Heat Capacity Data

Introduction

Heat capacity tables tell how much a substance's energy changes as temperature changes. A single number at 298 K may be convenient over a narrow range, but wider temperature spans require the temperature dependence of Cp or Cv. Phase changes add separate enthalpy effects that cannot be captured by one smooth heat-capacity curve.

Core explanation

Molar heat capacity at constant pressure Cp,m is (∂Hm/∂T)p; at constant volume Cv,m is (∂Um/∂T)V. They are different quantities and use units such as J mol⁻¹ K⁻¹. For an ideal gas, Cp,m − Cv,m = R, but this relation does not apply universally to arbitrary liquids or strongly nonideal conditions. Choose the quantity matching the process and the energy function being calculated. The NIST Chemistry WebBook publishes heat-capacity data and fitted temperature functions for many species.

For a single phase at constant pressure, ΔHm = ∫Cp,m(T)dT from T₁ to T₂. If Cp is nearly constant over a small interval, ΔH ≈ CpΔT is useful. For larger spans, use tabulated values or a fitted function. If Cp(T) = a + bT over a stated range, integration gives a(T₂−T₁) + (b/2)(T₂²−T₁²). The fitted coefficients have units that make Cp consistent; a is J mol⁻¹ K⁻¹ and b is J mol⁻¹ K⁻² in this simple form. Do not extrapolate a polynomial far beyond its fitted interval.

If a substance melts or boils between T₁ and T₂, split the calculation: heat the starting phase to the transition, add the latent enthalpy, then heat the new phase. A heat-capacity table for liquid water is not valid through a range containing vaporization without a phase-change term. NIST's WebBook guide distinguishes gas and condensed-phase heat-capacity entries and their temperature ranges.

For a chemical reaction, ΔrCp = ΣνproductsCp,m − ΣνreactantsCp,m. Kirchhoff's relation gives ΔrH(T₂) = ΔrH(T₁) + ∫ΔrCp(T)dT, assuming phases and reference conventions are treated consistently. An analogous entropy correction integrates ΔrCp/T. This connects formation-data calculations at one reference temperature to another process temperature. In high-precision work, also track pressure effects and uncertainty in fitted coefficients.

Step-by-step reasoning

1. Identify whether the calculation concerns H at fixed pressure or U at fixed volume. 2. Select Cp or Cv for the correct phase and temperature interval. 3. Check whether a constant approximation is justified or an integral is needed. 4. Insert any phase-transition enthalpies as separate steps. 5. Apply the resulting correction to a species or reaction with consistent units.

Visual explanation

Plot Cp,m against temperature for a liquid and a gas, with a vertical marker at boiling. Shade the area under Cp from T₁ to boiling and from boiling to T₂; these areas are sensible-heating contributions. A separate vertical jump at boiling represents latent enthalpy. The picture shows why one continuous area under a liquid Cp curve would miss vaporization.

Real-world analogy

Climbing a hill with varying slope requires adding each segment's rise, not multiplying the starting slope by the whole distance. Integrating Cp(T) sums the small energy changes across temperature. Crossing a cliff is like a phase transition: a separate jump must be included.

Real-world example

A reactor feed enters at 300 K and leaves at 700 K. Using one room-temperature Cp value for all gases can bias the energy balance. Engineers use temperature-dependent Cp correlations for each species and integrate them. If one feed component vaporizes before reaction, they include its latent heat and the relevant phase-specific Cp values.

Why?

Why can a reaction enthalpy change with temperature even when the reaction equation is unchanged? Reactants and products gain enthalpy at different rates if their total heat capacities differ. The difference ΔrCp accumulates as temperature changes. A reaction exothermic at one temperature can become less or more exothermic elsewhere, though the exact change requires data.

Common misconception

“Cp equals Cv for every substance” is false. “A 298 K heat capacity is exact over any temperature span” ignores dependence. “A phase transition is just a very high ordinary Cp value in a single-phase table” misses latent heat. “One polynomial fit can be extrapolated indefinitely” ignores its stated range.

Worked example

Suppose Cp,m(T) = 20 + 0.010T J mol⁻¹ K⁻¹ from 300 to 500 K with no phase change. Then ΔHm = 20(500−300) + (0.010/2)(500²−300²) = 4,000 + 800 = 4,800 J/mol. The room-temperature approximation using Cp at 300 K gives Cp = 23 J mol⁻¹ K⁻¹ and ΔH ≈ 4,600 J/mol, 200 J/mol lower. Both numbers are illustrative, and the integral is preferred when the stated correlation is valid. If a phase transition occurred at 400 K, this calculation would need to be split and a latent enthalpy added.

Quick check

1. Why is a liquid Cp correlation insufficient for heating through boiling? Answer: It does not include vaporization enthalpy or the gas-phase heat capacity after boiling.

Exam focus

State definitions and units of Cp and Cv, integrate a simple Cp(T), and split calculations at phase transitions. Use ΔrCp to explain temperature corrections to reaction enthalpy. Check the validity range of fitted coefficients before extrapolating.

Advanced insight

Many databases use multiparameter functions fitted over selected temperature ranges. Continuity of H and S across adjoining ranges should be checked when using piecewise coefficients. Numerical precision in the coefficients does not remove uncertainty in the underlying experiments or errors caused by using the wrong phase or pressure regime.

Summary

Heat capacities describe temperature-dependent changes in enthalpy or internal energy. Integrals replace constant-Cp shortcuts over broad ranges, and phase transitions require additional enthalpy terms. Reaction heat-capacity differences correct reaction enthalpies from one temperature to another.

Practice questions

1. What is the ideal-gas relation between Cp,m and Cv,m? Answer: Cp,m − Cv,m = R for an ideal gas. 2. Integrate Cp = 20 + 0.010T from 300 to 500 K. Answer: 4,800 J/mol under the stated single-phase correlation. 3. What is ΔrCp for a reaction? Answer: The stoichiometric product heat-capacity sum minus the reactant sum. 4. What contribution is added when a substance vaporizes during heating? Answer: The enthalpy of vaporization at the transition condition, plus phase-specific sensible-heating terms.