Internal energy

From Wikipedia, the free encyclopedia

 

Internal energy
Common symbols	U
SI unit	J
In SI base units	m2*kg/s2
Derivations from other quantities	${\displaystyle U=\sum _{i}E_{i}\!}$

In thermodynamics, the internal energy of a system is the energy contained within the system, excluding the kinetic energy of motion of the system as a whole and the potential energy of the system as a whole due to external force fields. It keeps account of the gains and losses of energy of the system that are due to changes in its internal state.^[1][2]

The internal energy of a system can be changed by transfers of matter or heat or by doing work.^[3] When matter transfer is prevented by impermeable containing walls, the system is said to be closed. Then the first law of thermodynamics states that the increase in internal energy is equal to the total heat added plus the work done on the system by its surroundings. If the containing walls pass neither matter nor energy, the system is said to be isolated and its internal energy cannot change. The first law of thermodynamics may be regarded as establishing the existence of the internal energy.

The internal energy is one of the two cardinal state functions of the state variables of a thermodynamic system.

Introduction

The internal energy of a given state of a system cannot be directly measured. It is determined through some convenient chain of thermodynamic operations and thermodynamic processes by which the given state can be prepared, starting with a reference state which is customarily assigned a reference value for its internal energy. Such a chain, or path, can be theoretically described by certain extensive state variables of the system, namely, its entropy, S, its volume, V, and its mole numbers, {N_j}. The internal energy, U(S,V,{N_j}), is a function of those. Sometimes, to that list are appended other extensive state variables, for example electric dipole moment. For practical considerations in thermodynamics and engineering it is rarely necessary or convenient to consider all energies belonging to the total intrinsic energy of a system, such as the energy given by the equivalence of mass. Customarily, thermodynamic descriptions include only items relevant to the processes under study. Thermodynamics is chiefly concerned only with changes in the internal energy, not with its absolute value.

The internal energy is a state function of a system, because its value depends only on the current state of the system and not on the path taken or processes undergone to prepare it. It is an extensive quantity. It is the one and only cardinal thermodynamic potential.^[4] Through it, by use of Legendre transforms, are mathematically constructed the other thermodynamic potentials. These are functions of variable lists in which some extensive variables are replaced by their conjugate intensive variables. Legendre transformation is necessary because mere substitutive replacement of extensive variables by intensive variables does not lead to thermodynamic potentials. Mere substitution leads to a less informative formula, an equation of state.

Though it is a macroscopic quantity, internal energy can be explained in microscopic terms by two theoretical virtual components. One is the microscopic kinetic energy due to the microscopic motion of the system's particles (translations, rotations, vibrations). The other is the potential energy associated with the microscopic forces, including the chemical bonds, between the particles; this is for ordinary physics and chemistry. If thermonuclear reactions are specified as a topic of concern, then the static rest mass energy of the constituents of matter is also counted. There is no simple universal relation between these quantities of microscopic energy and the quantities of energy gained or lost by the system in work, heat, or matter transfer.

The SI unit of energy is the joule (J). Sometimes it is convenient to use a corresponding density called specific internal energy which is internal energy per unit of mass (kilogram) of the system in question. The SI unit of specific internal energy is J/kg. If the specific internal energy is expressed relative to units of amount of substance (mol), then it is referred to as molar internal energy and the unit is J/mol.

From the standpoint of statistical mechanics, the internal energy is equal to the ensemble average of the sum of the microscopic kinetic and potential energies of the system.

Cardinal functions

The internal energy, U(S,V,{N_j}), expresses the thermodynamics of a system in the energy-language, or in the energy representation. Its arguments are exclusively extensive variables of state. Alongside the internal energy, the other cardinal function of state of a thermodynamic system is its entropy, as a function, S(U,V,{N_j}), of the same list of extensive variables of state, except that the entropy, S, is replaced in the list by the internal energy, U. It expresses the entropy representation.^[4][5][6]

Each cardinal function is a monotonic function of each of its natural or canonical variables. Each provides its characteristic or fundamental equation, for example U = U(S,V,{N_j}), that by itself contains all thermodynamic information about the system. The fundamental equations for the two cardinal functions can in principle be interconverted by solving, for example, U = U(S,V,{N_j}) for S, to get S = S(U,V,{N_j}).

In contrast, Legendre transforms are necessary to derive fundamental equations for other thermodynamic potentials and Massieu functions. The entropy as a function only of extensive state variables is the one and only cardinal function of state for the generation of Massieu functions. It is not itself customarily designated a 'Massieu function', though rationally it might be thought of as such, corresponding to the term 'thermodynamic potential', which includes the internal energy.^[5][7][8]

For real and practical systems, explicit expressions of the fundamental equations are almost always unavailable, but the functional relations exist in principle. Formal, in principle, manipulations of them are valuable for the understanding of thermodynamics.

