Biological thermodynamics

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Biological thermodynamics is the quantitative study of the energy transductions that occur in or between living organisms, structures, and cells and of the nature and function of the chemical processes underlying these transductions. Biological thermodynamics may address the question of whether the benefit associated with any particular phenotypic trait is worth the energy investment it requires.

History

German-British medical doctor and biochemist Hans Krebs' 1957 book Energy Transformations in Living Matter (written with Hans Kornberg) was the first major publication on the thermodynamics of biochemical reactions. In addition, the appendix contained the first-ever published thermodynamic tables, written by Kenneth Burton, to contain equilibrium constants and Gibbs free energy of formations for chemical species, able to calculate biochemical reactions that had not yet occurred.

Non-equilibrium thermodynamics has been applied for explaining how biological organisms can develop from disorder. Ilya Prigogine developed methods for the thermodynamic treatment of such systems. He called these systems dissipative systems, because they are formed and maintained by the dissipative processes that exchange energy between the system and its environment, and because they disappear if that exchange ceases. It may be said that they live in symbiosis with their environment. Energy transformations in biology are dependent primarily on photosynthesis. The total energy captured by photosynthesis in green plants from the solar radiation is about 2 x 10²³ joules of energy per year. Annual energy captured by photosynthesis in green plants is about 4% of the total sunlight energy that reaches Earth. The energy transformations in biological communities surrounding hydrothermal vents are exceptions; they oxidize sulfur, obtaining their energy via chemosynthesis rather than photosynthesis.

The focus of thermodynamics in biology

The field of biological thermodynamics is focused on principles of chemical thermodynamics in biology and biochemistry. Principles covered include the first law of thermodynamics, the second law of thermodynamics, Gibbs free energy, statistical thermodynamics, reaction kinetics, and on hypotheses of the origin of life. Presently, biological thermodynamics concerns itself with the study of internal biochemical dynamics as: ATP hydrolysis, protein stability, DNA binding, membrane diffusion, enzyme kinetics, and other such essential energy controlled pathways. In terms of thermodynamics, the amount of energy capable of doing work during a chemical reaction is measured quantitatively by the change in the Gibbs free energy. The physical biologist Alfred Lotka attempted to unify the change in the Gibbs free energy with evolutionary theory. 

Energy transformation in biological systems

The sun is the primary source of energy for living organisms. Some living organisms like plants need sunlight directly while other organisms like humans can acquire energy from the sun indirectly. There is however evidence that some bacteria can thrive in harsh environments like Antarctica as evidence by the blue-green algae beneath thick layers of ice in the lakes. No matter what the type of living species, all living organisms must capture, transduce, store, and use energy to live. 

The relationship between the energy of the incoming sunlight and its wavelength λ or frequency ν is given by

${\displaystyle E={\frac {hc}{\lambda }}=h\nu ,}$

where h is the Planck constant (6.63x10⁻³⁴Js) and c is the speed of light (2.998x10⁸ m/s). Plants trap this energy from the sunlight and undergo photosynthesis, effectively converting solar energy into chemical energy. To transfer the energy once again, animals will feed on plants and use the energy of digested plant materials to create biological macromolecules.

Thermodynamic Theory of Evolution

The biological evolution may be explained through a thermodynamic theory. The four laws of thermodynamics are used to frame the biological theory behind evolution. The first law of thermodynamics states that states that energy can not be created or destroyed. No life can create energy but must obtain it through its environment. The second law of thermodynamics states that energy can be transformed and that occurs everyday in lifeforms. As organisms take energy from their environment they can transform it into useful energy. This is the foundation of tropic dynamics.

The general example is that the open system can be defined as any ecosystem that moves toward maximizing the dispersal of energy. All things strive towards maximum entropy production, which in terms of evolution, occurs in changes in DNA to increase biodiversity. Thus, diversity can be linked to the second law of thermodynamics. Diversity can also be argued to be a diffusion process that diffuses toward a dynamic equilibrium to maximize entropy. Therefore, thermodynamics can explain the direction and rate of evolution along with the direction and rate of succession.

