Thermochemistry Formulae

Introduction

Thermochemistry formulae translate temperature measurements into heat flows and combine known reaction enthalpies into unknown ones. The central equations are q = mcΔT for a simple sample, q reaction = −q surroundings for an insulated calorimeter, and ΔH total = ΣΔH steps for a reaction cycle. Each has conditions: heat capacity may change with temperature, a calorimeter may absorb heat, and reaction enthalpy belongs to a specified equation and state.

Core explanation

For a sample with approximately constant specific heat capacity c over a modest temperature interval, q = mcΔT. Here m is mass, c has units J g⁻¹ K⁻¹ if mass is in grams, and ΔT = T final − T initial. A positive q means the designated sample gained heat under that sign convention. For a whole object, use q = CΔT with heat capacity C in J K⁻¹. The equation assumes no phase change and that c is acceptably constant; during melting, added heat can change phase without raising temperature.

In calorimetry, define the system and surroundings explicitly. If a reaction is the system and the solution is the surroundings, energy conservation gives q reaction + q solution + q calorimeter ≈ 0 for an adequately insulated apparatus. In a simplified coffee-cup problem, q calorimeter may be neglected, so q reaction ≈ −mcΔT solution. A rising solution temperature means the reaction released heat, giving a negative reaction heat.

At constant pressure, if pressure–volume work is the relevant work, q p = ΔH for the process. At constant volume, a bomb calorimeter measures a heat more closely related to ΔU. These are not interchangeable in every reaction, particularly when gas moles change. Reaction enthalpy is usually quoted per reaction as written; doubling all coefficients doubles ΔH, and reversing the equation changes its sign.

Hess's law works because enthalpy is a state function. If reaction A can be formed by adding reactions B and C, then ΔH A = ΔH B + ΔH C. Manipulate the chemical equations first: reverse a step and reverse its ΔH, multiply a step and multiply its ΔH. Cancel intermediate species, including phase labels. A liquid water product and gaseous water product differ in enthalpy.

Standard formation enthalpies give Δ rH° = Σν products Δ fH° − Σν reactants Δ fH°. Standard formation enthalpy of an element in its reference state is zero by convention, not because that element has no internal energy. The temperature and standard-state definitions must match the tabulated values. Bond-enthalpy estimates use average gas-phase bond energies and can be useful approximations, but they are less precise for particular molecules and phases.

Heat capacity itself can vary with temperature. Over a broad range, q = m∫c(T)dT is more appropriate than one constant-c product. If a reaction heats the solution strongly, the calorimeter's own heat capacity and heat loss to the room should be calibrated or modeled. A high-precision enthalpy claim cannot come from a simplified equation with large unmeasured losses.

Step-by-step reasoning

State which body gains heat and its sign convention. Convert mass and temperature units, then calculate q for solution and calorimeter. Use conservation to obtain reaction heat. For a molar enthalpy, divide by reaction extent in moles. For Hess's law, combine balanced equations and their signed enthalpies before calculating.

Visual explanation

Draw a reaction vessel inside an insulated box. An arrow labeled heat leaves an exothermic reaction and enters solution and calorimeter. Below, draw a two-path energy diagram with identical start and end states; the sum of step enthalpies equals the direct path.

Real-world analogy

Climbing a mountain from one point to another changes elevation by the same amount whether one follows a direct trail or several detours. Hess's law similarly concerns state difference, not route. Heat lost along a physical trail is not represented by elevation, which reminds us to separate heat from state function.

Real-world example

An acid–base neutralization warms a coffee-cup calorimeter solution. The measured ΔT and solution heat capacity estimate heat absorbed by the solution; the opposite sign estimates reaction heat. A calibrated cup correction improves the result when the cup absorbs a meaningful fraction.

Why?

These formulae connect accessible temperature measurements to chemical energy changes and let difficult reaction enthalpies be inferred from measured cycles. They are fundamental to fuel, materials and reaction-design calculations.

Common misconception

“Positive solution q means positive reaction ΔH” reverses the system boundary. Another error adds reaction enthalpies without reversing or scaling them when the chemical equations are manipulated.

Worked example

A reaction heats 100.0 g solution by 2.0 K. Take c = 4.18 J g⁻¹ K⁻¹ and neglect the cup. q solution = 100.0 × 4.18 × 2.0 = 836 J. The reaction released about 836 J, so q reaction ≈ −836 J. If 0.0100 mol reaction occurred as written, ΔH ≈ −83.6 kJ mol⁻¹. Heat loss would make this magnitude an underestimate.

Quick check

1. If a solution warms during an insulated reaction, what is the reaction heat sign? Answer: Negative under the convention that heat released by the reaction is negative.

Exam focus

Label system and surroundings. Use q = mcΔT only when its assumptions fit. Reverse and scale enthalpies with their equations, and include physical states in Hess cycles.

Advanced insight

Kirchhoff's law relates reaction enthalpy at different temperatures through the difference in reactant and product heat capacities. This explains why a standard enthalpy tabulated at 298 K cannot always be used unchanged at very high temperatures.

Summary

Calorimetry uses heat capacity and energy conservation to estimate reaction heat. At suitable constant pressure, that heat is ΔH. Hess's law combines enthalpy changes independently of route, provided equations, signs and physical states are handled consistently.

Practice questions

1. How much heat raises 50 g water by 5 K if c = 4.18 J g⁻¹ K⁻¹? Answer: q = 50 × 4.18 × 5 = 1045 J. 2. What happens to ΔH when a reaction is reversed? Answer: Its sign reverses. 3. What happens to ΔH when all coefficients are doubled? Answer: Its value doubles for the doubled reaction extent. 4. Why can a phase label matter in a Hess cycle? Answer: Different phases have different enthalpies, so an uncanceled phase change contributes heat.

Sources

- OpenStax Chemistry 2e: Enthalpy and Hess's law. - OpenStax Chemistry 2e: Calorimetry.