic reactions to drive endothermic reactions or to preheat feed streams, rather than paying for external utilities like steam or cooling water.

Effective heat integration requires knowing:

• The heat of reaction for each reactive unit
• The temperature at which that heat is available or needed
• The heat capacity flow rates of the process streams

Getting these values right reduces operating costs, improves energy efficiency, and lowers the environmental footprint of the process. The $\Delta H_{rxn}$ calculations you've learned here are the starting point for all of that downstream work.