Examples

First Law of Thermodynamics

The First Law of Thermodynamics is a statement of the conservation of energy; though it can be changed from one form to another, energy can be neither created nor destroyed. From the first law, a principle called Hess's Law arises. Hess’s Law states that the heat absorbed or evolved in a given reaction must always be constant and independent of the manner in which the reaction takes place. Although some intermediate reactions may be endothermic and others may be exothermic, the total heat exchange is equal to the heat exchange had the process occurred directly. This principle is the basis for the calorimeter, a device used to determine the amount of heat in a chemical reaction. Since all incoming energy enters the body as food and is ultimately oxidized, the total heat production may be estimated by measuring the heat produced by the oxidation of food in a calorimeter. This heat is expressed in kilocalories, which are the common unit of food energy found on nutrition labels.

Second Law of Thermodynamics

The Second Law of Thermodynamics is concerned primarily with whether or not a given process is possible. The Second Law states that no natural process can occur unless it is accompanied by an increase in the entropy of the universe. Stated differently, an isolated system will always tend to disorder. Living organisms are often mistakenly believed to defy the Second Law because they are able to increase their level of organization. To correct this misinterpretation, one must refer simply to the definition of systems and boundaries. A living organism is an open system, able to exchange both matter and energy with its environment. For example, a human being takes in food, breaks it down into its components, and then uses those to build up cells, tissues, ligaments, etc. This process increases order in the body, and thus decreases entropy. However, humans also 1) conduct heat to clothing and other objects they are in contact with, 2) generate convection due to differences in body temperature and the environment, 3) radiate heat into space, 4) consume energy-containing substances (i.e., food), and 5) eliminate waste (e.g., carbon dioxide, water, and other components of breath, urine, feces, sweat, etc.). When taking all these processes into account, the total entropy of the greater system (i.e., the human and her/his environment) increases. When the human ceases to live, none of these processes (1-5) take place, and any interruption in the processes (esp. 4 or 5) will quickly lead to morbidity and/or mortality. 

Gibbs Free Energy

In biological systems, in general energy and entropy change together. Therefore, it is necessary to be able to define a state function that accounts for these changes simultaneously. This state function is the Gibbs Free Energy, G.

G = H − TS

where:

• H is the enthalpy (SI unit: joule)
• T is the temperature (SI unit: kelvin)
• S is the entropy (SI unit: joule per kelvin)

The change in Gibbs Free Energy can be used to determine whether a given chemical reaction can occur spontaneously. If ∆G is negative, the reaction can occur spontaneously. Likewise, if ∆G is positive, the reaction is nonspontaneous. Chemical reactions can be “coupled” together if they share intermediates. In this case, the overall Gibbs Free Energy change is simply the sum of the ∆G values for each reaction. Therefore, an unfavorable reaction (positive ∆G₁) can be driven by a second, highly favorable reaction (negative ∆G₂ where the magnitude of ∆G₂ > magnitude of ∆G₁). For example, the reaction of glucose with fructose to form sucrose has a ∆G value of +5.5 kcal/mole. Therefore, this reaction will not occur spontaneously. The breakdown of ATP to form ADP and inorganic phosphate has a ∆G value of -7.3 kcal/mole. These two reactions can be coupled together, so that glucose binds with ATP to form glucose-1-phosphate and ADP. The glucose-1-phosphate is then able to bond with fructose yielding sucrose and inorganic phosphate. The ∆G value of the coupled reaction is -1.8 kcal/mole, indicating that the reaction will occur spontaneously. This principle of coupling reactions to alter the change in Gibbs Free Energy is the basic principle behind all enzymatic action in biological organisms.

Intensive and extensive properties

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Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one whose magnitude is independent of the size of the system whereas an extensive quantity is one whose magnitude is additive for subsystems. This reflects the corresponding mathematical ideas of mean and measure, respectively.

An intensive property is a bulk property, meaning that it is a local physical property of a system that does not depend on the system size or the amount of material in the system. Examples of intensive properties include temperature, T; refractive index, n; density, ρ; and hardness of an object, η.