Description and definition

The internal energy U of a given state of the system is determined relative to that of a standard state of the system, by adding up the macroscopic transfers of energy that accompany a change of state from the reference state to the given state:

${\displaystyle \Delta U=\sum _{i}E_{i}\,}$

where ΔU denotes the difference between the internal energy of the given state and that of the reference state, and the E_i are the various energies transferred to the system in the steps from the reference state to the given state. It is the energy needed to create the given state of the system from the reference state.

From a non-relativistic microscopic point of view, it may be divided into microscopic potential energy, U_{micro pot}, and microscopic kinetic energy, U_{micro kin}, components:

${\displaystyle U=U_{\mathrm {micro\,pot} }+U_{\mathrm {micro\,kin} }}$

The microscopic kinetic energy of a system arises as the sum of the motions of all the system's particles with respect to the center-of-mass frame, whether it be the motion of atoms, molecules, atomic nuclei, electrons, or other particles. The microscopic potential energy algebraic summative components are those of the chemical and nuclear particle bonds, and the physical force fields within the system, such as due to internal induced electric or magnetic dipole moment, as well as the energy of deformation of solids (stress-strain). Usually, the split into microscopic kinetic and potential energies is outside the scope of macroscopic thermodynamics.

Internal energy does not include the energy due to motion or location of a system as a whole. That is to say, it excludes any kinetic or potential energy the body may have because of its motion or location in external gravitational, electrostatic, or electromagnetic fields. It does, however, include the contribution of such a field to the energy due to the coupling of the internal degrees of freedom of the object with the field. In such a case, the field is included in the thermodynamic description of the object in the form of an additional external parameter.

For practical considerations in thermodynamics or engineering, it is rarely necessary, convenient, nor even possible, to consider all energies belonging to the total intrinsic energy of a sample system, such as the energy given by the equivalence of mass. Typically, descriptions only include components relevant to the system under study. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.^[9] Therefore, a convenient null reference point may be chosen for the internal energy.

The internal energy is an extensive property: it depends on the size of the system, or on the amount of substance it contains.

At any temperature greater than absolute zero, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an isolated system (cf. table). In the classical picture of thermodynamics, kinetic energy vanishes at zero temperature and the internal energy is purely potential energy. However, quantum mechanics has demonstrated that even at zero temperature particles maintain a residual energy of motion, the zero point energy. A system at absolute zero is merely in its quantum-mechanical ground state, the lowest energy state available. At absolute zero a system of given composition has attained its minimum attainable entropy.

The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. Statistical mechanics relates the pseudo-random kinetic energy of individual particles to the mean kinetic energy of the entire ensemble of particles comprising a system. Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. This energy is often referred to as the thermal energy of a system,^[10] relating this energy, like the temperature, to the human experience of hot and cold.

Statistical mechanics considers any system to be statistically distributed across an ensemble of N microstates. Each microstate has an energy E_i and is associated with a probability p_i. The internal energy is the mean value of the system's total energy, i.e., the sum of all microstate energies, each weighted by their probability of occurrence:

${\displaystyle U=\sum _{i=1}^{N}p_{i}\,E_{i}\ .}$

This is the statistical expression of the first law of thermodynamics.

Internal energy changes

Interactions of thermodynamic systems Type of system Mass flow Work Heat Open Closed Thermally isolated Mechanically isolated Isolated

Thermodynamics is chiefly concerned only with the changes, ΔU, in internal energy.

For a closed system, with matter transfer excluded, the changes in internal energy are due to heat transfer Q and due to work. The latter can be split into two kinds, pressure-volume work W_{pressure-volume}, and frictional and other kinds, such as electrical polarization, which do not alter the volume of the system, and are called isochoric, W_isochoric. Accordingly, the internal energy change ΔU for a process may be written^[3]

${\displaystyle \Delta U=Q+W_{\mathrm {pressure-volume} }+W_{\mathrm {isochoric} }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm {(closed\,\,system,\,\,no\,\,transfer\,\,of\,\,matter)} .}$

When a closed system receives energy as heat, this energy increases the internal energy. It is distributed between microscopic kinetic and microscopic potential energies. In general, thermodynamics does not trace this distribution. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be sensible.

A second mechanism of change of internal energy of a closed system is the doing of work on the system, either in mechanical form by changing pressure or volume, or by other perturbations, such as directing an electric current through the system.

