Molar heat capacity

The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Alternatively, it is the heat capacity of a sample of the substance divided by the amount of substance of the sample, or the specific heat capacity of the substance times its molar mass. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K⁻¹⋅mol⁻¹.
Like specific heat, the measured molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated than when it is heated in a closed vessel that prevents expansion. The ratio between the two, however, is the same heat capacity ratio obtained from the corresponding specific heat capacities.
This property is most relevant in chemistry, when amounts of substances are often specified in moles rather than by mass or volume. The molar heat capacity generally increases with the molar mass, often varies with temperature and pressure, and is different for each state of matter. For example, at atmospheric pressure, the molar heat capacity of water just above the melting point is about 76 J⋅K⁻¹⋅mol⁻¹, but that of ice just below that point is about 37.84 J⋅K⁻¹⋅mol⁻¹. While the substance is undergoing a phase transition, such as melting or boiling, its molar heat capacity is technically infinite, because the heat goes into changing its state rather than raising its temperature. The concept is not appropriate for substances whose precise composition is not known, or whose molar mass is not well defined, such as polymers and oligomers of indeterminate molecular size.
A closely related property of a substance is the heat capacity per mole of atoms, or atom-molar heat capacity, in which the heat capacity of the sample is divided by the number of moles of atoms instead of moles of molecules. So, for example, the atom-molar heat capacity of water is 1/3 of its molar heat capacity, namely 25.3 J⋅K⁻¹⋅mol⁻¹.
In informal chemistry contexts, the molar heat capacity may be called just "heat capacity" or "specific heat". However, international standards now recommend that "specific heat capacity" always refer to capacity per unit of mass, to avoid possible confusion. Therefore, the word "molar", not "specific", should always be used for this quantity.

Definition

The molar heat capacity of a substance, which may be denoted by c_m, is the heat capacity C of a sample of the substance divided by the amount n of the substance in the sample:
where Q is the amount of heat needed to raise the temperature of the sample by ΔT. Obviously, this parameter cannot be computed when n is not known or defined.
Like the heat capacity of an object, the molar heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T of the sample and the pressure P applied to it. Therefore, it should be considered a function c_m of those two variables.
These parameters are usually specified when giving the molar heat capacity of a substance. For example, "H₂O: 75.338 J⋅K⁻¹⋅mol⁻¹ " When not specified, published values of the molar heat capacity c_m generally are valid for some standard conditions for temperature and pressure.
However, the dependency of c_m on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one can usually omit the qualifier and approximate the molar heat capacity by a constant c_m suitable for those ranges.
Since the molar heat capacity of a substance is the specific heat c times the molar mass of the substance M/''N'', its numerical value in SI units is generally smaller than that of the specific heat. Paraffin wax, for example, has a specific heat of about but a molar heat capacity of about.
The molar heat capacity is an "intensive" property of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration.

Variations

The injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured molar heat capacity, even for the same starting pressure P and starting temperature T. Two particular choices are widely used:

If the pressure is kept constant, and the sample is allowed to expand, the expansion generates work as the force from the pressure displaces the enclosure. That work must come from the heat energy provided. The value thus obtained is said to be the molar heat capacity at constant pressure, and is often denoted c_P,m, c_p,m, c_P,m, etc.
On the other hand, if the expansion is prevented — for example by a sufficiently rigid enclosure, or by increasing the external pressure to counteract the internal one — no work is generated, and the heat energy that would have gone into it must instead contribute to the internal energy of the object, including raising its temperature by an extra amount. The value obtained this way is said to be the molar heat capacity at constant volume and denoted c_V,m, c_v,m, c_v,m, etc.

The value of c_V,m is always less than the value of c_P,m. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.
All methods for the measurement of specific heat apply to molar heat capacity as well.

Units

The SI unit of molar heat capacity heat is joule per kelvin per mole, J/. Since an increment of temperature of one degree Celsius is the same as an increment of one kelvin, that is the same as joule per degree Celsius per mole.
In chemistry, heat amounts are still often measured in calories. Confusingly, two units with that name, denoted "cal" or "Cal", have been commonly used to measure amounts of heat:

the "small calorie" is 4.184 J, exactly.
The "grand calorie" is 1000 small calories, that is, 4184 J, exactly.

When heat is measured in these units, the unit of specific heat is usually
The molar heat capacity of a substance has the same dimension as the heat capacity of an object; namely, L²⋅M⋅T⁻²⋅Θ⁻¹, or M²/Θ. Therefore, the SI unit J⋅K⁻¹⋅mol⁻¹ is equivalent to kilogram metre squared per second squared per kelvin per mole.

Physical basis

Monatomic gases

The temperature of a sample of a substance reflects the average kinetic energy of its constituent particles relative to its center of mass. Quantum mechanics predicts that, at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy. Therefore, when a certain number N of atoms of a monatomic gas receives an input Q of heat energy, in a container of fixed volume, the kinetic energy of each atom will increase by Q/''N, independently of the atom's mass. This assumption is the foundation of the theory of ideal gases.
In other words, that theory predicts that the molar heat capacity at constant volume c''_V,m of all monatomic gases will be the same; specifically,
where R is the ideal gas constant, about 8.31446 J⋅K⁻¹⋅mol⁻¹. And, indeed, the experimental values of c_V,m for the noble gases helium, neon, argon, krypton, and xenon are all 12.5 J⋅K⁻¹⋅mol⁻¹, which is R; even though their atomic weights range from 4 to 131.
The same theory predicts that the molar heat capacity of a monatomic gas at constant pressure will be
This prediction matches the experimental values, which, for helium through xenon, are 20.78, 20.79, 20.85, 20.95, and 21.01 J⋅K⁻¹⋅mol⁻¹, respectively; very close to the theoretical R = 20.78 J⋅K⁻¹⋅mol⁻¹.
Therefore, the specific heat of a monatomic gas will be inversely proportional to its relative atomic mass A. That is, approximately,

Polyatomic gases

Degrees of freedom

A polyatomic molecule can store heat energy in other forms besides its kinetic energy. These forms include rotation of the molecule, and vibration of the atoms relative to its center of mass.
These extra degrees of freedom contribute to the molar heat capacity of the substance. Namely, when heat energy is injected into a gas with polyatomic molecules, only part of it will go into increasing their kinetic energy, and hence the temperature; the rest will go to into those other degrees of freedom. Thus, in order to achieve the same increase in temperature, more heat energy will have to be provided to a mol of that substance than to a mol of a monatomic gas. Substances with high atomic count per molecule, like octane, can therefore have a very large heat capacity per mole, and yet a relatively small specific heat.
If the molecule could be entirely described using classical mechanics, then the theorem of equipartition of energy could be used to predict that each degree of freedom would have an average energy in the amount of kT, where k is the Boltzmann constant, and T is the temperature. If the number of degrees of freedom of the molecule is f, then each molecule would be holding, on average, a total energy equal to fkT. Then the molar heat capacity would be
where R is the ideal gas constant. According to Mayer's relation, the molar heat capacity at constant pressure would be
Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas.
In particular, each molecule of a monatomic gas has only f = 3 degrees of freedom, namely the components of its velocity vector; therefore c_V,m = R and c_P,m = R.