By contrast, extensive properties such as the mass, volume and entropy of systems are additive for subsystems because they increase and decrease as they grow larger and smaller, respectively.

These two categories are not exhaustive, since some physical properties are neither exclusively intensive nor extensive. For example, the electrical impedance of two subsystems is additive when — and only when — they are combined in series; whilst if they are combined in parallel, the resulting impedance is less than that of either subsystem.

The terms intensive and extensive quantities were introduced by Richard C. Tolman in 1917.

Intensive properties

An intensive property is a physical quantity whose value does not depend on the amount of the substance for which it is measured. For example, the temperature of a system in thermal equilibrium is the same as the temperature of any part of it. If the system is divided, the temperature of each subsystem is identical. The same applies to the density of a homogeneous system; if the system is divided in half, the mass and the volume are both divided in half and the density remains unchanged. Additionally, the boiling point of a substance is another example of an intensive property. For example, the boiling point of water is 100 °C at a pressure of one atmosphere, which remains true regardless of quantity.

The distinction between intensive and extensive properties has some theoretical uses. For example, in thermodynamics, according to the state postulate: "The state of a simple compressible system is completely specified by two independent, intensive properties". Other intensive properties are derived from those two variables. 

Examples

Examples of intensive properties include:

• chemical potential, μ
• color
• concentration, c
• density, ρ (or specific gravity)
• magnetic permeability, μ
• melting point and boiling point
• molality, m or b
• pressure, p
• Specific conductance (or electrical conductivity)
• specific heat capacity, c_p
• specific internal energy, u
• specific rotation, [α]
• specific volume, v
• standard reduction potential, E°
• surface tension
• temperature, T
• thermal conductivity
• viscosity

Extensive properties

An extensive property is a physical quantity whose value is proportional to the size of the system it describes, or to the quantity of matter in the system. For example, the mass of a sample is an extensive quantity; it depends on the amount of substance. The related intensive quantity is density which is independent of the amount. The density of water is approximately 1g/mL whether you consider a drop of water or a swimming pool, but the mass is different in the two cases. 

Dividing one extensive property by another extensive property generally gives an intensive value—for example: mass (extensive) divided by volume (extensive) gives density (intensive). 

Examples

Examples of extensive properties include:

• amount of substance, n
• energy, E
• enthalpy, H
• entropy, S
• Gibbs energy, G
• heat capacity, C_p
• Helmholtz energy, A or F
• internal energy, U
• mass, m
• volume, V

Composite properties

The ratio of two extensive properties of the same object or system is an intensive property. For example, the ratio of an object's mass and volume, which are two extensive properties, is density, which is an intensive property.

More generally properties can be combined to give new properties, which may be called derived or composite properties. For example, the base quantities mass and volume can be combined to give the derived quantity density. These composite properties can also be classified as intensive or extensive. Suppose a composite property ${\displaystyle F}$ is a function of a set of intensive properties ${\displaystyle \{a_{i}\}}$ and a set of extensive properties ${\displaystyle \{A_{j}\}}$, which can be shown as ${\displaystyle F(\{a_{i}\},\{A_{j}\})}$. If the size of the system is changed by some scaling factor, ${\displaystyle \alpha }$, only the extensive properties will change, since intensive properties are independent of the size of the system. The scaled system, then, can be represented as ${\displaystyle F(\{a_{i}\},\{\alpha A_{j}\})}$. 

Intensive properties are independent of the size of the system, so the property F is an intensive property if for all values of the scaling factor, ${\displaystyle \alpha }$,

${\displaystyle F(\{a_{i}\},\{\alpha A_{j}\})=F(\{a_{i}\},\{A_{j}\}).\,}$

(This is equivalent to saying that intensive composite properties are homogeneous functions of degree 0 with respect to ${\displaystyle \{A_{j}\}}$.) 

