The Gibbs–Helmholtz equation is a

thermodynamic Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of t ...

equation used for calculating changes in the

Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work (physics), work that may be performed by a closed system, thermodynamically closed system a ...

of a system as a function of

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on ...

. It was originally presented in an 1882 paper entitled " Die Thermodynamik chemischer Vorgange" by

Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...

. It describes how the Gibbs free energy, which was presented originally by

Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...

, varies with temperature. The equation is:Physical chemistry,

P. W. Atkins Peter William Atkins (born 10 August 1940) is an English chemist and a Fellow of Lincoln College at the University of Oxford. He retired in 2007. He is a prolific writer of popular chemistry textbooks, including ''Physical Chemistry'', ''Ino ...

, Oxford University Press, 1978, where ''H'' is the

enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...

, ''T'' the

absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...

and ''G'' the

of the system, all at constant

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...

''p''. The equation states that the change in the ''G/T'' ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor ''H/T''².

Chemical reactions and work

The typical applications of this equation are to

chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and break ...

s. The equation reads: :

\left( \frac  \right)_p = - \frac

with Δ''G'' as the change in Gibbs energy due to reaction, and Δ''H'' as the

enthalpy of reaction The standard enthalpy of reaction (denoted \Delta_ H^\ominus or \Delta H_^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. This can i ...

(often, but not necessarily, assumed to be independent of temperature). The o denotes the use of

standard states Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object t ...

, and particularly the choice of a particular standard pressure (1 bar). Integrating with respect to ''T'' (again ''p'' is constant) it becomes: :

\frac - \frac = \Delta H^\ominus \left(\frac - \frac\right)

This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature ''T''₂ with knowledge of just the

standard Gibbs free energy change of formation In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...

and the

standard enthalpy change of formation In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with ...

for the individual components. Also, using the reaction isotherm equation, that is :

\frac = -R \ln K

which relates the Gibbs energy to a chemical

equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...

, the

van 't Hoff equation The Van 't Hoff equation relates the change in the equilibrium constant, , of a chemical reaction to the change in temperature, ''T'', given the standard enthalpy change, , for the process. It was proposed by Dutch chemist Jacobus Henricus van ...

can be derived.Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, Since the change in a system's Gibbs energy is equal to the maximum amount of non-expansion work that the system can do in a process, the Gibbs-Helmholtz equation may be used to estimate how much non-expansion work can be done by a chemical process as a function of temperature. For example, the capacity of rechargeable electric batteries can be estimated as a function of temperature using the Gibbs-Helmholtz equation.

Derivation

Background

The definition of the Gibbs function is

H = G + ST

where is the enthalpy defined by:

H = U + pV

Taking differentials of each definition to find and , then using the

fundamental thermodynamic relation In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...

(always true for reversible or

irreversible Irreversible may refer to: * Irreversible process, in thermodynamics, a process that is not reversible *'' Irréversible'', a 2002 film * ''Irréversible'' (soundtrack), soundtrack to the film ''Irréversible'' * An album recorded by hip-hop artis ...

processes A process is a series or set of activities that interact to produce a result; it may occur once-only or be recurrent or periodic. Things called a process include: Business and management *Business process, activities that produce a specific se ...

dU = T\,dS - p\,dV

where is the

entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

, is

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...

, (minus sign due to reversibility, in which : work other than pressure-volume may be done and is equal to ) leads to the "reversed" form of the initial fundamental relation into a new master equation:

dG = - S\,dT + V\,dp

This is the

for a closed system. The Gibbs–Helmholtz equation can be derived by this second master equation, and the

chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...

for

partial derivatives In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

Sources

External links

Link
- Gibbs–Helmholtz equation, by W. R. Salzman (2004)

Gibbs-Helmholtz Equation, by P. Mander (accessed 17 March 2022) {{DEFAULTSORT:Gibbs-Helmholtz equation Thermodynamic equations Hermann von Helmholtz