Average or Individual Molar Heat Capacity?

Molar heat capacity can refer to two related but different quantities: the heat capacity of one mole of a particular substance, or an average value for a mixture or temperature interval. The right choice depends on what system you are heating and how accurately the heat capacity must be represented.

The key distinction is that heat capacity is extensive, while molar heat capacity is intensive. A sample’s total heat capacity depends on how much material is present; its molar heat capacity is a property expressed per mole.

What molar heat capacity means

The molar heat capacity at constant pressure, for a fixed amount and composition, is defined as

$$C_{p,m}=\frac{1}{n}\left(\frac{\partial H}{\partial T}\right)_p$$

For a closed system of fixed composition undergoing reversible heating at constant pressure with only pressure–volume work, this is related to heat by

$$\delta q_p^{\mathrm{rev}}=nC_{p,m}\,dT.$$

At constant volume, the corresponding quantity is

$$C_{v,m}=\frac{1}{n}\left(\frac{\partial U}{\partial T}\right)_V.$$

Here, $n$ is the amount of substance in moles, and $C_{p,m}$ and $C_{v,m}$ usually have units of $\mathrm{J\,mol^{-1}\,K^{-1}}$. In chemistry, tabulated heat capacities are often $C_{p,m}$ values because many laboratory processes occur approximately at constant atmospheric pressure.

For a sample containing $n$ moles, the total heat capacity is approximately

$$C_p=nC_{p,m}$$

if the molar heat capacity is constant over the temperature range. The heat required for a temperature change is then

$$q_p=nC_{p,m}(T_2-T_1).$$

When to use an individual value

Use the individual molar heat capacity of the substance being heated. For example, if you heat $2.00\ \mathrm{mol}$ of liquid water and use

$$C_{p,m}=75.3\ \mathrm{J\ mol^{-1}\ K^{-1}},$$

through a temperature increase of $10.0\ \mathrm{K}$, then

$$\begin{aligned} q_p&=nC_{p,m}\Delta T\\ &=(2.00\ \mathrm{mol})(75.3\ \mathrm{J\ mol^{-1}\ K^{-1}})(10.0\ \mathrm{K})\\ &=1506\ \mathrm{J}. \end{aligned}$$

The relevant value is water’s molar heat capacity. You should not replace it with an average of the heat capacities of hydrogen and oxygen atoms. The heat capacity of a compound depends on its molecular structure, bonding, phase, and accessible motions, so it is not generally obtained by simply averaging or adding the elemental values.

How to calculate a mixture-average value

Suppose a nonreacting mixture contains components $i$, with amounts $n_i$ and individual molar heat capacities $C_{p,m,i}$. For an ideal mixture, or when excess heat-capacity effects are negligible, and assuming the component heat capacities are appropriate for the mixture’s composition and conditions, the total constant-pressure heat capacity is

$$C_{p,\mathrm{total}}=\sum_i n_iC_{p,m,i}.$$

If you want the mixture’s molar heat capacity per mole of total mixture, divide by the total amount:

$$C_{p,m,\mathrm{mix}}=\frac{\sum_i n_iC_{p,m,i}}{\sum_i n_i}.$$

This is a mole-weighted average, not usually an unweighted average.

Worked example

A gas mixture contains $2.00\ \mathrm{mol}$ of nitrogen and $1.00\ \mathrm{mol}$ of oxygen. At the temperature of interest, take

$$C_{p,m}(\mathrm{N_2})=29.1\ \mathrm{J\ mol^{-1}\ K^{-1}},$$

and

$$C_{p,m}(\mathrm{O_2})=29.4\ \mathrm{J\ mol^{-1}\ K^{-1}}.$$

The total heat capacity is

$$\begin{aligned} C_{p,\mathrm{total}}&=(2.00)(29.1)+(1.00)(29.4)\\ &=87.6\ \mathrm{J\ K^{-1}}. \end{aligned}$$

