In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as pressure/volume or temperature/entropy. In fact all thermodynamic potentials are expressed in terms of conjugate pairs. In thermodynamics, there are a large number of equations relating the various thermodynamic quantities. ... The laws of Thermodynamics in principle describe the specifics for the transport of heat and work in thermodynamic processes. ... Pressure (symbol: p) quantifies the intensity of a force acting on a surface in a direction perpendicular to that surface. ... Volume, also called capacity, is a quantification of how much space an object occupies. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... For other senses of the term entropy, see entropy (disambiguation). ... The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. ... The particle number, N, is the number of so called elementary particles (or elementary constituents) of a thermodynamical system. ... In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials: where T = temperature, S = entropy, P = pressure, V = volume // Differential definitions Any differential change in the internal energy U of a system can be written as the sum of heat flowing into the system and work done... Maxwells relations are a set of equations in Thermodynamics which are derivable from the definitions of the four thermodynamic potentials. ... In Thermodynamics, Bridgmans Thermodynamic equations is actually a method of generating a large number of thermodynamic identities involving a number of thermodynamic quantities. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β... Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of temperature on physical systems at the macroscopic scale. ... The internal energy of a system (abbreviated E or U) is the total kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the total potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. ... In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials: where T = temperature, S = entropy, P = pressure, V = volume // Differential definitions Any differential change in the internal energy U of a system can be written as the sum of heat flowing into the system and work done...

For a mechanical system, a small increment of energy is the product of a force times a small displacement. A very 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.

For example, for a system with two types of particle, a small change in the internal energy is given by:

where U is internal energy, T is temperature, P is pressure, V is volume, μ_i is the chemical potential of the i-th particle, and N_i is the number of i-type particles in the system. In the above equation, the equal sign holds for a reversible change in the energy.

The pressure/volume pair

For 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. Pressure (symbol: p) quantifies the intensity of a force acting on a surface in a direction perpendicular to that surface. ... Volume, also called capacity, is a quantification of how much space an object occupies. ... Work (abbreviated W) is the energy transferred by a force to a moving object. ...

The temperature/entropy pair

In a similar way, temperature differences drive changes in entropy, and their product is the energy transferred by heating. Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... For other senses of the term entropy, see entropy (disambiguation). ... A red-hot iron rod cooling after being worked by a blacksmith. ...

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. The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. ... The particle number, N, is the number of so called elementary particles (or elementary constituents) of a thermodynamical system. ...