# Heat capacity

In thermodynamics, heat capacity is a value specific for each body defined as the ratio of the quantity of heat δQ exchanged by the system with the surroundings to the observed temperature change dT: $c = \frac{\delta Q}{dT}$

The smaller the temperature change in a body caused by the transfer of a given quantity of heat, the greater its capacity. 

History
In c.1770, Scottish chemist Joseph Black, in his lectures and experiments, is generally credited with being the inventor of heat capacity (he also later invented the concept of latent heat); the following is one synopsis of this:

“The concept of ‘heat capacity’, though imperfect, was fruitful, and destined in time to evolve into the concept of ‘specific heat’. Only a rigorous purist would deny Black the credit for this pioneering work. According to some of his lecture notes, he mentioned by way of illustration the heat capacities of four or five common substances. One determination of the heat capacities is as follows: a pound of gold at 190 degrees is put into a pound of water at 50 degrees and the final temperature of the mixture is found to be 55 degrees. As the gold has been cooled by 135 degrees and the water heated by only 5 degrees it is evident that, weight for weight, water has a much greater capacity for heat than has gold, the ratio being 19 to 1.”
— Donald Caldwell (1971), From Watt to Clausius (pg. 37) 

French chemist Antoine Lavoisier is also attributed as a second inventor of heat capacity. 

In 1780, Joao Magellan, based on data supplied by Richard Kirwan, gave a clearly stated definition of heat capacity, together with the first table of specific heat values. 

Synonyms
The terms "specific fire" (Richard Kirwan, c.1777), "specific heat" (Joao Magellan, 1780), "thermal capacity", and "specific heat capacity" all seem to be synonyms; in the same way that perfect gas, ideal perfect gas, and ideal gas are synonyms, each having a specific etymological history.

In 1805, English science writer Jane Marcet, in her sand and marble model, seemed to be visualizing things in terms of the “capacity of a body for caloric”, the sand being the caloric (or heat), the marbles being the atoms of a body. 

The term "capacity" seems to have its origin in the phlogiston theory and caloric theory of heat, to the effect that each body may had a certain "capacity" to hold heat particles (phlogiston or caloric). The modern heat-as-movement view, however, indicates that no body can contain heat or hold it they way a bottle has a certain "capacity" for an amount of liquid.

Dulong-Petit law
In 1819, French chemists Pierre Dulong and Alexis Petit defined ‘specific heat’ as the heat required to raise the temperature of a small quantity of substance by a fraction of 1 degree; using this definition, they determined the specific heat of all sorts of materials. With this data, they produced a law, now called the Dulong-Petit law, which is an equation which predicts the specific heat capacity of common materials such as lead and copper; albeit a law that was later found not to hold at the extreme ends of the temperature range. The specific heat capacity of copper at 20 K, for instance, drops to 3 percent of its room temperature value. 

Constant volume heat capacity
When heat capacity is typically measured constant volume (isochoric), the following formula can be derived: $c_V = \frac{\delta Q_V}{dT} = \left( \frac{\part U}{\part T} \right)_V$

which is the isochoric heat capacity.

Constant pressure heat capacity
When heat capacity is typically measured constant pressure (isobaric), the following formula can be derived: $c_P = \frac{\delta Q_P}{dT} = \left( \frac{\part H}{\part T} \right)_P$
or $c_P = \bigg ( \frac{ \part U}{\part T} \bigg )_P + P \bigg ( \frac{\part V}{\part T} \bigg )_P \,$

which is the isobaric heat capacity. The second term on the right-hand side represents the effect of thermal capacity of the work performed during the expansion.

Subsequently, heat capacities at constant volume can be measured by the partial derivative of internal energy U with respect to temperature and heat capacities at constant pressure can be measured by the partial derivative of enthalpy H with respect to temperature.

Other
In 1936, Italian physicist Enrico Fermi defined what he calls "thermal capacity" by the formula: $\frac{dQ}{dT} \,$

which he says is the ratio of the infinitesimal amount of heat dQ absorbed a body to the infinitesimal increase in the temperature dT. Fermi states that the thermal capacity of one gram of substance is called the "specific heat" of that substance; and the thermal capacity of one mole is called the "molecular heat" of that substance. 

Heat
Once these values have been determined via experiment, the calculation of the amount of heat Q released or absorbed by a body can be calculated by the following expression: $Q = n c \Delta T\,$

where n is the number of particles in the system and ΔT is the temperature change.

Social heat capacity

See main: Social heat capacity
In 1981, a South Korean social thinker outlined the following view: 

“Second, major concepts of thermodynamicstemperature, heat, mass, pressure, entropy, free energy, and heat capacity—must have correspondingly defined social thermodynamics conceptions. These are social temperature, social heat, social mass, social pressure, etc. These latter concepts must carry their original (thermodynamic) meaning as well as reflecting the unique characteristics of social phenomena.”

Quotes
The following are related quotes:

Black’s theory led to vague, indistinct and inaccurate notions on the subject; nor was the word ‘capacity’ well chosen. The term ‘specific heat’ is that which is more approved by later writes, particularly Dalton and John Leslie [1766-1832] (Ѻ).”
— Robertson Buchanan (1810), Practical and Descriptive Essays on the Economy of Fuel and the Management of Heat (pg. 43); cited by Donald Cardwell (1971) in From Watt to Clausius (pg. 61)

“It was well known to some philosophers that the capacity or ‘equilibrium of heat’ as we then called it, was much smaller in mercury and tin than in water.”
James Watt (1814). “Letter to David Brewster”, May

References
1. Perrot, Pierre. (1998). A to Z of Thermodynamics (pgs. 139-41). Oxford University Press.
2. Smith, J.M. Van Ness, H.C., and Abbott, M.M. (2005). Introduction to Chemical Engineering Thermodynamics (2.11: Heat Capacity, pgs. 40-44). McGraw-Hill Book Co. Inc.
3. Thomson, Thomas. (1840). An Outline of the Sciences of Heat and Electricity (pg. 54). H. Bailliere.
4. (a) de Magellan, J.H. (1780). “Essai sur law Nouvelle Theorie du Feu Elementaire, et de la Chaleur des Corps.” London.
(b) Fenby, David V. (1987). “Heat: Its measurement from Galileo to Lavoisier.” Pure & Appl. Chem., 59: 91-100.
(c) Salje, Ekhard K.H. (1988). Physical Properties and Thermodynamics Behavior of Minerals (pg. 433). Springer.
5. Shachtman, Tom. (1999). Absolute Zero and the Quest for Absolute Cold (specific heat, pg. 195). Mariner Books.
6. Fermi, Enrico. (1936). Thermodynamics (pg. 20). Prentice Hall.
7. (a) Marcet, Jane. (1805). Conversations on Chemistry (pg. 66). Philadelphia: Grigg & Elliot, 1846.
(b) Jane Marcet – Wikipedia.
8. Author. (1981). “Article”, (Korean) Social Science Journal, Vol 6-8 (thermodynamics, social phenomena, pg. 89; entropy, 8+ pgs.). Korean National Commission for Unesco.
9. (a) Black, Joseph. (1786). Lectures on the Elements of Chemistry (editor: John Robinson). Edinburgh, 1803.
(b) Talbot, Geoffroy. R. (1967). Origins and Solutions of Some Problems of Heat in the Eighteenth Century (PhD Dissertation) (Ѻ)(Ѻ) (pgs. 13.27-13.30). Manchester University.
(c) Cardwell, Donald S.L. (1971). From Watt to Clausius: the Rise of Thermodynamics in the Early Industrial Age (pg. 37). Cornell University Press.