The mixture contains $3.00\ \mathrm{mol}$ in total, so its molar heat capacity is

$$C_{p,m,\mathrm{mix}}=\frac{87.6}{3.00}=29.2\ \mathrm{J\ mol^{-1}\ K^{-1}}.$$

For a $20.0\ \mathrm{K}$ temperature increase, the heat supplied is

$$q_p=C_{p,\mathrm{total}}\Delta T=(87.6)(20.0)=1752\ \mathrm{J}.$$

For an ideal gas mixture, this mole-weighted result is especially straightforward. Real mixtures can have excess heat-capacity effects, so the exact mixture property may not be a simple weighted sum, particularly at high pressures or for strongly interacting liquids.

Average heat capacity over a temperature range

A substance’s heat capacity often changes with temperature. The exact heat required at constant pressure is

$$q_p=n\int_{T_1}^{T_2}C_{p,m}(T)\,dT.$$

An average molar heat capacity over that interval can be defined by

$$\overline{C}_{p,m}=\frac{1}{T_2-T_1}\int_{T_1}^{T_2}C_{p,m}(T)\,dT.$$

Then the heat calculation becomes

$$q_p=n\overline{C}_{p,m}(T_2-T_1).$$

If the heat capacity changes approximately linearly between the endpoints, a useful approximation is

$$\overline{C}_{p,m}\approx\frac{C_{p,m}(T_1)+C_{p,m}(T_2)}{2}.$$

For example, if a substance has $C_{p,m}=40.0\ \mathrm{J\ mol^{-1}\ K^{-1}}$ at $300\ \mathrm{K}$ and $44.0\ \mathrm{J\ mol^{-1}\ K^{-1}}$ at $400\ \mathrm{K}$, the linear approximation gives

$$\overline{C}_{p,m}\approx\frac{40.0+44.0}{2}=42.0\ \mathrm{J\ mol^{-1}\ K^{-1}}.$$

For $1.50\ \mathrm{mol}$ heated across this range,

$$q_p\approx(1.50)(42.0)(100)=6300\ \mathrm{J}.$$

If a phase change occurs, do not use one continuous heat-capacity formula across it. Add the latent heat separately, and use the heat capacity appropriate to each phase on either side.

Heat capacities in chemical reactions

For a reaction, use the heat capacity of each individual species, multiplied by its stoichiometric coefficient. The reaction heat-capacity change is

$$\Delta_rC_p=\sum_i\nu_iC_{p,m,i},$$

where products have positive stoichiometric coefficients and reactants have negative coefficients.

For example, for

$$\mathrm{A+2B\rightarrow C},$$

one obtains

$$\Delta_rC_p=C_{p,m}(\mathrm{C})-C_{p,m}(\mathrm{A})-2C_{p,m}(\mathrm{B}).$$

This is not an average of the three listed heat capacities. Stoichiometry determines how many moles of each species are formed or consumed.

Common mistakes

• Using an arithmetic average for a mixture: Weight each value by the number of moles. Equal weighting is correct only when the components are present in equal amounts.
• Confusing molar and specific heat capacity: Molar heat capacity uses $\mathrm{J\,mol^{-1}\,K^{-1}}$; specific heat capacity uses $\mathrm{J\,g^{-1}\,K^{-1}}$ or $\mathrm{J\,kg^{-1}\,K^{-1}}$.
• Ignoring pressure or volume conditions: $C_p$ and $C_v$ are different quantities. For an ideal gas, $C_{p,m}-C_{v,m}=R$.
• Treating heat capacity as constant over any range: Check whether the table gives a temperature-dependent expression or whether a phase transition occurs.
• Adding atomic heat capacities to obtain a molecular value: Molecular heat capacity is not generally a simple sum of elemental or atomic values.

Thus, use an individual molar heat capacity for a specified substance, form a mole-weighted average for a mixture, and use the integral average when the value varies significantly with temperature.