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Heat capacity or thermal capacity is a physical property of matter, defined as the amount of
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
to be supplied to an object to produce a unit change in its
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
. The
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
of heat capacity is
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...
per
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
(J/K). Heat capacity is an
extensive property Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
. The corresponding intensive property is the specific heat capacity, found by dividing the heat capacity of an object by its mass. Dividing the heat capacity by the amount of substance in moles yields its molar heat capacity. The volumetric heat capacity measures the heat capacity per
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
. In
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...
and
civil engineering Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...
, the heat capacity of a building is often referred to as its
thermal mass In building design, thermal mass is a property of the mass of a building that enables it to store heat and provide inertia against temperature fluctuations. It is sometimes known as the thermal flywheel effect. The thermal mass of heavy structur ...
.

# Definition

## Basic definition

The heat capacity of an object, denoted by $C$, is the limit : $C = \lim_\frac,$ where $\Delta Q$ is the amount of heat that must be added to the object (of mass ''M'') in order to raise its temperature by $\Delta T$. The value of this parameter usually varies considerably depending on the starting temperature $T$ of the object and the pressure $P$ applied to it. In particular, it typically varies dramatically with
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states o ...
s such as melting or vaporization (see enthalpy of fusion and
enthalpy of vaporization The enthalpy of vaporization (symbol ), also known as the (latent) heat of vaporization or heat of evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance to transform a quantity of that substance into a gas. T ...
). Therefore, it should be considered a function $C\left(P,T\right)$ of those two variables.

## Variation with temperature 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 of
iron Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in f ...
weighing one pound is about 204 J/K when measured from a starting temperature ''T'' = 25 °C and ''P'' = 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, ''δQ'' = ''dU'' + ''PdV'' (isobaric process In thermodynamics, an isobaric process is a type of thermodynamic process in which the pressure of the system stays constant: Δ''P'' = 0. The heat transferred to the system does work, but also changes the internal energy (''U'') of th ...)

At constant pressure, heat supplied to the system contributes to both the
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal ...
done and the change in
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
, according to the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amou ...
. The heat capacity is called $C_P.$

### At constant volume, ''dV'' = 0, ''δQ'' = ''dU'' (isochoric process In thermodynamics, an isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which the volume Volume is a measure of occupied three-dimensional space. It ...)

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 $C_V.$ The value of $C_V$ is always less than the value of $C_P.$ ($C_V$ < $C_P.$)

### Calculating ''CP'' and ''CV'' for an ideal gas

Mayer's relation: : $C_P - C_V = nR.$ : $C_P/C_V = \gamma,$ where : $n$ is the number of moles of the gas, : $R$ is the universal gas constant, : $\gamma$ is the heat capacity ratio (which can be calculated by knowing the number of
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
of the gas molecule). Using the above two relations, the specific heats can be deduced as follows: : $C_V = \frac,$ : $C_P = \gamma \frac.$

### At constant temperature ( Isothermal process)

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 an infinite 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.

### At the time of phase change (Phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states o ...)

Heat capacity of a system undergoing phase transition is infinite, because the heat is utilized in changing the state of the material rather than raising the overall temperature.

## 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 An electric motor is an electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a wire winding to generate forc ...
, 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 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 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 $C_\mathrm$ and $C_\mathrm$. For complex thermodynamic systems with several interacting parts and
state variables A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
, 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 as
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
(energy of motion) and
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
(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 In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
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 Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of t ...
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 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.

# 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 In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
L2⋅M⋅T−2⋅Θ−1. Therefore, the SI unit J/K is equivalent to
kilogram The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. ...
meter The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pre ...
squared per
second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
squared per
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
(kg⋅m2⋅s−2⋅K−1 ).