It follows, for example, that the ratio of two extensive properties is an intensive property. To illustrate, consider a system having a certain mass, ${\displaystyle m}$, and volume, ${\displaystyle V}$. The density, ${\displaystyle \rho }$ is equal to mass (extensive) divided by volume (extensive): ${\displaystyle \rho ={\frac {m}{V}}}$. If the system is scaled by the factor ${\displaystyle \alpha }$, then the mass and volume become ${\displaystyle \alpha m}$ and ${\displaystyle \alpha V}$, and the density becomes ${\displaystyle \rho ={\frac {\alpha m}{\alpha V}}}$; the two ${\displaystyle \alpha }$s cancel, so this could be written mathematically as ${\displaystyle \rho (\alpha m,\alpha V)=\rho (m,V)}$, which is analogous to the equation for ${\displaystyle F}$ above. 

The property ${\displaystyle F}$ is an extensive property if for all ${\displaystyle \alpha }$,

${\displaystyle F(\{a_{i}\},\{\alpha A_{j}\})=\alpha F(\{a_{i}\},\{A_{j}\}).\,}$

(This is equivalent to saying that extensive composite properties are homogeneous functions of degree 1 with respect to ${\displaystyle \{A_{j}\}}$.) It follows from Euler's homogeneous function theorem that

${\displaystyle F(\{a_{i}\},\{A_{j}\})=\sum _{j}A_{j}\left({\frac {\partial F}{\partial A_{j}}}\right),}$

where the partial derivative is taken with all parameters constant except ${\displaystyle A_{j}}$. This last equation can be used to derive thermodynamic relations. 

Specific properties

A specific property is the intensive property obtained by dividing an extensive property of a system by its mass. For example, heat capacity is an extensive property of a system. Dividing heat capacity, C_p, by the mass of the system gives the specific heat capacity, c_p, which is an intensive property. When the extensive property is represented by an upper-case letter, the symbol for the corresponding intensive property is usually represented by a lower-case letter. Common examples are given in the table below.

Specific properties derived from extensive properties

Extensive property	Symbol	SI units	Intensive (specific) property	Symbol	SI units	Intensive (molar) property	Symbol	SI units
Volume	V	m3 or L	Specific volume*	v	m3/kg or L/kg	Molar volume	Vm	m3/mol or L/mol
Internal energy	U	J	Specific internal energy	u	J/kg	Molar internal energy	Um	J/mol
Enthalpy	H	J	Specific enthalpy	h	J/kg	Molar enthalpy	Hm	J/mol
Gibbs free energy	G	J	Specific Gibbs free energy	g	J/kg	Chemical potential	Gm or µ	J/mol
Entropy	S	J/K	Specific entropy	s	J/(kg·K)	Molar entropy	Sm	J/(mol·K)
Heat capacity at constant volume	CV	J/K	Specific heat capacity at constant volume	cV	J/(kg·K)	Molar heat capacity at constant volume	CV,m	J/(mol·K)
Heat capacity at constant pressure	CP	J/K	Specific heat capacity at constant pressure	cP	J/(kg·K)	Molar heat capacity at constant pressure	CP,m	J/(mol·K)

*Specific volume is the reciprocal of density.

If the amount of substance in moles can be determined, then each of these thermodynamic properties may be expressed on a molar basis, and their name may be qualified with the adjective molar, yielding terms such as molar volume, molar internal energy, molar enthalpy, and molar entropy. The symbol for molar quantities may be indicated by adding a subscript "m" to the corresponding extensive property. For example, molar enthalpy is H_m. Molar Gibbs free energy is commonly referred to as chemical potential, symbolized by μ, particularly when discussing a partial molar Gibbs free energy μ_i for a component i in a mixture.

For the characterization of substances or reactions, tables usually report the molar properties referred to a standard state. In that case an additional superscript ° is added to the symbol. Examples:

• V_m° = 22.41L/mol is the molar volume of an ideal gas at standard conditions for temperature and pressure.
• C_P,m° is the standard molar heat capacity of a substance at constant pressure.
• Δ_rH_m° is the standard enthalpy variation of a reaction (with subcases: formation enthalpy, combustion enthalpy...).
• E° is the standard reduction potential of a redox couple, i.e. Gibbs energy over charge, which is measured in volt = J/C.

