Conjugate variables (thermodynamics)

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Conjugate variables (thermodynamics)

In thermodynamics

Thermodynamics

In physics, thermodynamics is the study of the conversion of heat energy into different forms of energy ; different energy conversions into heat energy; and its relation to macroscopic variables such as temperature, pressure, and volume....
, the internal energy

Internal energy

Thermodynamic potentials

A thermodynamic potential is a scalar potential function used to represent the thermodynamic state of a physical system. One main thermodynamic potential which has a physical interpretation is the internal energy, U....
are expressed in terms of conjugate pairs.

For a mechanical system, a small increment of energy is the product of a force times a small displacement. A similar situation exists in thermodynamics. An increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" which, when imbalanced, cause certain generalized "displacements"

Generalized coordinates

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In thermodynamics

Thermodynamics

Internal energy

Thermodynamic potentials

Generalized coordinates

By deriving equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any coordinate system that is ultimately specified." The name is a holdover from a period when Cartesian coordinates were the standard system....
, and the product of the two is the energy transferred as a result. These forces and their associated displacements are called conjugate variables. The thermodynamic force is always an intensive variable and the displacement is always an extensive variable, yielding an extensive energy transfer. The intensive (force) variable is the derivative of the internal energy with respect to the extensive (displacement) variable, while all other extensive variables are held constant.

Example

The most commonly considered conjugate thermodynamic variables are (with corresponding SI

Si, si, or SI may refer to :...
units):

Thermal parameters:

Temperature
Temperature

In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
: T (K
Kelvin

The kelvin is a Units of measurement of temperature and is one of the seven SI base units. The Kelvin scale is a Thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero ....
)
Entropy
Entropy

In many branches of science, entropy is a measure of the disorder of a system. The concept of entropy is particularly notable as it is applied across physics, information theory and mathematics....
: S (J K^-1)

Mechanical parameters:

Pressure
Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
: P (Pa
Pascal (unit)

The pascal is the SI derived unit of pressure, stress , Young's modulus and tensile strength. It is a measure of force per unit area i.e. equivalent to one newton per square meter or one joule per cubic meter....
= J m^-3)
Volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
: V (m³ = J Pa^-1)

or, more generally,

Stress
Stress (physics)

In continuum mechanics, stress is a measure of the average amount of force exerted per unit area. It is a measure of the intensity of the total internal forces acting within a body across imaginary internal surfaces, as a reaction to external applied forces and body forces....
: (Pa
Pascal (unit)

The pascal is the SI derived unit of pressure, stress , Young's modulus and tensile strength. It is a measure of force per unit area i.e. equivalent to one newton per square meter or one joule per cubic meter....
= J m^-3)
V × Strain: (m³ = J Pa^-1)

Material parameters:

chemical potential
Chemical potential

In thermodynamics, physics and chemistry, chemical potential, symbolized by ?, is a term introduced by the American engineer, chemist and mathematical physicist Willard Gibbs, which he defined as follows:...
: µ (J)
particle number
Particle number

The particle number, N, is the number of constituent particles in a Thermodynamics. The particle number is a fundamental parameter in thermodynamics and it is Conjugate variables to the chemical potential....
: N (particles or mole)

For a system with different types of particles, a small change in the internal energy is given by:

where U is internal energy, T is temperature, S is entropy, P is pressure, V is volume, is the chemical potential of the i-th particle type, and is the number of i-type particles in the system.

Here, the temperature, pressure, and chemical potential are the generalized forces, which drive the generalized changes in entropy, volume, and particle number respectively. These parameters all affect the internal energy

Internal energy

In thermodynamics, the internal energy of a thermodynamic system, or a physical body with well-defined dimension, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules and the potential energy associated with the vibrational and electricity energy of atoms within molecules or crysta...
of a thermodynamic system. A small change in the internal energy of the system is given by the sum of the flow of energy across the boundaries of the system due to the corresponding conjugate pair. These concepts will be expanded upon in the following sections.