## English (Imperial) engineering units

Professionals in
construction Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and ...
,
civil engineering Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...
,
chemical engineering Chemical engineering is an engineering field which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials int ...
, and other technical disciplines, especially in the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
, may use the so-called English Engineering units, that include the pound (lb = 0.45359237 kg) as the unit of mass, the degree Fahrenheit or Rankine (°K, about 0.55556 °K) as the unit of temperature increment, and the British thermal unit (BTU ≈ 1055.06 J), Published under the auspices of the ''Verein Deutscher Ingenieure'' (VDI). as the unit of heat. In those contexts, the unit of heat capacity is 1 BTU/°R ≈ 1900 J/°K. 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 and the calorie (below).

## Calories

In chemistry, heat amounts are often measured in
calorie The calorie is a unit of energy. For historical reasons, two main definitions of "calorie" are in wide use. The large calorie, food calorie, or kilogram calorie was originally defined as the amount of heat needed to raise the temperature of on ...
s. 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 The gram (originally gramme; SI unit symbol g) is a unit of mass in the International System of Units (SI) equal to one one thousandth of a kilogram. Originally defined as of 1795 as "the absolute weight of a volume of pure water equal to th ...
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, 4184 J, exactly. 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 :: 1 cal/°C = 4.184 J/K :: 1 kcal/°C = 4184 J/K

# Negative heat capacity

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.Landau, L. D.; Lifshitz, E. M. (reprinted 2011). ''Statistical Physics Part 1'', Ch.II §21, 3rd edition, Elsevier ISBN 978-0-7506-3372-7 However, even though it can seem paradoxical at first, there are some systems for which the heat capacity $Q$/$\Delta T$ is ''negative''. Examples include a reversibly and nearly adiabatically expanding ideal gas, which cools, $\Delta T$< 0, while a small amount of heat $Q$ > 0 is put in, or combusting methane with increasing temperature, $\Delta T$> 0, and giving off heat, $Q$ < 0. 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 some nano-scale clusters of a few tens of atoms close to a phase transition. A negative heat capacity can result in a
negative temperature Certain systems can achieve negative thermodynamic temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. This should be distinguished from temperatures expressed as negative numbers ...
.

## 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 ''U''pot and the average kinetic energy ''U''kin are locked together in the relation :$U_\text = -2 U_\text.$ The total energy ''U'' (= ''U''pot + ''U''kin) therefore obeys :$U = - U_\text.$ 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.See e.g., Section 4 and onwards. A more extreme version of this occurs with
black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
s. 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 Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...
, it will become hotter and hotter until it boils away.

## Consequences

According to the
Second Law of Thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unless ...
, 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 a statistical point of view). Therefore, if such systems have equal temperatures, they are at
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
. However, this equilibrium is stable only if the systems have ''positive'' 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 with ''negative'' 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 is unstable. For example, according to theory, the smaller (less massive) a black hole is, the smaller its Schwarzschild radius will be and therefore the greater the curvature of its
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact o ...
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.

* Quantum statistical mechanics * Heat capacity ratio * Statistical mechanics * Thermodynamic equations * Thermodynamic databases for pure substances *
Heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
* Heat transfer coefficient * Heat of mixing *
Latent heat Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process — usually a first-order phase transition. Latent heat can be unders ...
*
Material properties (thermodynamics) The thermodynamic properties of materials are intensive thermodynamic parameters which are specific to a given material. Each is directly related to a second order differential of a thermodynamic potential. Examples for a simple 1-component syste ...
* Joback method (estimation of heat capacities) * Specific heat of melting (enthalpy of fusion) * Specific heat of vaporization (enthalpy of vaporization) * Volumetric heat capacity *
Thermal mass In building design, thermal mass is a property of the mass of a building that enables it to store heat and provide inertia against temperature fluctuations. It is sometimes known as the thermal flywheel effect. The thermal mass of heavy structur ...
*
R-value (insulation) In the context of construction, the R-value is a measure of how well a two-dimensional barrier, such as a layer of insulation, a window or a complete wall or ceiling, resists the conductive flow of heat. R-value is the temperature difference pe ...
*
Storage heater A storage heater or heat bank (Australia) is an electrical heater which thermal energy storage, stores thermal energy during the evening, or at night when electricity is available at lower cost, and releases the heat during the day as required. ...
* Frenkel line * Table of specific heat capacities