Heat capacity orthermal capacity is aphysical property ofmatter, defined as the amount ofheat that must be supplied to an object to produce a unit change in itstemperature.[1] TheSI unit of heat capacity isjoule perkelvin (J/K). It quantifies the ability of a material or system to store thermal energy.
The heat capacity of an object, denoted by, is the limitwhere is the amount of heat that must be added to the object (of massM) in order to raise its temperature by.
The value of this parameter usually varies considerably depending on the starting temperature of the object and the pressure applied to it. In particular, it typically varies dramatically withphase transitions such as melting or vaporization (seeenthalpy of fusion andenthalpy of vaporization). Therefore, it should be considered a function of those two variables.
The variation can be ignored in contexts when working with objects in narrow ranges of temperature and pressure. For example, the heat capacity of a block ofiron weighing onepound is about 204 J/K when measured from a starting temperatureT = 25 °C andP = 1 atm of pressure. That approximate value is adequate for temperatures between 15 °C and 35 °C, and surrounding pressures from 0 to 10 atmospheres, because the exact value varies very little in those ranges. One can trust that the same heat input of 204 J will raise the temperature of the block from 15 °C to 16 °C, or from 34 °C to 35 °C, with negligible error.
Heat capacities of a homogeneous system undergoing different thermodynamic processes
At constant pressure, heat supplied to the system contributes to both thework done and the change ininternal energy, according to thefirst law of thermodynamics. The heat capacity is called and defined as:
A system undergoing a process at constant volume implies that no expansion work is done, so the heat supplied contributes only to the change in internal energy. The heat capacity obtained this way is denoted The value of is always less than the value of. (.)
Expressing the inner energy as a function of the variables and gives:
Using the above two relations, the specific heats can be deduced as follows:
Following from theequipartition of energy, it is deduced that an ideal gas has the isochoric heat capacity
where is the number ofdegrees of freedom of each individual particle in the gas, and is the number ofinternal degrees of freedom, where the number 3 comes from the three translational degrees of freedom (for a gas in 3D space). This means that amonoatomic ideal gas (with zero internal degrees of freedom) will have isochoric heat capacity.
No change in internal energy (as the temperature of the system is constant throughout the process) leads to only work done by the total supplied heat, and thus aninfinite amount of heat is required to increase the temperature of the system by a unit temperature, leading to infinite or undefined heat capacity of the system.
Heat capacity of a system undergoing phase transition isinfinite, because the heat is utilized in changing the state of the material rather than raising the overall temperature.
The heat capacity may be well-defined even for heterogeneous objects, with separate parts made of different materials; such as anelectric motor, acrucible with some metal, or a whole building. In many cases, the (isobaric) heat capacity of such objects can be computed by simply adding together the (isobaric) heat capacities of the individual parts.
However, this computation is valid only when all parts of the object are at the same external pressure before and after the measurement. That may not be possible in some cases. For example, when heating an amount of gas in an elastic container, its volumeand pressure will both increase, even if the atmospheric pressure outside the container is kept constant. Therefore, the effective heat capacity of the gas, in that situation, will have a value intermediate between its isobaric and isochoric capacities and.
For complexthermodynamic systems with several interacting parts andstate variables, or for measurement conditions that are neither constant pressure nor constant volume, or for situations where the temperature is significantly non-uniform, the simple definitions of heat capacity above are not useful or even meaningful. The heat energy that is supplied may end up askinetic energy (energy of motion) andpotential energy (energy stored in force fields), both at macroscopic and atomic scales. Then the change in temperature will depend on the particular path that the system followed through itsphase space between the initial and final states. Namely, one must somehow specify how the positions, velocities, pressures, volumes, etc. changed between the initial and final states; and use the general tools ofthermodynamics to predict the system's reaction to a small energy input. The "constant volume" and "constant pressure" heating modes are just two among infinitely many paths that a simple homogeneous system can follow.
The heat capacity can usually be measured by the method implied by its definition: start with the object at a known uniform temperature, add a known amount of heat energy to it, wait for its temperature to become uniform, and measure the change in its temperature. This method can give moderately accurate values for many solids; however, it cannot provide very precise measurements, especially for gases.