If the system is not closed, the third mechanism that can increase the internal energy is transfer of matter into the system. This increase, ΔU_matter cannot be split into heat and work components. If the system is so set up physically that heat and work can be done on it by pathways separate from and independent of matter transfer, then the transfers of energy add to change the internal energy:

${\displaystyle \Delta U=Q+W_{\mathrm {pressure-volume} }+W_{\mathrm {isochoric} }+\Delta U_{\mathrm {matter} }}$

${\displaystyle \mathrm {(separate\,\,pathway\,\,for\,\,matter\,\,transfer\,\,from\,\,heat\,\,and\,\,work\,\,transfer\,\,pathways)} .}$

If a system undergoes certain phase transformations while being heated, such as melting and vaporization, it may be observed that the temperature of the system does not change until the entire sample has completed the transformation. The energy introduced into the system while the temperature did not change is called a latent energy, or latent heat, in contrast to sensible heat, which is associated with temperature change.

Internal energy of the ideal gas

Thermodynamics often uses the concept of the ideal gas for teaching purposes, and as an approximation for working systems. The ideal gas is a gas of particles considered as point objects that interact only by elastic collisions and fill a volume such that their free mean path between collisions is much larger than their diameter. Such systems are approximated by the monatomic gases, helium and the other noble gases. Here the kinetic energy consists only of the translational energy of the individual atoms. Monatomic particles do not rotate or vibrate, and are not electronically excited to higher energies except at very high temperatures.

Therefore, internal energy changes in an ideal gas may be described solely by changes in its kinetic energy. Kinetic energy is simply the internal energy of the perfect gas and depends entirely on its pressure, volume and thermodynamic temperature.

The internal energy of an ideal gas is proportional to its mass (number of moles) n and to its temperature T

${\displaystyle U=cnT,}$

where c is the heat capacity (at constant volume) of the gas. The internal energy may be written as a function of the three extensive properties S, V, n (entropy, volume, mass) in the following way ^[11][12]

${\displaystyle U(S,V,n)=const\cdot e^{\frac {S}{cn}}V^{\frac {-R}{c}}n^{\frac {R+c}{c}},}$

where const is an arbitrary positive constant and where R is the universal gas constant. It is easily seen that U is a linearly homogeneous function of the three variables (that is, it is extensive in these variables), and that it is weakly convex. Knowing temperature and pressure to be the derivatives

${\displaystyle }$
${\displaystyle T={\frac {\partial U}{\partial S}},}$ ${\displaystyle p=-{\frac {\partial U}{\partial V}},}$ the ideal gas law ${\displaystyle PV=nRT}$ immediately follows.

Internal energy of a closed thermodynamic system

This above summation of all components of change in internal energy assume that a positive energy denotes heat added to the system or work done on the system, while a negative energy denotes work of the system on the environment.

Typically this relationship is expressed in infinitesimal terms using the differentials of each term. Only the internal energy is an exact differential. For a system undergoing only thermodynamics processes, i.e. a closed system that can exchange only heat and work, the change in the internal energy is

${\displaystyle dU=\delta Q+\delta W\,}$

which constitutes the first law of thermodynamics.^{[note 1]} It may be expressed in terms of other thermodynamic parameters. Each term is composed of an intensive variable (a generalized force) and its conjugate infinitesimal extensive variable (a generalized displacement).

For example, for a non-viscous fluid, the mechanical work done on the system may be related to the pressure p and volume V. The pressure is the intensive generalized force, while the volume is the extensive generalized displacement:

${\displaystyle \delta W=-p\mathrm {d} V\,}$.

This defines the direction of work, W, to be energy flow from the working system to the surroundings, indicated by a negative term.^{[note 1]} Taking the direction of heat transfer Q to be into the working fluid and assuming a reversible process, the heat is

${\displaystyle \delta Q=T\mathrm {d} S\,}$.

${\displaystyle T}$ is temperature

${\displaystyle S}$ is entropy

and the change in internal energy becomes

${\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V\!}$

Changes due to temperature and volume

The expression relating changes in internal energy to changes in temperature and volume is

${\displaystyle dU=C_{V}dT+\left[T\left({\frac {\partial p}{\partial T}}\right)_{V}-p\right]dV\,\,{\text{ (1)}}.\,}$

This is useful if the equation of state is known.

In case of an ideal gas, we can derive that ${\displaystyle dU=C_{V}dT}$, i.e. the internal energy of an ideal gas can be written as a function that depends only on the temperature.

Changes due to temperature and pressure

When dealing with fluids or solids, an expression in terms of the temperature and pressure is usually more useful:

${\displaystyle dU=\left(C_{p}-\alpha pV\right)dT+\left(\beta _{T}p-\alpha T\right)Vdp\,}$

where it is assumed that the heat capacity at constant pressure is related to the heat capacity at constant volume according to:

${\displaystyle C_{p}=C_{V}+VT{\frac {\alpha ^{2}}{\beta _{T}}}\,}$

Changes due to volume at constant temperature

The internal pressure is defined as a partial derivative of the internal energy with respect to the volume at constant temperature:

${\displaystyle \pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}}$

Internal energy of multi-component systems

In addition to including the entropy S and volume V terms in the internal energy, a system is often described also in terms of the number of particles or chemical species it contains:

