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 the ...

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 that may be performed by a thermodynamically closed system at constant temperature and ...

of a system as a function of

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...

. 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, 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 w ...

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 and e ...

''p''. The equation states that the change in the ''G/T'' ratio at constant pressure as a result of an

infinitesimally In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...

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 IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...

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

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 processes):

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 thermodynam ...

, 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 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)=f(g(x)) for every , ...

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). Pa ...

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