The SI unit for heat capacity of an object is joule per kelvin (J/K or J⋅K−1). Since an increment of temperature of onedegree Celsius is the same as an increment of one kelvin, that is the same unit as J/°C.
The heat capacity of an object is an amount of energy divided by a temperature change, which has thedimension L2⋅M⋅T−2⋅Θ−1. Therefore, the SI unit J/K is equivalent tokilogrammeter squared persecond squared perkelvin (kg⋅m2⋅s−2⋅K−1 ).
Professionals inconstruction,civil engineering,chemical engineering, and other technical disciplines, especially in theUnited States, may use the so-calledEnglish Engineering units, that include thepound (lb = 0.45359237 kg) as the unit of mass, thedegree Fahrenheit orRankine (5/9K, about 0.55556 K) as the unit of temperature increment, and theBritish thermal unit (BTU ≈ 1055.06 J),[3][4] as the unit of heat. In those contexts, the unit of heat capacity is 1 BTU/°R ≈ 1900 J/K.[5] The BTU was in fact defined so that the average heat capacity of one pound of water would be 1 BTU/°F. In this regard, with respect to mass, note conversion of 1 Btu/lb⋅°R ≈ 4,187 J/kg⋅K[6] and the calorie (below).
In chemistry, heat amounts are often measured incalories. Confusingly, two units with that name, denoted "cal" or "Cal", have been commonly used to measure amounts of heat:
The "small calorie" (or "gram-calorie", "cal") is exactly 4.184 J. It was originally defined so that the heat capacity of 1 gram of liquid water would be 1 cal/°C.
The "grand calorie" (also "kilocalorie", "kilogram-calorie", or "food calorie"; "kcal" or "Cal") is 1000 cal, that is, exactly 4184 J. It was originally defined so that the heat capacity of 1 kg of water would be 1 kcal/°C.
With these units of heat energy, the units of heat capacity are
Most physical systems exhibit a positive heat capacity; constant-volume and constant-pressure heat capacities, rigorously defined as partial derivatives, are always positive for homogeneous bodies.[7] However, even though it can seem paradoxical at first,[8][9] there are some systems for which the heat capacity isnegative. Examples include a reversibly and nearly adiabatically expanding ideal gas, which cools,, while a small amount of heat is put in, or combusting methane with increasing temperature,, and giving off heat,. Others are inhomogeneous systems that do not meet the strict definition of thermodynamic equilibrium. They include gravitating objects such as stars and galaxies, and also somenano-scale clusters of a few tens of atoms close to a phase transition.[10] A negative heat capacity can result in anegative temperature.
According to thevirial theorem, for a self-gravitating body like a star or an interstellar gas cloud, the average potential energyUpot and the average kinetic energyUkin are locked together in the relation
The total energyU (=Upot +Ukin) therefore obeys
If the system loses energy, for example, by radiating energy into space, the average kinetic energy actually increases. If a temperature is defined by the average kinetic energy, then the system therefore can be said to have a negative heat capacity.[11]
A more extreme version of this occurs withblack holes. According toblack-hole thermodynamics, the more mass and energy a black hole absorbs, the colder it becomes. In contrast, if it is a net emitter of energy, throughHawking radiation, it will become hotter and hotter until it boils away.
According to thesecond law of thermodynamics, when two systems with different temperatures interact via a purely thermal connection, heat will flow from the hotter system to the cooler one (this can also be understood from astatistical point of view). Therefore, if such systems have equal temperatures, they are atthermal equilibrium. However, this equilibrium is stable only if the systems havepositive heat capacities. For such systems, when heat flows from a higher-temperature system to a lower-temperature one, the temperature of the first decreases and that of the latter increases, so that both approach equilibrium. In contrast, for systems withnegative heat capacities, the temperature of the hotter system will further increase as it loses heat, and that of the colder will further decrease, so that they will move farther from equilibrium. This means that the equilibrium isunstable.
For example, according to theory, the smaller (less massive) a black hole is, the smaller itsSchwarzschild radius will be, and therefore the greater thecurvature of itsevent horizon will be, as well as its temperature. Thus, the smaller the black hole, the more thermal radiation it will emit and the more quickly it will evaporate byHawking radiation.