Potential sources of confusion

The use of the term intensive is potentially confusing. The meaning here is "something within the area, length, or size of something", and often constrained by it, as opposed to "extensive", "something without the area, more than that". 

Limitations

The general validity of the division of physical properties into extensive and intensive kinds has been addressed in the course of science. Redlich noted that, although physical properties and especially thermodynamic properties are most conveniently defined as either intensive or extensive, these two categories are not all-inclusive and some well-defined physical properties conform to neither definition. Redlich also provides examples of mathematical functions that alter the strict additivity relationship for extensive systems, such as the square or square root of volume, which may occur in some contexts, albeit rarely used.

Other systems, for which standard definitions do not provide a simple answer, are systems in which the subsystems interact when combined. Redlich pointed out that the assignment of some properties as intensive or extensive may depend on the way subsystems are arranged. For example, if two identical galvanic cells are connected in parallel, the voltage of the system is equal to the voltage of each cell, while the electric charge transferred (or the electric current) is extensive. However, if the same cells are connected in series, the charge becomes intensive and the voltage extensive. The IUPAC definitions do not consider such cases.

Some intensive properties do not apply at very small sizes. For example, viscosity is a macroscopic quantity and is not relevant for extremely small systems. Likewise, at a very small scale color is not independent of size, as shown by quantum dots, whose color depends on the size of the "dot". 

Complex systems and entropy production

Ilya Prigogine’s  ground breaking work shows that every form of energy is made up of an intensive variable and an extensive variable. Measuring these two factors and taking the product of these two variables gives us an amount for that particular form of energy. If we take the energy of expansion the intensive variable is pressure (P) and the extensive variable is volume (V) we get PxV this is then the energy of expansion. Likewise one can do this for density/mass movement where density and velocity (intensive) and volume (extensive) essentially describe the energy of the movement of mass.

Other energy forms can be derived from this relationship also such as electrical, thermal, sound, springs. Within the quantum realm it appears that energy is made up of intensive factors mainly. For example frequency is intensive. It appears that as one pass to the subatomic realms the intensive factor is more dominant. The example is the quantum dot where color (intensive variable) is dictated by size, size is normally an extensive variable. There appears to be integration of these variables. This then appears as the basis of the quantum effect.

The key insight to all this is that the difference in the intensive variable gives us the entropic force and the change in the extensive variable gives us the entropic flux for a particular form of energy. A series of entropy production formula can be derived.

∆S _heat= [(1/T)_a-(1/T)_b] x ∆ thermal energy

∆S _expansion= [(pressure/T)_a-(pressure/T)_b] x ∆ volume

∆S _electric = [(voltage/T)_a-(voltage/T)_b] x ∆ current

These equations have the form

∆S_s = [(intensive)_a -(intensive)_b] x ∆ extensive
where the a and b are two different regions.

This is the long version of Prigogine’s equation

∆S_s = X_sJ_s
where X_s is the entropic force and J_s is the entropic flux.

It is possible to derive a number of different energy forms from Prigogine’s equation. 

Note that in thermal energy in the entropy production equation the intensive factor’s numerator is 1. Whilst the other equations we have a numerators of pressure and voltage and the denominator is still temperature. This means lower than the level of molecules there are no definite stable units.

Conjugate variables (thermodynamics)

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https://en.wikipedia.org/wiki/Conjugate_variables_(thermodynamics)

In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature and entropy or pressure and volume. In fact, all thermodynamic potentials are expressed in terms of conjugate pairs. The product of two quantities that are conjugate has units of energy or sometimes power.

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" that, when unbalanced, 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.

The thermodynamic square can be used as a tool to recall and derive some of the thermodynamic potentials based on conjugate variables.

In the above description, the product of two conjugate variables yields an energy. In other words, the conjugate pairs are conjugate with respect to energy. In general, conjugate pairs can be defined with respect to any thermodynamic state function. Conjugate pairs with respect to entropy are often used, in which the product of the conjugate pairs yields an entropy. Such conjugate pairs are particularly useful in the analysis of irreversible processes, as exemplified in the derivation of the Onsager reciprocal relations.

