# Heat capacity

Heat capacity or thermal capacity is a physical property of a material object, defined as the amount of energy (in the form of heat) that must be added to (or removed from) the object in order to achieve a small change in its temperature, divided by the magnitude of that change.[1] Informally, it is the amount of heat energy that must be provided to the object in order to uniformly raise its temperature by one unit.

The SI unit of heat capacity is joule per kelvin (J/K). However, several other units of measure have been used for this quantity in the past, and are still used in certain contexts.

While some part of the object is undergoing a phase transition, such as melting or boiling, its heat capacity is technically infinite, because the heat goes into changing the state of that part, rather than raising the overall temperature.

Heat capacity is an extensive property of matter, meaning that it is proportional to the size of the object. To express the corresponding intensive property of a substance, the heat capacity of a sample is divided by the mass of the sample, yielding the specific heat capacity (or "specific heat"). Dividing by the amount of substance in moles yields its molar heat capacity. The volumetric heat capacity measures the heat capacity per volume.

The concept is also called thermal mass. This term is often used in architecture and civil engineering to refer to the heat capacity of a building.

## Definition

The heat capacity of an object, usually denoted by ${\displaystyle C}$ , is the limit

${\displaystyle C=\lim _{\Delta T\to 0}{\frac {\Delta Q}{\Delta T}},}$

where ${\displaystyle \Delta Q}$  is the amount of heat that must be added to the object in order to raise its temperature by ${\displaystyle \Delta T}$ .

The value of this parameter usually varies considerably depending on the starting temperature ${\displaystyle T}$  of the object and the pressure ${\displaystyle P}$  applied to it. Therefore, it should be considered a function ${\displaystyle C(P,T)}$  of those two variables.

However, the variation can often be ignored in practical contexts, e.g. when working with solid objects in narrow ranges of temperature and pressure. In those contexts one usually omits the qualifier ${\displaystyle (P,T)}$ , and approximates the function by a constant ${\displaystyle C}$  suitable for those ranges.

For example, the heat capacity ${\displaystyle C(P,T)}$  of a block of iron weighing one pound is about 204 J/K when measured from a starting temperature ${\displaystyle T={}}$ 25°C and ${\displaystyle P={}}$ 1 atm of ambient pressure. That approximate value is quite adequate for all temperatures between, say, 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 capacity is an "extensive" property, meaning it depends on the extent or size of the object or the amount of matter considered. A block of iron twice as big as another block requires the transfer of twice the amount of heat (${\displaystyle 2\Delta Q}$ ) to achieve the same change in temperature (${\displaystyle \Delta T}$ ).

### Variations

The injection of heat energy into a material object (such as a sample of some substance), besides raising its temperature, usually causes an increase in its volume and/or its pressure, depending on how the object is confined. The choice made about the latter affects the capacity measured, even for the same starting pressure ${\displaystyle P}$  and starting temperature ${\displaystyle T}$ . For a simple homogeneous object (such as a sample of some substance or material), two particular choices are widely used:

• If the external pressure is kept constant (for instance, at the ambient atmospheric pressure), and the object is allowed to expand, the expansion generates work as the force from the pressure displaces the enclosure or the surrounding air. That work must come from the heat energy provided. The heat capacity thus obtained is said to be measured at constant pressure (or isobaric), and is often denoted ${\displaystyle C_{P}}$ , ${\displaystyle C_{p}}$ , ${\displaystyle C_{\mathrm {p} }}$ , 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 expansion — 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 heat capacity obtained this way is said to be measured at constant volume (or isochoric) and denoted ${\displaystyle C_{V}}$ , ${\displaystyle C_{v}}$  ${\displaystyle C_{\mathrm {v} }}$ , etc. The value of ${\displaystyle C_{V}}$  is usually less than the value of ${\displaystyle C_{P}}$ .

If an object is homogeneous, its heat capacity can be obtained by multiplying its specific heat capacity ${\displaystyle c}$  by its mass ${\displaystyle M}$ :

${\displaystyle C_{V}=c_{V}M\quad \quad \quad \quad C_{P}=c_{P}M}$

### Heterogeneous objects

The heat capacity may be well-defined even for heterogeneous objects, with separate parts made of different materials; such as an electric motor, a crucible 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 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 volume and 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 ${\displaystyle C_{P}}$  and ${\displaystyle C_{V}}$ .

For complex thermodynamic systems with several interacting parts and state variables, or for measurement condiditions 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 as kinetic energy (energy of motion) and potential energy (energy stored in force fields), both at macroscopic and atomic scales. Then the change in temperature will depends on the particular path that the system followed through its phase 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 of thermodynamics 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.

## Measurement

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 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.

## Units

### International system

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 one degree 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 the dimension L2·M·T−2·Θ−1. Therefore, the SI unit J/K is equivalent to kilogram metre squared per second squared per kelvin (kg m2 s−2 K−1 ).

### English (Imperial) engineering units

Professionals in construction, civil engineering, chemical engineering, and other technical disciplines, especially in the United States, may use the so-called English Engineering units, that include the Imperial pound (lb = 0.45459237 kg) as the unit of mass, the degree Fahrenheit or Rankine (5/9 K, about 0.55556 K) as the unit of temperature increment, and the British thermal unit (BTU ≈ 1055.06 J),[2][3] as the unit of heat. In those contexts, the unit of heat capacity is BTU/°F ≈ 1900 J. The BTU was in fact defined so that the average heat capacity of one pound of water would be 1 BTU/F°.

### Calories

In chemistry, heat amounts are 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" (or "gram-calorie", "cal") is 4.184 J, exactly. 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 small calories, that is, 4184 J, exactly. It was originallly 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

1 cal/°C ("small calorie") = 4.184 J/K
1 kcal/°C ("large calorie") = 4184 J/K

## Negative heat capacity

Most physical systems exhibit a positive heat capacity. However, even though it can seem paradoxical at first,[4][5] there are some systems for which the heat capacity is negative. These are inhomogeneous systems that do not meet the strict definition of thermodynamic equilibrium. They include gravitating objects such as stars and galaxies, and also sometimes some nano-scale clusters of a few tens of atoms, close to a phase transition.[6] A negative heat capacity can result in a negative temperature.

### Stars and black holes

According to the virial theorem, for a self-gravitating body like a star or an interstellar gas cloud, the average potential energy Upot and the average kinetic energy Ukin are locked together in the relation

${\displaystyle U_{\text{pot}}=-2U_{\text{kin}}.}$

The total energy U (= Upot + Ukin) therefore obeys

${\displaystyle U=-U_{\text{kin}}.}$

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.[7]

A more extreme version of this occurs with black holes. According to black-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, through Hawking radiation, it will become hotter and hotter until it boils away.