While dealing with processes in which systems exchange matter or energy, classical thermodynamics is not concerned with the rate

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
at which such processes take place, termed kinetics

Kinetics

Kinetics, derived from the Greek language word ????s?? meaning movement or the act of moving, may refer to:...
. For this reason, the term thermodynamics is usually used synonymously with equilibrium thermodynamics. A central notion for this connection is that of quasistatic process

Quasistatic process

In thermodynamics, a quasistatic process is a thermodynamic process that happens infinitely slowly. In practice, such processes can be approximated by performing them "very slowly"....
es, namely idealized, "infinitely slow" processes. Time-dependent thermodynamic processes far away from equilibrium are studied by non-equilibrium thermodynamics

Non-equilibrium thermodynamics

Non-equilibrium thermodynamics is a branch of thermodynamics concerned with studying time-dependent thermodynamic systems, irreversible transformations and Open system ....
. This can be done through linear or non-linear analysis of irreversible processes, allowing systems near and far away from equilibrium to be studied, respectively.

The pressure/volume and stress/strain pair

As an example, consider the PV conjugate pair. The pressure

Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
acts as a generalized force - pressure differences force a change in volume

Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
, and their product is the energy lost by the system due to mechanical work

Mechanical work

In physics, mechanical work is the amount of energy transferred by a force acting through a distance. Like energy, it is a scalar quantity, with SI of joules....
. Pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables.

The above holds true only for non-viscous fluids. In the case of viscous fluids

Viscosity

Viscosity is a measure of the Drag of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness"....
, plastic

Plasticity (physics)

In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces....
and elastic

Elasticity (physics)

In physics, elasticity is the physical property of a material when it deforms under stress , but returns to its original shape when the stress is removed....
solids, the pressure force is generalized to the stress tensor

Stress tensor

For the stress tensor in classical physics, see the article* stress .For the stress tensor in theory of relativity theories, see* stress-energy tensor....
, and changes in volume are generalized to the volume multiplied by the strain tensor . These then form a conjugate pair. If is the ij component of the stress tensor, and is the ij component of the strain tensor, then the mechanical work done as the result of a stress-induced infinitesimal strain is:

or, using Einstein notation

Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulas....
for the tensors, in which repeated indices are assumed to be summed:

In the case of pure compression (i.e. no shearing forces), the stress tensor is simply the negative of the pressure times the unit tensor

Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a Function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise....
so that

The trace

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of the strain tensor is just the fractional change in volume so that the above reduces to as it should.

The temperature/entropy pair

In a similar way, temperature

Temperature

In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
differences drive changes in entropy

Entropy

In many branches of science, entropy is a measure of the disorder of a system. The concept of entropy is particularly notable as it is applied across physics, information theory and mathematics....
, and their product is the energy transferred by heat

Heat

In physics and thermodynamics, heat is any transfer of energy from one body or thermodynamic system to another due to a difference in temperature....
ing. We should note that this is the only heat term, the other terms are essentially all various forms of work.

The chemical potential/particle number pair

The chemical potential

Chemical potential

In thermodynamics, physics and chemistry, chemical potential, symbolized by ?, is a term introduced by the American engineer, chemist and mathematical physicist Willard Gibbs, which he defined as follows:...
is like a force which pushes an increase in particle number

Particle number

The particle number, N, is the number of constituent particles in a Thermodynamics. The particle number is a fundamental parameter in thermodynamics and it is Conjugate variables to the chemical potential....
. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example if a container holds water and water vapor, there will be a chemical potential (which is negative) for the liquid pushing water molecules into the vapor (evaporation) and a chemical potential for the vapor, pushing vapor molecules into the liquid (condensation). Only when these "forces" equilibrate is equilibrium obtained.

Other conjugate variables

There are many other types of conjugate variables, depending on the type of work a certain system is doing (or is being subjected to). Notations vary somewhat, but following are common.

electrical work: Ede (E= electromotive force
Electromotive force

Electromotive force is a term used to characterize electrical devices, such as voltaic cells, Thermoelectric effects, electrical generators and transformers, and even resistors....
; e amount of charge)
magnetic work MdH (M= magnetization; H = field)
surface energy: ?dA (? = surface tension ; A = surface area)
elastic stretching: FdL (F = elastic force; L length stretched)
gravitational potential energy: ?dm (? = gravitational potential; m = mass)

Conjugate variables (thermodynamics)

Example

The pressure/volume and stress/strain pair

The temperature/entropy pair

The chemical potential/particle number pair

Other conjugate variables

See also