Overview

Just as a small increment of energy in a mechanical system is the product of a force times a small displacement, so an increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" which, when unbalanced, cause certain generalized "displacements" to occur, with their product being the energy transferred as a result. These forces and their associated displacements are called conjugate variables.(Alberty 2001, p. 1353) For example, consider the ${\displaystyle pV}$ conjugate pair. The pressure ${\displaystyle p}$ acts as a generalized force: Pressure differences force a change in volume ${\displaystyle \mathrm {d} V}$, and their product is the energy lost by the system due to work. Here, pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables. In a similar way, temperature differences drive changes in entropy, and their product is the energy transferred by heat transfer. The thermodynamic force is always an intensive variable and the displacement is always an extensive variable, yielding an extensive energy. The intensive (force) variable is the derivative of the (extensive) internal energy with respect to the extensive (displacement) variable, with all other extensive variables held constant.

The theory of thermodynamic potentials is not complete until one considers the number of particles in a system as a variable on par with the other extensive quantities such as volume and entropy. The number of particles is, like volume and entropy, the displacement variable in a conjugate pair. The generalized force component of this pair is the chemical potential. The chemical potential may be thought of as a force which, when imbalanced, pushes an exchange of particles, either with the surroundings, or between phases inside the system. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example, if a container holds liquid water and water vapor, there will be a chemical potential (which is negative) for the liquid which pushes the 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, and the chemical potential of each phase is equal, is equilibrium obtained. 

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

Thermal parameters:

• Temperature: ${\displaystyle T}$  (K)
• Entropy: ${\displaystyle S}$  (J K⁻¹)

Mechanical parameters:

• Pressure: ${\displaystyle p}$  (Pa= J m⁻³)
• Volume: ${\displaystyle V}$  (m³ = J Pa⁻¹)

or, more generally,

• Stress: ${\displaystyle \sigma _{ij}\,}$ (Pa= J m⁻³)
• Volume × Strain: ${\displaystyle V\times \varepsilon _{ij}}$ (m³ = J Pa⁻¹)

Material parameters:

• chemical potential: ${\displaystyle \mu }$ (J)
• particle number: ${\displaystyle N}$   (particles or mole)

For a system with different types ${\displaystyle i}$ of particles, a small change in the internal energy is given by:

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

where ${\displaystyle U}$ is internal energy, ${\displaystyle T}$ is temperature, ${\displaystyle S}$ is entropy, ${\displaystyle p}$ is pressure, ${\displaystyle V}$ is volume, ${\displaystyle \mu _{i}}$ is the chemical potential of the ${\displaystyle i}$-th particle type, and ${\displaystyle N_{i}}$ is the number of ${\displaystyle i}$-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 ${\displaystyle \mathrm {d} U}$ 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. 

Pressure/volume and stress/strain pairs

As an example, consider the ${\displaystyle pV}$ 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 ${\displaystyle \sigma _{ij}}$ is the ij component of the stress tensor, and ${\displaystyle \varepsilon _{ij}}$ is the ij component of the strain tensor, then the mechanical work done as the result of a stress-induced infinitesimal strain ${\displaystyle \mathrm {\varepsilon } _{ij}}$ is:

${\displaystyle \delta w=V\sum _{ij}\sigma _{ij}\,\mathrm {d} \varepsilon _{ij}}$

or, using Einstein notation for the tensors, in which repeated indices are assumed to be summed:

${\displaystyle \delta w=V\sigma _{ij}\,\mathrm {d} \varepsilon _{ij}}$

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

${\displaystyle \delta w=V\,(-p\delta _{ij})\,\mathrm {d} \varepsilon _{ij}=-\sum _{k}pV\,\mathrm {d} \varepsilon _{kk}}$

The trace of the strain tensor (${\displaystyle \varepsilon _{kk}}$) is the fractional change in volume so that the above reduces to ${\displaystyle \delta w=-p\mathrm {d} V}$ as it should. 

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 driving force, entropy is the associated displacement, and the two form a pair of conjugate variables. The temperature/entropy pair of conjugate variables is the only heat term; the other terms are essentially all various forms of work. 

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.