In
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, activity (symbol ) is a measure of the "effective concentration" of a
species
A species () is often defined as the largest group of organisms in which any two individuals of the appropriate sexes or mating types can produce fertile offspring, typically by sexual reproduction. It is the basic unit of Taxonomy (biology), ...
in a mixture, in the sense that the species'
chemical potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
depends on the activity of a real solution in the same way that it would depend on
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', '' number concentration'', ...
for an
ideal solution. The term "activity" in this sense was coined by the American chemist
Gilbert N. Lewis
Gilbert Newton Lewis (October 23 or October 25, 1875 – March 23, 1946) was an American physical chemist and a dean of the college of chemistry at University of California, Berkeley. Lewis was best known for his discovery of the covalent bon ...
in 1907.
By convention, activity is treated as a
dimensionless quantity
Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...
, although its value depends on customary choices of
standard state for the species. The activity of pure substances in condensed phases (solids and liquids) is taken as = 1. Activity depends on temperature, pressure and composition of the mixture, among other things. For gases, the activity is the effective partial pressure, and is usually referred to as
fugacity.
The difference between activity and other measures of concentration arises because the interactions between different types of
molecule
A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s in non-ideal
gases or
solutions are different from interactions between the same types of molecules. The activity of an
ion is particularly influenced by its surroundings.
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 ...
s should be defined by activities but, in practice, are often defined by
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', '' number concentration'', ...
s instead. The same is often true of equations for
reaction rates. However, there are circumstances where the activity and the concentration are ''significantly different'' and, as such, it is not valid to approximate with concentrations where activities are required. Two examples serve to illustrate this point:
*In a solution of
potassium hydrogen iodate KH(IO
3)
2 at 0.02
M the activity is 40% lower than the calculated hydrogen ion concentration, resulting in a much higher
pH than expected.
*When a 0.1 M
hydrochloric acid
Hydrochloric acid, also known as muriatic acid or spirits of salt, is an aqueous solution of hydrogen chloride (HCl). It is a colorless solution with a distinctive pungency, pungent smell. It is classified as a acid strength, strong acid. It is ...
solution containing
methyl green indicator
Indicator may refer to:
Biology
* Environmental indicator of environmental health (pressures, conditions and responses)
* Ecological indicator of ecosystem health (ecological processes)
* Health indicator, which is used to describe the health o ...
is added to a 5 M solution of
magnesium chloride, the color of the indicator changes from green to yellow—indicating increasing acidity—when in fact the acid has been diluted. Although at low ionic strength (< 0.1 M) the
activity coefficient approaches unity, this coefficient can actually increase with ionic strength in a high ionic strength regime. For hydrochloric acid solutions, the minimum is around 0.4 M.
Definition
The relative activity of a species , denoted , is defined
as:
:
where is the (molar)
chemical potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
of the species under the conditions of interest, is the (molar) chemical potential of that species under some defined set of standard conditions, is the
gas constant, is the
thermodynamic temperature
Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion.
Thermodynamic temperature is typically expres ...
and is
the exponential constant.
Alternatively, this equation can be written as:
:
In general, the activity depends on any factor that alters the chemical potential. Such factors may include: concentration, temperature, pressure, interactions between chemical species, electric fields, etc. Depending on the circumstances, some of these factors, in particular concentration and interactions, may be more important than others.
The activity depends on the choice of standard state such that changing the standard state will also change the activity. This means that activity is a relative term that describes how "active" a compound is compared to when it is under the standard state conditions. In principle, the choice of standard state is arbitrary; however, it is often chosen out of mathematical or experimental convenience. Alternatively, it is also possible to define an "absolute activity", , which is written as:
:
Note that this definition corresponds to setting as standard state the solution of
, if the latter exists.
Activity coefficient
The activity coefficient , which is also a dimensionless quantity, relates the activity to a measured
mole fraction
In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...
(or in the gas phase),
molality
In chemistry, molality is a measure of the amount of solute in a solution relative to a given mass of solvent. This contrasts with the definition of '' molarity'' which is based on a given volume of solution.
A commonly used unit for molality ...
,
mass fraction ,
molar concentration
Molar concentration (also called molarity, amount concentration or substance concentration) is the number of moles of solute per liter of solution. Specifically, It is a measure of the concentration of a chemical species, in particular, of a so ...
(molarity) or
mass concentration :
[ McQuarrie, D. A.; Simon, J. D. ''Physical Chemistry – A Molecular Approach'', p. 990 & p. 1015 (Table 25.1), University Science Books, 1997. ]
:
The division by the standard molality (usually 1 mol/kg) or the standard molar concentration (usually 1 mol/L) is necessary to ensure that both the activity and the activity coefficient are dimensionless, as is conventional.