${\displaystyle U=U(S,V,N_{1},\ldots ,N_{n})\,}$

where N_j are the molar amounts of constituents of type j in the system. The internal energy is an extensive function of the extensive variables S, V, and the amounts N_j, the internal energy may be written as a linearly homogeneous function of first degree:

${\displaystyle U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots )=\alpha U(S,V,N_{1},N_{2},\ldots )\,}$

where α is a factor describing the growth of the system. The differential internal energy may be written as

${\displaystyle \mathrm {d} U={\frac {\partial U}{\partial S}}\mathrm {d} S+{\frac {\partial U}{\partial V}}\mathrm {d} V+\sum _{i}\ {\frac {\partial U}{\partial N_{i}}}\mathrm {d} N_{i}\ =T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i}\,}$

which shows (or defines) temperature T to be the partial derivative of U with respect to entropy S and pressure p to be the negative of the similar derivative with respect to volume V

${\displaystyle T={\frac {\partial U}{\partial S}},}$
${\displaystyle p=-{\frac {\partial U}{\partial V}},}$

and where the coefficients ${\displaystyle \mu _{i}}$ are the chemical potentials for the components of type i in the system. The chemical potentials are defined as the partial derivatives of the energy with respect to the variations in composition:

${\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}}$

As conjugate variables to the composition ${\displaystyle \lbrace N_{j}\rbrace }$, the chemical potentials are intensive properties, intrinsically characteristic of the qualitative nature of the system, and not proportional to its extent. Under conditions of constant T and p, because of the extensive nature of U and its independent variables, using Euler's homogeneous function theorem, the differential dU may be integrated and yields an expression for the internal energy:

${\displaystyle U=TS-pV+\sum _{i}\mu _{i}N_{i}\,}$.

The sum over the composition of the system is the Gibbs free energy:

${\displaystyle G=\sum _{i}\mu _{i}N_{i}\,}$

that arises from changing the composition of the system at constant temperature and pressure. For a single component system, the chemical potential equals the Gibbs energy per amount of substance, i.e. particles or moles according to the original definition of the unit for ${\displaystyle \lbrace N_{j}\rbrace }$.

Internal energy in an elastic medium

For an elastic medium the mechanical energy term of the internal energy must be replaced by the more general expression involving the stress ${\displaystyle \sigma _{ij}}$ and strain ${\displaystyle \varepsilon _{ij}}$. The infinitesimal statement is:

${\displaystyle \mathrm {d} U=T\mathrm {d} S+V\sigma _{ij}\mathrm {d} \varepsilon _{ij}}$

where Einstein notation has been used for the tensors, in which there is a summation over all repeated indices in the product term. The Euler theorem yields for the internal energy:^[13]

${\displaystyle U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}}$

For a linearly elastic material, the stress is related to the strain by:

${\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl}}$

where the C_ijkl are the components of the 4th-rank elastic constant tensor of the medium.

History

James Joule studied the relationship between heat, work, and temperature. He observed that if he did mechanical work on a fluid, such as water, by agitating the fluid, its temperature increased. He proposed that the mechanical work he was doing on the system was converted to thermal energy. Specifically, he found that 4185.5 joules of energy were needed to raise the temperature of a kilogram of water by one degree Celsius.^[14]

Work (thermodynamics)

From Wikipedia, the free encyclopedia

 
In thermodynamics, work performed by a system is the energy transferred by the system to its surroundings, that is fully accounted for solely by macroscopic forces exerted on the system by factors external to it, that is to say, factors in its surroundings. Thermodynamic work is a version of the concept of work in physics.

The external factors may be electromagnetic,^[1][2][3] gravitational,^[4] or pressure/volume or other simply mechanical constraints.^[5] Thermodynamic work is defined to be measurable solely from knowledge of such external macroscopic forces. These forces are associated with macroscopic state variables of the system that always occur in conjugate pairs, for example pressure and volume^[5] or magnetic flux density and magnetization.^[2] In the SI system of measurement, work is measured in joules (symbol: J). The rate at which work is performed is power.

History

1824

Work, i.e. "weight lifted through a height", was originally defined in 1824 by Sadi Carnot in his famous paper Reflections on the Motive Power of Fire, where he used the term motive power for work. Specifically, according to Carnot:

We use here motive power to express the useful effect that a motor is capable of producing. This effect can always be likened to the elevation of a weight to a certain height. It has, as we know, as a measure, the product of the weight multiplied by the height to which it is raised.

1845

Joule's apparatus for measuring the mechanical equivalent of heat

In 1845, the English physicist James Joule wrote a paper On the mechanical equivalent of heat for the British Association meeting in Cambridge.^[6] In this paper, he reported his best-known experiment, in which the mechanical power released through the action of a "weight falling through a height" was used to turn a paddle-wheel in an insulated barrel of water.

