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In
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 the ...
, an isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a
thermodynamic process Classical thermodynamics considers three main kinds of thermodynamic process: (1) changes in a system, (2) cycles in a system, and (3) flow processes. (1)A Thermodynamic process is a process in which the thermodynamic state of a system is change ...
during which the
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 ...
of the closed system undergoing such a process remains constant. An isochoric process is exemplified by the heating or the cooling of the contents of a sealed, inelastic container: The thermodynamic process is the addition or removal of heat; the isolation of the contents of the container establishes the closed system; and the inability of the container to deform imposes the constant-volume condition. The isochoric process here should be a quasi-static process.

# Formalism

An isochoric thermodynamic quasi-static process is characterized by constant
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 ...
, i.e., . The process does no pressure-volume
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 ...
, since such work is defined by $W = P \Delta V ,$ where is pressure. The sign convention is such that positive work is performed by the system on the environment. If the process is not quasi-static, the work can perhaps be done in a volume constant thermodynamic process. For a reversible process, 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 amoun ...
gives the change in the system's internal energy: $dU = dQ - dW$ Replacing
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 ...
with a change in volume gives $dU = dQ - P \, dV$ Since the process is isochoric, , the previous equation now gives $dU = dQ$ Using the definition of
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
at constant volume, , where is the mass of the gas, we get $dQ = m c_\mathrm \, dT$ Integrating both sides yields $\Delta Q\ = m \int_^ \! c_\mathrm \, dT,$ where is the specific heat capacity at constant volume, is the initial temperature and is the final temperature. We conclude with: $\Delta Q\ = m c_\mathrm \Delta T$ On a
pressure volume diagram Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, an isochoric process appears as a straight vertical line. Its thermodynamic conjugate, an
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'') o ...
would appear as a straight horizontal line.

## Ideal gas

If an ideal gas is used in an isochoric process, and the quantity of
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
stays constant, then the increase in
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
is proportional to an increase in temperature and pressure. For example a gas heated in a rigid container: the pressure and temperature of the gas will increase, but the volume will remain the same.

# Ideal Otto cycle

The ideal
Otto cycle An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. It is the thermodynamic cycle most commonly found in automobile engines. The Otto cycle is a description of what hap ...
is an example of an isochoric process when it is assumed that the burning of the
gasoline Gasoline (; ) or petrol (; ) (see ) is a transparent, petroleum-derived flammable liquid that is used primarily as a fuel in most spark-ignited internal combustion engines (also known as petrol engines). It consists mostly of organic com ...
-air mixture in an internal combustion engine car is instantaneous. There is an increase in the temperature and the pressure of the gas inside the cylinder while the volume remains the same.

# Etymology

The noun "isochor" and the adjective "isochoric" are derived from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
words ἴσος (''isos'') meaning "equal", and χώρα (''khṓra'') meaning "space."