heat capacity - Definition, Heat capacity of compressible bodies, Specific heat capacity, Dimensionless heat capacity

The quantity of heat needed to produce a temperature rise of one kelvin (or 1°C) in some material. Loosely, it measures the ability of a substance to get hot while absorbing energy. Specific heat capacity c (also called specific heat), units J/(kg.K), is the heat capacity per kilogram of material. For water, c = 4180 J/(kg.K).

Heat capacity (usually denoted by a capital C, often with subscripts) is a measurable physical quantity that characterizes the ability of a body to store heat as it changes in temperature. Dividing heat capacity by the body's mass yields a specific heat capacity (also called more properly "mass-specific heat capacity" or more loosely "specific heat"), which is an "intensive quantity," meaning it is no longer dependent on amount of material, and is now more dependent on the type of material, as well as the physical conditions of heating.

Definition

Heat capacity is mathematically defined as the ratio of a small amount of heat δQ added to the body, to the corresponding small increase in its temperature dT:

For thermodynamic systems with more than one physical dimension, the above definition does not give a single, unique quantity unless a particular infinitesimal path through the system's phase space has been defined (this means that one needs to know at all times where all parts of the system are, how much mass they have, and how fast they are moving).

Heat capacity of compressible bodies

The state of a simple compressible body with fixed mass is described by two thermodynamic parameters such as temperature T and pressure P. Therefore as mentioned above, one may distinguish between heat capacity at constant volume, CV, and heat capacity at constant pressure, CP:

where

δQ is the infinitesimal amount of heat added, dT is the subsequent rise in temperature.

The increment of internal energy is the heat added and the work added:

So the heat capacity at constant volume is

The enthalpy is defined by H = U + PV. The increment of enthalpy is

which, after replacing dU with the equation above and cancelling the PdV terms reduces to:

So the heat capacity at constant pressure is

Note that this last "definition" is a bit circular, since the concept of "enthalpy" itself was invented to be a measure of heat absorbed or produced at constant pressures (the conditions in which chemists usually work).

Specific heat capacity

The specific heat capacity of a material is

which in the absence of phase transitions is equivalent to

where

C is the heat capacity of a body made of the material in question (J·K−1) m is the mass of the body (kg) V is the volume of the body (m3) ρ = mV)

For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include isobaric (constant pressure, dP = 0) and isochoric (constant volume, dV = 0) processes, and one conventionally writes for gases:

Units shown are SI units but, of course, any consistent set of units may be used. Of course, from the above relationships, for solids one writes:

Dimensionless heat capacity

The dimensionless heat capacity of a material is

where

C is the heat capacity of a body made of the material in question (J·K−1) n is the amount of matter in the body (mol) R is the gas constant (J·K) nR=Nk is the amount of matter in the body (J·K−1) N is the number of molecules in the body. This gives:

where

CV is the heat capacity at constant volume of the gas CV,m is the molar heat capacity at constant volume of the gas N is the total number of atoms present in the container n is the number of moles of atoms present in the container (n is the ratio of N and Avogadro's number) R is the ideal gas constant, (8.314570[70] J K). R is equal to the product of Boltzmann's constant kB and Avogadro's number

The following table shows experimental molar constant volume heat capacity measurements taken for each noble monatomic gas (at 1 atm and 25 °C):

Monatomic gas	CV, m (J K)	CV, m/R
He	12.5	1.50
Ne	12.5	1.50
Ar	12.5	1.50
Kr	12.5	1.50
Xe	12.5	1.50

It is apparent from the table that the experimental heat capacities of the monatomic noble gases agrees with this simple application of statistical mechanics to a very high degree. There are three degrees of translational freedom, and two degrees of rotational freedom, therefore

Each rotational and translational degree of freedom will contribute R/2 in the total molar heat capacity of the gas. Therefore, we expect that a diatomic molecule would have a molar constant-volume heat capacity of

where the terms originate from the translational, rotational, and vibrational degrees of freedom, respectively.

The following is a table of some molar constant-volume heat capacities of various diatomic gasses

Diatomic gas	CV, m (J K)	CV, m / R
H2	20.18	2.427
CO	20.2	2.43
N2	19.9	2.39
Cl2	24.1	2.90
Br2	32.0	3.84

From the above table, clearly there is a problem with the above theory. In fact, if it is assumed that the molecules remain in their lowest energy vibrational state because the inter-level energy spacings are large, the predicted molar constant volume heat capacity for a diatomic molecule becomes

which is a fairly close approximation of the heat capacities of the lighter molecules in the above table.

Solid phase

For matter in a crystalline solid phase, the Dulong-Petit law, which was discovered empirically, states that the dimensionless specific heat capacity assumes the value 3.

The theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong-Petit limit of 3R, so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contibution that comes from potential energy that cannot be stored between separate molecules in a gas.

Heat capacity at absolute zero

From the definition of entropy

we can calculate the absolute entropy by integrating from zero temperature to the final temperature Tf

The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the third law of thermodynamics.