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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 ...
, an isothermal process is a type of thermodynamic process in which the
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 ...
''T'' of a
system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...
remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside
thermal reservoir A thermal reservoir, also thermal energy reservoir or thermal bath, is a thermodynamic system with a heat capacity so large that the temperature of the reservoir changes relatively little when a much more significant amount of heat is added or ex ...
, and a change in the system occurs slowly enough to allow the system to be continuously adjusted to the temperature of the reservoir through
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 ...
exchange (see quasi-equilibrium). In contrast, an ''
adiabatic process In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
'' is where a system exchanges no heat with its surroundings (''Q'' = 0). Simply, we can say that in an isothermal process * $T = \text$ * $\Delta T = 0$ * $dT = 0$ * For ideal gases only, internal energy $\Delta U = 0$ while in adiabatic processes: * $Q = 0.$

# Etymology

The adjective "isothermal" is derived from the Greek words "ἴσος" ("isos") meaning "equal" and "θέρμη" ("therme") meaning "heat".

# Examples

Isothermal processes can occur in any kind of system that has some means of regulating the temperature, including highly structured machines, and even
living Living or The Living may refer to: Common meanings *Life, a condition that distinguishes organisms from inorganic objects and dead organisms ** Living species, one that is not extinct *Personal life, the course of an individual human's life * H ...
cells. Some parts of the cycles of some
heat engine In thermodynamics and engineering, a heat engine is a system that converts heat to mechanical energy, which can then be used to do mechanical work. It does this by bringing a working substance from a higher state temperature to a lower stat ...
s are carried out isothermally (for example, in the
Carnot cycle A Carnot cycle is an ideal thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. By Carnot's theorem, it provides an upper limit on the efficiency of any classical thermodyna ...
). In the thermodynamic analysis of
chemical reactions A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
, it is usual to first analyze what happens under isothermal conditions and then consider the effect of temperature. Phase changes, such as
melting Melting, or fusion, is a physical process that results in the phase transition of a substance from a solid to a liquid. This occurs when the internal energy of the solid increases, typically by the application of heat or pressure, which in ...
or
evaporation Evaporation is a type of vaporization that occurs on the surface of a liquid as it changes into the gas phase. High concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when h ...
, are also isothermal processes when, as is usually the case, they occur at constant pressure. Isothermal processes are often used as a starting point in analyzing more complex, non-isothermal processes. Isothermal processes are of special interest for ideal gases. This is a consequence of Joule's second law which states that the internal energy of a fixed amount of an ideal gas depends only on its temperature. Thus, in an isothermal process the internal energy of an ideal gas is constant. This is a result of the fact that in an ideal gas there are no intermolecular forces. Note that this is true only for ideal gases; the internal energy depends on pressure as well as on temperature for liquids, solids, and real gases. In the isothermal compression of a gas there is work done on the system to decrease the volume and increase the pressure. Doing work on the gas increases the internal energy and will tend to increase the temperature. To maintain the constant temperature energy must leave the system as heat and enter the environment. If the gas is ideal, the amount of energy entering the environment is equal to the work done on the gas, because internal energy does not change. For isothermal expansion, the energy supplied to the system does work on the surroundings. In either case, with the aid of a suitable linkage the change in gas volume can perform useful mechanical work. For details of the calculations, see calculation of work. For an
adiabatic process In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
, in which no heat flows into or out of the gas because its container is well insulated, ''Q'' = 0. If there is also no work done, i.e. a free expansion, there is no change in internal energy. For an ideal gas, this means that the process is also isothermal. Thus, specifying that a process is isothermal is not sufficient to specify a unique process.

# Details for an ideal gas

For the special case of a gas to which Boyle's law applies, the product ''pV'' (''p'' for gas pressure and ''V'' for gas volume) is a constant if the gas is kept at isothermal conditions. The value of the constant is ''nRT'', where ''n'' is the number of
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain *The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People * Abraham Moles, French engin ...
of the present gas and ''R'' is the ideal gas constant. In other words, the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
''pV'' = ''nRT'' applies. Therefore: : $p = =$ holds. The family of curves generated by this equation is shown in the graph in Figure 1. Each curve is called an isotherm, meaning a curve at a same temperature ''T''. Such graphs are termed indicator diagrams and were first used by James Watt and others to monitor the efficiency of engines. The temperature corresponding to each curve in the figure increases from the lower left to the upper right.

# Calculation of work

In thermodynamics, the reversible work involved when a gas changes from state ''A'' to state ''B'' is :$W_ = -\int_^p\,dV$ where ''p'' for gas pressure and ''V'' for gas volume. For an isothermal (constant temperature ''T''), reversible process, this integral equals the area under the relevant PV (pressure-volume) isotherm, and is indicated in purple in Figure 2 for an ideal gas. Again, ''p'' =  applies and with ''T'' being constant (as this is an isothermal process), the expression for work becomes: :$W_ = -\int_^p\,dV = -\int_^\fracdV = -nRT\int_^\fracdV = -nRT\ln$ In
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
convention, work is defined as work on a system by its surroundings. If, for example, the system is compressed, then the work is done on the system by the surrounding so the work is positive and the internal energy of the system increases. Conversely, if the system expands (i.e., system surrounding expansion, so free expansions not the case), then the work is negative as the system does work on the surroundings and the internal energy of the system decreases. It is also worth noting that for ideal gases, if the temperature is held constant, the internal energy of the system ''U'' also is constant, and so Δ''U'' = 0. Since 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 am ...
states that Δ''U'' = ''Q'' + ''W'' in
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
convention, it follows that ''Q'' = −''W'' for the isothermal compression or expansion of ideal gases.

