Conjugate variables (thermodynamics)
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In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume. In fact all thermodynamic potentials 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", 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.
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[edit] Example
The most commonly considered conjugate thermodynamic variables are (with corresponding SI units):
- Thermal parameters:
- Temperature: T (K)
- Entropy: S (J K-1)
-
- or, more generally,
(
(m3 = J Pa-1)- Material parameters:
- chemical potential: μ (J)
- particle number: N (particles or mole)
For a system with different types i 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, μi is the chemical potential of the i-th particle type, and Ni 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 of a thermodynamic system. A small change dU 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 at which such processes take place, termed kinetics. For this reason, the term thermodynamics is usually used synonymously with equilibrium thermodynamics. A central notion for this connection is that of quasistatic processes, namely idealized, "infinitely slow" processes. Time-dependent thermodynamic processes far away from equilibrium are studied by non-equilibrium thermodynamics. 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.
[edit] The pressure/volume and stress/strain pair
As an example, consider the PV conjugate pair. The pressure acts as a generalized force - pressure differences force a change in volume, and their product is the energy lost by the system due to mechanical work. 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, plastic and elastic solids, the pressure force is generalized to the stress tensor, and changes in volume are generalized to the volume multiplied by the strain tensor (Landau & Lifshitz 1986). These then form a conjugate pair. If σij 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 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 so that
The trace of the strain tensor (
) is just the fractional change in volume so that the above reduces to δw = PdV as it should.
[edit] The temperature/entropy pair
In a similar way, temperature differences drive changes in entropy, and their product is the energy transferred by heating. We should note that this is the only heat term, the other terms are essentially all various forms of work.
[edit] The chemical potential/particle number pair
The chemical potential is like a force which pushes an increase in particle number. 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.
[edit] 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; e amount of charge)
- magnetic work MdH (M= magnetization; H = field)
- surface energy: γdA (γ = surface tension ; A = surface area)
- elastic stretching: FdL (F = elastic force; L length stretched)
- gravitational potential energy: ψdm (ψ = gravitational potential; m = mass)
[edit] See also
[edit] References
- Alberty, R. A. (2001). "Use of Legendre transforms in chemical thermodynamics". Pure Appl. Chem. Vol. 73 (8): 1349–1380. doi:.
- Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo (1961). Thermodynamics, 2nd Edition, New York, NY USA: McGraw-Hill Book Co.. ISBN 0-07-113809-9.
- Landau, L. D.; Lifshitz, E. M. (1986). Theory of Elasticity (Course of Theoretical Physics Volume 7), (Translated from Russian by J.B. Sykes and W.H. Reid), Third ed., Boston, MA: Butterworth Heinemann. ISBN 0-7506-2633-X.