In this experiment, the friction and agitation of the paddle-wheel on the body of water caused heat to be generated which, in turn, increased the temperature of water. Both the temperature change ∆T of the water and the height of the fall ∆h of the weight mg were recorded. Using these values, Joule was able to determine the mechanical equivalent of heat. Joule estimated a mechanical equivalent of heat to be 819 ft•lbf/Btu (4.41 J/cal). The modern day definitions of heat, work, temperature, and energy all have connection to this experiment.

Overview

Thermodynamic work is performed by actions such as compression, and including shaft work, stirring, and rubbing. A simple case is work due to change of volume against a resisting pressure. Work without change of volume is known as isochoric work, for example when an outside agency, in the surroundings of the system, drives a frictional action on the surface of the system. In this case the dissipation is usually not confined to the system, and the quantity of energy so transferred as work must be estimated through the overall change of state of the system as measured by both its mechanically and externally measurable deformation variables (such as its volume), and its corresponding non-deformation variable (such as its pressure). In a process of transfer of energy as work, the change of internal energy of the system is then defined in theory by the amount of adiabatic work that would have been necessary to reach the final from the initial state, such adiabatic work being measurable only through the externally measurable mechanical or deformation variables of the system, that provide full information about the forces exerted by the surroundings on the system during the process. In the case of some of Joule's measurements, the process was so arranged that heat produced outside the system (in the paddles) by the frictional process was practically entirely transferred into the system during the process, so that the quantity of work done by the surrounds on the system could be calculated as shaft work, an external mechanical variable.^[7][8]

The amount of energy transferred as work is measured through quantities defined externally to the system of interest, and thus belonging to its surroundings. In an important sign convention, work that adds to the internal energy of the system is counted as positive. Nevertheless, on the other hand, for historical reasons, an oft-encountered sign convention is to consider work done by the system on its surroundings as positive. Although all real physical processes entail some dissipation of kinetic energy, it is a matter of definition in thermodynamics that the dissipation that results from transfer of energy as work occurs only inside the system. Energy dissipated outside the system, in the process of transfer of energy, is not counted as thermodynamic work, because it is not fully accounted for by macroscopic forces exerted on the system by external factors. Thermodynamic work does not account for any energy transferred between systems as heat or through transfer of matter.

All the various mechanical and non-mechanical forms of work can be converted into each other with no fundamental limitation due to the laws of thermodynamics, so that the energy conversion efficiency can approach 100% in some cases.^[9] In particular, all forms of work can be converted into the mechanical work of lifting a weight, which was the original form of thermodynamic work considered by Carnot and Joule (see History section above). Some authors have considered this equivalence to the lifting of a weight as a defining characteristic of work.^{[10][11][12][13]} In contrast, the conversion of heat into work in a heat engine can never exceed the Carnot efficiency, as a consequence of the second law of thermodynamics.

For a closed thermodynamic system, the first law of thermodynamics relates changes in the internal energy to two forms of energy transfer, as heat and as work. In theory, heat is properly defined for a process in a closed system (no transfer of matter) by the amount of adiabatic work that would be needed to effect the change occasioned by the process. In practice it is often estimated calorimetrically, through change of temperature of a known quantity of calorimetric material substance; it is of the essence of heat transfer that it is not mediated by the externally defined forces variables that define work. This distinction between work and heat is essential to thermodynamics.

Beyond the conceptual scope of thermodynamics proper, heat is transferred by the microscopic thermal motions of particles and their associated inter-molecular potential energies,^[14] or by radiation.^[15][16] There are two forms of macroscopic heat transfer by direct contact between a closed system and its surroundings: conduction,^[17] and radiation. There are several forms of dissipative transduction of energy that can occur internally within a system at a microscopic level, such as friction including bulk and shear viscosity,^[18] chemical reaction,^[1] unconstrained expansion as in Joule expansion and in diffusion, and phase change;^[1] these are not transfers of heat between systems.

Convection of internal energy is a form a transport of energy but is in general not, as sometimes mistakenly supposed (a relic of the caloric theory of heat), a form of transfer of energy as heat, because convection is not in itself a microscopic motion of microscopic particles or their intermolecular potential energies, or photons; nor is it a transfer of energy as work. Nevertheless, if the wall between the system and its surroundings is thick and contains fluid, in the presence of a gravitational field, convective circulation within the wall can be considered as indirectly mediating transfer of energy as heat between the system and its surroundings, though they are not in direct contact.

For an open system, the first law of thermodynamics admits three forms of energy transfer, as work, as heat, and as energy associated with matter that is transferred. The latter cannot be split uniquely into heat and work components.

Formal definition

In thermodynamics, the quantity of work done by a closed system on its surroundings is defined by factors strictly confined to the interface of the surroundings with the system and to the surroundings of the system, for example an extended gravitational field in which the system sits, that is to say, to things external to the system. There are a few especially important kinds of thermodynamic work.

