Heat capacity is a property of a thermodynamic system that measures the amount of heat needed to produce a given increase in temperature under specified conditions. Denoted by (C), it is a central quantity in thermodynamics. A system with a larger heat capacity requires more heat for the same temperature rise. Heat capacity depends on the amount of material, its physical state, and the constraints imposed during heating, particularly whether volume or pressure remains constant. (goldbook.iupac.org)
Definition and units
For an infinitesimal temperature change along a specified heating process,
[ C=\frac{\delta Q}{dT}, ]
where (\delta Q) is the heat supplied. The notation (\delta Q), rather than (dQ), emphasizes that heat is energy transferred during a process, not a state function stored in a body. Consequently, the heating conditions must be specified before this ratio defines a material property. (goldbook.iupac.org)
In the International System of Units, heat capacity is measured in joules per kelvin, (\mathrm{J,K^{-1}}). It is an extensive property: doubling the quantity of a homogeneous material under the same conditions approximately doubles its heat capacity. The related specific heat capacity, (c=C/m), is heat capacity per unit mass, with units (\mathrm{J,kg^{-1},K^{-1}}). Molar heat capacity, (C_m=C/n), is heat capacity per mole, with units (\mathrm{J,mol^{-1},K^{-1}}). These normalized quantities allow comparisons between different substances. (openstax.org)
When heat capacity is effectively constant over a temperature interval, the transferred heat is approximated by (Q=C\Delta T=mc\Delta T). If it varies appreciably, the corresponding expression is
[ Q=\int_{T_1}^{T_2}C(T),dT, ]
with the same constraints maintained throughout and any phase-change contributions treated separately. (openstax.org)
Constant-volume and constant-pressure heat capacities
Two standard heat capacities are defined through partial derivatives:
[ C_V=\left(\frac{\partial U}{\partial T}\right)_V, \qquad C_p=\left(\frac{\partial H}{\partial T}\right)_p, ]
where (U) is internal energy and (H=U+pV) is enthalpy. Composition and amount of substance are held fixed. For a closed system doing only pressure–volume work, the first law of thermodynamics identifies heat supplied at constant volume with the change in internal energy, and heat supplied at constant pressure with the change in enthalpy. (goldbook.iupac.org)
For an ideal gas, heating at constant pressure causes expansion. Some supplied heat therefore supports work on the surroundings, so more heat is needed than at constant volume for the same temperature increase. The resulting relation is
[ C_p-C_V=nR, ]
or (C_{p,m}-C_{V,m}=R) on a molar basis, where (R) is the gas constant. This relation does not generally apply to real gases, liquids, or solids. (openstax.org)
Microscopic interpretation
Statistical mechanics explains heat capacity through changes in the average energy of microscopic constituents. Heating can increase translational motion, molecular rotation, vibration, or other excitations. The number and accessibility of these energy-storage mechanisms determine the temperature response. (openstax.org)
In the classical limit, the equipartition theorem assigns an average energy (k_BT/2) to each independent quadratic contribution to energy. A monatomic ideal gas has three translational contributions, giving
[ C_{V,m}=\frac32R,\qquad C_{p,m}=\frac52R. ]
For many diatomic gases near room temperature, two rotational contributions also participate, yielding approximately (C_{V,m}=5R/2). At higher temperatures, molecular vibration can increase the heat capacity further. (openstax.org)
Quantum mechanics explains why classical predictions fail at sufficiently low temperatures. When the spacing between excitation energies is large compared with (k_BT), thermal excitation is suppressed. Such modes contribute little to heat capacity; their contributions become significant as temperature rises. (openstax.org)
Solids and temperature dependence
The classical Dulong–Petit law predicts a molar constant-volume heat capacity approaching (3R) for a monatomic crystalline solid. This follows from the kinetic and potential energy contributions of atomic vibration in three dimensions. Many elemental solids approach this value at sufficiently high temperatures, but substantial departures occur at low temperatures. (openstax.org)
The Einstein model represents atomic vibrations by oscillators with a common frequency. The Debye model instead includes a spectrum of collective vibrations, whose quantized excitations are phonons. For ordinary three-dimensional insulating crystals at temperatures well below the characteristic Debye temperature, the lattice contribution follows approximately (C_V\propto T^3). Both models recover the classical high-temperature limit, while the Debye model better describes the low-temperature lattice behavior. (ocw.mit.edu)
A phase transition requires special treatment. During equilibrium melting or boiling of a pure substance at fixed pressure, heat can be absorbed without a temperature increase. This latent heat is distinct from the heat that raises temperature within a single phase; a finite, constant heat capacity cannot describe the entire transition. (openstax.org)
Measurement and uses
Heat capacity is measured by calorimetry, relating a known heat input to the resulting temperature change. Measurements must account for the apparatus’s own heat capacity and heat exchange with the surroundings. In differential scanning calorimetry, a sample and reference are subjected to a controlled temperature program, and their differential thermal response is recorded. Calibrated measurements provide heat-capacity data and reveal thermal transitions. (openstax.org)
Heat-capacity data are used to calculate heating requirements and interpret thermal behavior. Liquid water, for example, has a specific heat capacity of approximately (4.18\ \mathrm{kJ,kg^{-1},K^{-1}}) near ordinary temperatures, considerably greater than that of many common metals. For equal masses receiving equal heat inputs without changing phase, water therefore undergoes a smaller temperature increase. Reliable tabulations specify the temperature range, physical phase, and measurement constraints. (openstax.org)