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Conservation of energy

A fundamental guiding principle of thermodynamics is the conservation of energy. The total energy of a system is the sum of its internal energy, of its potential energy as a whole system in an external force field, such as gravity, and of its kinetic energy as a whole system in motion. Thermodynamics has special concern with transfers of energy, from a body of matter, such as, for example a cylinder of steam, to the surroundings of the body, by mechanisms through which the body exerts macroscopic forces on its surroundings so as to lift a weight there; such mechanisms are the ones that are said to mediate thermodynamic work.

Besides transfer of energy as work, thermodynamics admits transfer of energy as heat. For a process in a closed (no transfer of matter) thermodynamic system, the first law of thermodynamics relates changes in the internal energy (or other cardinal energy function, depending on the conditions of the transfer) of the system to those two modes of energy transfer, as work, and as heat. Adiabatic work is done without matter transfer and without heat transfer. In principle, in thermodynamics, for a process in a closed system, the quantity of heat transferred is defined by the amount of adiabatic work that would be needed to effect the change in the system that is occasioned by the heat transfer. In experimental practice, heat transfer is often estimated calorimetrically, through change of temperature of a known quantity of calorimetric material substance.

Energy can also be transferred to or from a system through transfer of matter. The possibility of such transfer defines the system as an open system, as opposed to a closed system. By definition, such transfer is neither as work nor as heat.

Changes in the potential energy of a body as a whole with respect to forces in its surroundings, and in the kinetic energy of the body moving as a whole with respect to its surroundings, are by definition excluded from the body's cardinal energy (examples are internal energy and enthalpy).

Nearly reversible transfer of energy by work in the surroundings

In the surroundings of a thermodynamic system, external to it, all the various mechanical and non-mechanical macroscopic forms of work can be converted into each other with no limitation in principle due to the laws of thermodynamics, so that the energy conversion efficiency can approach 100% in some cases; such conversion is required to be frictionless, and consequently adiabatic. In particular, in principle, all macroscopic 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. For example, with the apparatus of Joule's experiment in which, through pulleys, a weight descending in the surroundings drives the stirring of a thermodynamic system, the descent of the weight can be diverted by a re-arrangement of pulleys, so that it lifts another weight in the surroundings, instead of stirring the thermodynamic system.

Such conversion may be idealized as nearly frictionless, though it occurs relatively quickly. It usually comes about through devices that are not simple thermodynamic systems (a simple thermodynamic system is a homogeneous body of material substances). For example, the descent of the weight in Joule's stirring experiment reduces the weight's total energy. It is described as loss of gravitational potential energy by the weight, due to change of its macroscopic position in the gravity field, in contrast to, for example, loss of the weight's internal energy due to changes in its entropy, volume, and chemical composition. Though it occurs relatively rapidly, because the energy remains nearly fully available as work in one way or another, such diversion of work in the surroundings may be idealized as nearly reversible, or nearly perfectly efficient.

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. Such energy conversion, through work done relatively rapidly, in a practical heat engine, by a thermodynamic system on its surroundings, cannot be idealized, not even nearly, as reversible.

Thermodynamic work done by a thermodynamic system on its surroundings is defined so as to comply with this principle. Historically, thermodynamics was about how a thermodynamic system could do work on its surroundings.

Work done by and on a simple thermodynamic system

Thermodynamic work and ordinary mechanical work are to be distinguished. Thermodynamic work is defined by the changes of the thermodynamic system's own internal state variables, such as volume, electric polarization, and magnetization, but excluding temperature and entropy. Ordinary mechanical work includes work done by compression, as well as shaft work, stirring, and rubbing; but shaft work, stirring, and rubbing are not thermodynamic work in so far as they do not change the volume of the system, though they change the temperature or entropy of the system. Work without change of volume is known as isochoric work, for example when friction acts on the surface or in the interior of the system.

In a process of transfer of energy from or to a thermodynamic system, the change of internal energy of the system is 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 some heating that occurred outside the system (in the substance of the paddles) by the frictional process also led to heat transfer from the paddles 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.

The amount of energy transferred as thermodynamic work is measured through quantities defined externally to the system of interest, that belong to its surroundings, but that are matched by internal state variables of the system, such as pressure. In an important sign convention, preferred in chemistry and by many physicists, thermodynamic work that adds to the internal energy of the system is counted as positive. On the other hand, for historical reasons, an oft-encountered sign convention, preferred in physics, is to consider thermodynamic work done by the system on its surroundings as positive.

Joule heating and rubbing

The definition of thermodynamic work is in terms of the changes of the system's extensive deformation (and chemical constitutive and certain other) state variables, such as volume, molar chemical constitution, or electric polarisation. Examples of state variables that are not extensive deformation or other such variables are temperature T and entropy S, as for example in the expression U = U(S, V, {Nj}). Changes of such variables are not actually physically measureable by use of a single simple adiabatic thermodynamic process; they are processes that occur neither by thermodynamic work nor by transfer of matter, and therefore are said occur by heat transfer. The quantity of thermodynamic work is defined as work done by the system on its surroundings. According to the second law of thermodynamics, such work is irreversible. To get an actual and precise physical measurement of a quantity of thermodynamic work, it is necessary to take account of the irreversibility by restoring the system to its initial condition by running a cycle, for example a Carnot cycle, that includes the target work as a step. The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle. A different cycle would be needed to actually measure the work done by the surroundings on the system. This is a reminder that rubbing the surface of a system appears to the rubbing agent in the surroundings as mechanical, though not thermodynamic, work done on the system, not as heat, but appears to the system as heat transferred to the system, not as thermodynamic work. The production of heat by rubbing is irreversible; historically, it was a piece of evidence for the rejection of the caloric theory of heat as a conserved substance. The irreversible process known as Joule heating also occurs through a change of a non-deformation extensive state variable.