A simple example of one of those important kinds is pressure–volume work. The pressure of concern is that exerted by the surroundings on the surface of the system, and the volume of interest is the negative of the increment of volume gained by the system from the surroundings. It is usually arranged that the pressure exerted by the surroundings on the surface of the system is well defined and equal to the pressure exerted by the system on the surroundings. This arrangement for transfer of energy as work can be varied in a particular way that depends on the strictly mechanical nature of pressure–volume work. The variation consists in letting the coupling between the system and surroundings be through a rigid rod that links pistons of different areas for the system and surroundings. Then for a given amount of work transferred, the exchange of volumes involves different pressures, inversely with the piston areas, for mechanical equilibrium. This cannot be done for the transfer of energy as heat because of its non-mechanical nature.^[19]

Another important kind of work is isochoric work, that is to say work that involves no eventual overall change of volume of the system between the initial and the final states of the process. Examples are friction on the surface of the system as in Rumford's experiment; shaft work such as in Joule's experiments; and slow vibrational action on the system that leaves its eventual volume unchanged, but involves friction within the system. Isochoric work for a body in its own state of internal thermodynamic equilibrium is done only by the surroundings on the body, not by the body on the surroundings, so that the sign of isochoric work with the present sign convention is always negative.

When work is done by a closed system that cannot pass heat in or out because it is adiabatically isolated, the work is referred to as being adiabatic in character. Adiabatic work can be of the pressure–volume kind or of the isochoric kind, or both.

Sign convention

Classically, a negative value of work indicates that a positive amount of work done by the system leads to energy being lost from the system. This sign convention has historically been used in many physics textbooks and will be used in the present article.^[20]

According to the first law of thermodynamics for a closed system, any net increase in the internal energy U must be fully accounted for, in terms of heat δQ entering the system and the work δW done by the system:^[14]

${\displaystyle dU=\delta Q-\delta W.\;}$ ^[21]

An alternate sign convention is to consider the work performed on the system by its surroundings as positive. This leads to a change in sign of the work, so that ${\displaystyle dU=\delta Q+\delta W\,}$. This convention has historically been used in chemistry, but has been adopted in several modern physics textbooks.^{[20][22][23][24]}

In the above, the letter d indicates an exact differential, expressing that internal energy U is a property of the state of the system; they depend only on the original state and the final state, and not upon the path taken. In contrast, the Greek deltas (δ's) in this equation reflect the fact that the heat transfer and the work transfer are not properties of the final state of the system. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. This can be summarized by saying that heat and work are not state functions of the system.^[14] This is in contrast to classical mechanics, where net work exerted by a particle is a state function.

Pressure–volume work

Pressure–volume work (or PV work) occurs when the volume V of a system changes. PV work is often measured in units of litre-atmospheres where 1L·atm = 101.325J. However, the litre-atmosphere is not a recognised unit in the SI system of units, which measures P in Pascal (Pa), V in m³, and PV in Joule (J), where 1 J = 1 Pa·m³. PV work is an important topic in chemical thermodynamics.

For a process in a closed system, occurring slowly enough for accurate definition of the pressure on the inside of the system's wall that moves and transmits force to the surroundings, described as quasi-static,^[25][26] work is represented by the following equation between differentials:

${\displaystyle \delta W=PdV\,}$

where denotes an infinitesimal increment of work done by the system, transferring energy to the surroundings;

${\displaystyle P}$ denotes the pressure inside the system, that it exerts on the moving wall that transmits force to the surroundings.^[27] In the alternative sign convention the right hand side has a negative sign.^[24]

${\displaystyle dV}$


denotes the infinitesimal increment of the volume of the system.

Moreover,

${\displaystyle W=\int _{V_{i}}^{V_{f}}P\,dV.}$

where denotes the work done by the system during the whole of the reversible process.
The first law of thermodynamics can then be expressed as

${\displaystyle dU=\delta Q-PdV\,.}$^[14]

(In the alternative sign convention where W = work done on the system, ${\displaystyle \delta W=-PdV\,}$. However, ${\displaystyle dU=\delta Q-PdV\,}$ is unchanged.)

Path dependence

As for all kinds of work, in general PV work is path-dependent and is therefore a thermodynamic process function. In general, the term P dV is not an exact differential.^[28] The statement that a process is reversible and adiabatic gives important information about the process, but does not determine the path uniquely, because the path can include several slow goings backward and forward in volume, as long as there is no transfer of energy as heat. The first law of thermodynamics states ${\displaystyle dU=\delta Q-\delta W}$. For an adiabatic process, ${\displaystyle \delta Q=0}$ and thus the integral amount work done is equal to minus the change in internal energy. For a reversible adiabatic process, the integral amount of work done during the process depends only on the initial and final states of the process, and is the one and the same for every intermediate path.