# Example of an isothermal process

The reversible expansion of an ideal gas can be used as an example of work produced by an isothermal process. Of particular interest is the extent to which heat is converted to usable work, and the relationship between the confining
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
and the extent of expansion. During isothermal expansion of an ideal gas, both and change along an isotherm with a constant product (i.e., constant ''T''). Consider a working gas in a cylindrical chamber 1 m high and 1 m2 area (so 1m3 volume) at 400 K in
static equilibrium In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is z ...
. The surroundings consist of air at 300 K and 1 atm pressure (designated as ). The working gas is confined by a piston connected to a mechanical device that exerts a force sufficient to create a working gas pressure of 2 atm (state ). For any change in state that causes a force decrease, the gas will expand and perform work on the surroundings. Isothermal expansion continues as long as the applied force decreases and appropriate heat is added to keep = 2 tm·m3(= 2 atm × 1 m3). The expansion is said to be internally reversible if the piston motion is sufficiently slow such that at each instant during the expansion the gas temperature and pressure is uniform and conform to the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
. Figure 3 shows the relationship for = 2 tm·m3for isothermal expansion from 2 atm (state ) to 1 atm (state ). The work done (designated $W_$) has two components. First, ''expansion'' work against the surrounding atmosphere pressure (designated as ), and second, usable ''mechanical'' work (designated as ). The output here could be movement of the piston used to turn a crank-arm, which would then turn a pulley capable of lifting water out of flooded salt mines. :$W_ = -p\,V\left\left(\ln\frac\right\right) = -W_ -W_$ The system attains state ( = 2 tm·m3with = 1 atm and = 2 m3) when the applied force reaches zero. At that point, $W_$ equals –140.5 kJ, and is –101.3 kJ. By difference, = –39.1 kJ, which is 27.9% of the heat supplied to the process (- 39.1 kJ / - 140.5 kJ). This is the maximum amount of usable mechanical work obtainable from the process at the stated conditions. The percentage of is a function of and , and approaches 100% as approaches zero. To pursue the nature of isothermal expansion further, note the red line on Figure 3. The fixed value of causes an exponential increase in piston rise vs. pressure decrease. For example, a pressure decrease from 2 to 0.6969 atm causes a piston rise of 0.0526 m. In comparison, a pressure decrease from 0.39 to 1 atm causes a piston rise of 0.418 m.

# Entropy changes

Isothermal processes are especially convenient for calculating changes in
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
since, in this case, the formula for the entropy change, Δ''S'', is simply :$\Delta S = \frac$ where ''Q''rev is the heat transferred (internally reversible) to the system and ''T'' is
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
. This formula is valid only for a hypothetical reversible process; that is, a process in which equilibrium is maintained at all times. A simple example is an equilibrium phase transition (such as melting or evaporation) taking place at constant temperature and pressure. For a phase transition at constant pressure, the heat transferred to the system is equal to the enthalpy of transformation, Δ''H''tr, thus ''Q'' = Δ''H''tr. At any given pressure, there will be a transition temperature, ''T''tr, for which the two phases are in equilibrium (for example, the normal
boiling point The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. The boiling point of a liquid varies depending upon the surrounding env ...
for vaporization of a liquid at one atmosphere pressure). If the transition takes place under such equilibrium conditions, the formula above may be used to directly calculate the entropy change :$\Delta S_\text = \frac$. Another example is the reversible isothermal expansion (or compression) of an ideal gas from an initial volume ''V''A and pressure ''P''A to a final volume ''V''B and pressure ''P''B. As shown in Calculation of work, the heat transferred to the gas is :$Q = -W = n R T \ln$. This result is for a reversible process, so it may be substituted in the formula for the entropy change to obtain :$\Delta S = n R \ln \frac$. Since an ideal gas obeys Boyle's Law, this can be rewritten, if desired, as :$\Delta S = n R \ln \frac$. Once obtained, these formulas can be applied to an
irreversible process In science, a process that is not reversible is called irreversible. This concept arises frequently in thermodynamics. All complex natural processes are irreversible, although a phase transition at the coexistence temperature (e.g. melting of ...
, such as the free expansion of an ideal gas. Such an expansion is also isothermal and may have the same initial and final states as in the reversible expansion. Since entropy is a
state function In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a system ...
(that depends on an equilibrium state, not depending on a path that the system takes to reach that state), the change in entropy of the system is the same as in the reversible process and is given by the formulas above. Note that the result ''Q'' = 0 for the free expansion can not be used in the formula for the entropy change since the process is not reversible. The difference between the reversible and irreversible is found in the entropy of the surroundings. In both cases, the surroundings are at a constant temperature, ''T'', so that Δ''S''sur = −; the minus sign is used since the heat transferred to the surroundings is equal in magnitude and opposite in sign to the heat ''Q'' transferred to the system. In the reversible case, the change in entropy of the surroundings is equal and opposite to the change in the system, so the change in entropy of the universe is zero. In the irreversible, ''Q'' = 0, so the entropy of the surroundings does not change and the change in entropy of the universe is equal to ΔS for the system.