Accordingly, in the opinion of Lavenda, work is not as primitive concept as is heat, which can be measured by calorimetry. This opinion does not negate the now customary thermodynamic definition of heat in terms of adiabatic work.

Known as a thermodynamic operation, the initiating factor of a thermodynamic process is, in many cases, a change in the permeability of a wall between the system and the surroundings. Rubbing is not a change in wall permeability. Kelvin's statement of the second law of thermodynamics uses the notion of an "inanimate material agency"; this notion is sometimes regarded as puzzling. The triggering of a process of rubbing can occur only in the surroundings, not in a thermodynamic system in its own state of internal thermodynamic equilibrium. Such triggering may be described as a thermodynamic operation.

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.

A main concern of thermodynamics is the properties of materials. Thermodynamic work is defined for the purposes of thermodynamic calculations about bodies of material, known as thermodynamic systems. Consequently, thermodynamic work is defined in terms of quantities that describe the states of materials, which appear as the usual thermodynamic state variables, such as volume, pressure, temperature, chemical composition, and electric polarization. For example, to measure the pressure inside a system from outside it, the observer needs the system to have a wall that can move by a measurable amount in response to pressure differences between the interior of the system and the surroundings. In this sense, part of the definition of a thermodynamic system is the nature of the walls that confine it.

Several kinds of thermodynamic work are especially important. One simple example 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.

Another important kind of work is isochoric work, i.e., 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; stirring of the system by a magnetic paddle inside it, driven by a moving magnetic field from the surroundings; and vibrational action on the system that leaves its eventual volume unchanged, but involves friction within the system. Isochoric mechanical 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 mechanical work with the physics sign convention is always negative.

When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. When mechanical work is done on such an adiabatically enclosed system by the surroundings, it can happen that friction in the surroundings is negligible, for example in the Joule experiment with the falling weight driving paddles that stir the system. Such work is adiabatic for the surroundings, even though it is associated with friction within the system. Such work may or may not be isochoric for the system, depending on the system and its confining walls. If it happens to be isochoric for the system (and does not eventually change other system state variables such as magnetization), it appears as a heat transfer to the system, and does not appear to be adiabatic for the system.

Path dependence

PV work is path-dependent and is, therefore, a thermodynamic process function. In general, the term

{\displaystyle P\,dV}

is not an exact differential. The statement that a process is quasi-static gives important information about the process but does not determine the P–V path uniquely, because the path can include several slow goings backwards and forward in volume, slowly enough to exclude friction within the system occasioned by departure from the quasi-static requirement. An adiabatic wall is one that does not permit passage of energy by conduction or radiation.

The first law of thermodynamics states that

−

{\displaystyle \Delta U=Q-W}

For a quasi-static adiabatic process,

{\displaystyle \delta Q=0}

so that

∫

{\displaystyle Q=\int \delta Q=0.}

Also

{\displaystyle \delta W=PdV}

so that

∫

{\displaystyle W=\int \delta W=\int P\,dV.}

It follows that

−

{\displaystyle dU=-\delta W}

so that

−

∫

{\displaystyle \Delta U=-\int P\,dV.}

Internal energy is a state function so its change depends only on the initial and final states of a process. For a quasi-static adiabatic process, the change in internal energy is equal to minus the integral amount of work done by the system, so the work also depends only on the initial and final states of the process and is one and the same for every intermediate path. As a result, the work done by the system also depends on the initial and final states.

If the process path is other than quasi-static and adiabatic, there are indefinitely many different paths, with significantly different work amounts, between the initial and final states. (Again the internal energy change depends only on the initial and final states as it is a state function).

In the current mathematical notation, the differential

{\displaystyle \delta W}

is an inexact differential.

In another notation, δW is written đW (with a horizontal 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.

Comparison with mechanical work by forces in contact

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 surroundings; the transfer of energy can then be considered as 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.

Comparison with shaft work

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. 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. 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

Electric field 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

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)

Magnetic work – where the force is defined by the surroundings' magnetic field strength and the generalized displacement is change of total magnetic dipole moment

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History

1824 - Carnot

1845 - Joule

1850-1865 - Clausius

Overview

Conservation of energy

Nearly reversible transfer of energy by work in the surroundings

Work done by and on a simple thermodynamic system

Processes not described by macroscopic work

Open systems

Fictively imagined reversible thermodynamic "processes"

Joule heating and rubbing

Formal definition

Sign convention

Pressure–volume work

Path dependence

Other mechanical types of work

Rotational work

Spring work

Work done on elastic solid bars

Work associated with the stretching of liquid film

Free energy and exergy

Non-mechanical forms of work

Comparison with mechanical work by forces in contact

Comparison with pressure–volume work

Comparison with shaft work

Gravitational work