If the process took a path other than an adiabatic path, the work would be different. This would only be possible if heat flowed into/out of the system. In a non-adiabatic process, there are indefinitely many paths between the initial and final states.

In the current mathematical notation, the differential ${\displaystyle \delta W}$ is an inexact differential.^[14]

In another notation, δW is written đW (with a line through the d). This notation indicates that đW is not an exact one-form. The line-through is merely a flag to warn us there is actually no function (0-form) W which is the potential of đW. If there were, indeed, this function W, we should be able to just use Stokes Theorem to evaluate this putative function, the potential of đW, at the boundary of the path, that is, the initial and final points, and therefore the work would be a state function. This impossibility is consistent with the fact that it does not make sense to refer to the work on a point in the PV diagram; work presupposes a path.

Other mechanical types of work

There are several ways of doing mechanical work, each in some way related to a force acting through a distance.^[29] In basic mechanics, the work done by a constant force F on a body displaced a distance s in the direction of the force is given by

${\displaystyle W=Fs\,}$

If the force is not constant, the work done is obtained by integrating the differential amount of work,

${\displaystyle W=\int _{1}^{2}F\,ds.}$

Shaft work

Energy transmission with a rotating shaft is very common in engineering practice. Often the torque T applied to the shaft is constant which means that the force F applied is constant. For a specified constant torque, the work done during n revolutions is determined as follows: A force F acting through a moment arm r generates a torque T

${\displaystyle T=Fr\,}$→${\displaystyle F={\frac {T}{r}}}$

This force acts through a distance s, which is related to the radius r by

${\displaystyle s=2\left(r\pi n\right)}$

The shaft work is then determined from:

${\displaystyle W_{s}=Fs=2\pi nT}$

The power transmitted through the shaft is the shaft work done per unit time, which is expressed as

${\displaystyle {\dot {W}}_{s}=2\pi T{\dot {n}}}$

Spring work

When a force is applied on a spring, and the length of the spring changes by a differential amount dx, the work done is

${\displaystyle \partial w_{s}=Fdx}$

For linear elastic springs, the displacement x is proportional to the force applied

${\displaystyle F=Kx}$,

where K is the spring constant and has the unit of N/m. The displacement x is measured from the undisturbed position of the spring (that is, X=0 when F=0). Substituting the two equations

${\displaystyle W_{s}={\frac {1}{2}}k\left(x_{1}^{2}-x_{2}^{2}\right)}$,

where x₁ and x₂ are the initial and the final displacement of the spring respectively, measured from the undisturbed position of the spring.

Work done on elastic solid bars

Solids are often modeled as linear springs because under the action of a force they contract or elongate, and when the force is lifted, they return to their original lengths, like a spring. This is true as long as the force is in the elastic range, that is, not large enough to cause permanent or plastic deformation. Therefore, the equations given for a linear spring can also be used for elastic solid bars. Alternately, we can determine the work associated with the expansion or contraction of an elastic solid bar by replacing the pressure P by its counterpart in solids, normal stress σ=F/A in the work expansion

${\displaystyle W=\int _{1}^{2}F\,dx.}$

${\displaystyle W=\int _{1}^{2}A\sigma \,dx.}$

where A is the cross sectional area of the bar.

Work associated with the stretching of liquid film

Consider a liquid film such as a soap film suspended on a wire frame. Some force is required to stretch this film by the movable portion of the wire frame. This force is used to overcome the microscopic forces between molecules at the liquid-air interface. These microscopic forces are perpendicular to any line in the surface and the force generated by these forces per unit length is called the surface tension σ whose unit is N/m. Therefore, the work associated with the stretching of a film is called surface tension work, and is determined from

${\displaystyle W_{s}=\int _{1}^{2}\sigma _{s}\,dA.}$

where dA=2b dx is the change in the surface area of the film. The factor 2 is due to the fact that the film has two surfaces in contact with air. The force acting on the moveable wire as a result of surface tension effects is F=2b σ, where σ is the surface tension force per unit length.

Free energy and exergy

The amount of useful work which may be extracted from a thermodynamic system is determined by the second law of thermodynamics. Under many practical situations this can be represented by the thermodynamic availability, or Exergy, function. Two important cases are: in thermodynamic systems where the temperature and volume are held constant, the measure of useful work attainable is the Helmholtz free energy function; and in systems where the temperature and pressure are held constant, the measure of useful work attainable is the Gibbs free energy.

Non-mechanical forms of work

Non-mechanical work in thermodynamics is work determined by long-range forces penetrating into the system as force fields. The action of such forces can be initiated by events in the surroundings of the system, or by thermodynamic operations on the shielding walls of the system. The long-range forces are forces in the ordinary physical sense of the word, not the so-called 'thermodynamic forces' of non-equilibrium thermodynamic terminology.

The non-mechanical work of long-range forces can have either positive or negative sign, work being done by the system on the surroundings, or vice versa. Work done by long-range forces can be done indefinitely slowly, so as to approach the fictive reversible quasi-static ideal, in which entropy is not created in the system by the process.

In thermodynamics, non-mechanical work is to be contrasted with mechanical work that is done by forces in immediate contact between the system and its surroundings. If the putative 'work' of a process cannot be defined as either long-range work or else as contact work, then sometimes it cannot be described by the thermodynamic formalism as work at all. Nevertheless, the thermodynamic formalism allows that energy can be transferred between an open system and its surroundings by processes for which work is not defined. An example is when the wall between the system and its surrounds is not considered as idealized and vanishingly thin, so that processes can occur within the wall, such as friction affecting the transfer of matter across the wall; in this case, the forces of transfer are neither strictly long-range nor strictly due to contact between the system and its surrounds; the transfer of energy can then be considered as by convection, and assessed in sum just as transfer of internal energy. This is conceptually different from transfer of energy as heat through a thick fluid-filled wall in the presence of a gravitational field, between a closed system and its surroundings; in this case there may convective circulation within the wall but the process may still be considered as transfer of energy as heat between the system and its surroundings; if the whole wall is moved by the application of force from the surroundings, without change of volume of the wall, so as to change the volume of the system, then it is also at the same time transferring energy as work. A chemical reaction within a system can lead to electrical long-range forces and to electric current flow, which transfer energy as work between system and surroundings, though the system's chemical reactions themselves (except for the special limiting case in which in they are driven through devices in the surroundings so as to occur along a line of thermodynamic equilibrium) are always irreversible and do not directly interact with the surroundings of the system.^[30]

Non-mechanical work contrasts with pressure–volume work. Pressure–volume work is one of the two mainly considered kinds of mechanical contact work. A force acts on the interfacing wall between system and surroundings. The force is that due to the pressure exerted on the interfacing wall by the material inside the system; that pressure is an internal state variable of the system, but is properly measured by external devices at the wall. The work is due to change of system volume by expansion or contraction of the system. If the system expands, in the present article it is said to do positive work on the surroundings. If the system contracts, in the present article it is said to do negative work on the surroundings. Pressure–volume work is a kind of contact work, because it occurs through direct material contact with the surrounding wall or matter at the boundary of the system. It is accurately described by changes in state variables of the system, such as the time courses of changes in the pressure and volume of the system. The volume of the system is classified as a "deformation variable", and is properly measured externally to the system, in the surroundings. Pressure–volume work can have either positive or negative sign. Pressure–volume work, performed slowly enough, can be made to approach the fictive reversible quasi-static ideal.

Non-mechanical work also contrasts with shaft work. Shaft work is the other of the two mainly considered kinds of mechanical contact work. It transfers energy by rotation, but it does not eventually change the shape or volume of the system. Because it does not change the volume of the system it is not measured as pressure–volume work, and it is called isochoric work. Considered solely in terms of the eventual difference between initial and final shapes and volumes of the system, shaft work does not make a change. During the process of shaft work, for example the rotation of a paddle, the shape of the system changes cyclically, but this does not make an eventual change in the shape or volume of the system. Shaft work is a kind of contact work, because it occurs through direct material contact with the surrounding matter at the boundary of the system. A system that is initially in a state of thermodynamic equilibrium cannot initiate any change in its internal energy. In particular, it cannot initiate shaft work. This explains the curious use of the phrase "inanimate material agency" by Kelvin in one of his statements of the second law of thermodynamics. Thermodynamic operations or changes in the surroundings are considered to be able to create elaborate changes such as indefinitely prolonged, varied, or ceased rotation of a driving shaft, while a system that starts in a state of thermodynamic equilibrium is inanimate and cannot spontaneously do that.^[31] Thus the sign of shaft work is always negative, work being done on the system by the surroundings. Shaft work can hardly be done indefinitely slowly; consequently it always produces entropy within the system, because it relies on friction or viscosity within the system for its transfer.^[32] The foregoing comments about shaft work apply only when one ignores that the system can store angular momentum and its related energy.

Examples of non-mechanical work modes include

• Electrical work – where the force is defined by the surroundings' voltage (the electrical potential) and the generalized displacement is change of spatial distribution of electrical charge
• Magnetic work – where the force is defined by the surroundings' magnetic field strength and the generalized displacement is change of total magnetic dipole moment
• Electrical polarization work – where the force is defined by the surroundings' electric field strength and the generalized displacement is change of the polarization of the medium (the sum of the electric dipole moments of the molecules)
• Gravitational work – where the force is defined by the surroundings' gravitational field and the generalized displacement is change of the spatial distribution of the matter within the system.