The activity depends on the chosen standard state and composition scale;
for instance, in the dilute limit it approaches the mole fraction, mass fraction, or numerical value of molarity, all of which are different. However, the activity coefficients are similar.
When the activity coefficient is close to 1, the substance shows almost ideal behaviour according to
Henry's law (but not necessarily in the sense of an
ideal solution). In these cases, the activity can be substituted with the appropriate dimensionless measure of composition , or . It is also possible to define an activity coefficient in terms of
Raoult's law
Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of ''liquids'' is ...
: the
International Union of Pure and Applied Chemistry
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
(IUPAC) recommends the symbol for this activity coefficient,
although this should not be confused with
fugacity.
:
Standard states
Gases
In most laboratory situations, the difference in behaviour between a real gas and an ideal gas is dependent only on the pressure and the temperature, not on the presence of any other gases. At a given temperature, the "effective" pressure of a gas is given by its
fugacity : this may be higher or lower than its mechanical pressure. By historical convention, fugacities have the dimension of pressure, so the dimensionless activity is given by:
:
where is the dimensionless fugacity coefficient of the species, is its
mole fraction
In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...
in the gaseous mixture ( for a pure gas) and is the total pressure. The value is the standard pressure: it may be equal to 1
atm (101.325
kPa) or 1
bar (100 kPa) depending on the source of data, and should always be quoted.
Mixtures in general
The most convenient way of expressing the composition of a generic mixture is by using the
mole fraction
In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...
s (written in the gas phase) of the different components (or chemical species: atoms or molecules) present in the system, where
:
: with , the number of moles of the component ''i'', and , the total number of moles of all the different components present in the mixture.
The standard state of each component in the mixture is taken to be the pure substance, ''i.e.'' the pure substance has an activity of one. When activity coefficients are used, they are usually defined in terms of
Raoult's law
Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of ''liquids'' is ...
,
:
where is the Raoult's law activity coefficient: an activity coefficient of one indicates ideal behaviour according to Raoult's law.
Dilute solutions (non-ionic)
A solute in dilute solution usually follows
Henry's law rather than Raoult's law, and it is more usual to express the composition of the solution in terms of the molar concentration (in mol/L) or the molality (in mol/kg) of the solute rather than in mole fractions. The standard state of a dilute solution is a hypothetical solution of concentration = 1 mol/L (or molality = 1 mol/kg) which shows ideal behaviour (also referred to as "infinite-dilution" behaviour). The standard state, and hence the activity, depends on which measure of composition is used. Molalities are often preferred as the volumes of non-ideal mixtures are not strictly additive and are also temperature-dependent: molalities do not depend on volume, whereas molar concentrations do.
The activity of the solute is given by:
:
Ionic solutions
When the solute undergoes ionic dissociation in solution (for example a salt), the system becomes decidedly non-ideal and we need to take the dissociation process into consideration. One can define activities for the cations and anions separately ( and ).
In a liquid solution the activity coefficient of a given
ion (e.g. Ca
2+) isn't measurable because it is experimentally impossible to independently measure the electrochemical potential of an ion in solution. (One cannot add cations without putting in anions at the same time). Therefore, one introduces the notions of
;mean ionic activity
:
;mean ionic molality
:
;mean ionic activity coefficient
:
where represent the stoichiometric coefficients involved in the ionic dissociation process
Even though and cannot be determined separately, is a measurable quantity that can also be predicted for sufficiently dilute systems using
Debye–Hückel theory. For electrolyte solutions at higher concentrations, Debye–Hückel theory needs to be extended and replaced, e.g., by a
Pitzer electrolyte solution model (see
external links
An internal link is a type of hyperlink on a web page to another page or resource, such as an image or document, on the same website or domain. It is the opposite of an external link, a link that directs a user to content that is outside its d ...
below for examples). For the activity of a strong ionic solute (complete dissociation) we can write:
:
Measurement
The most direct way of measuring the activity of a volatile species is to measure its equilibrium partial
vapor pressure
Vapor pressure or equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The equilibrium vapor pressure is an indicat ...
. For water as solvent, the
water activity ''a
w'' is the equilibrated
relative humidity
Humidity is the concentration of water vapor present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation (meteorology), precipitation, dew, or fog t ...
. For non-volatile components, such as
sucrose
Sucrose, a disaccharide, is a sugar composed of glucose and fructose subunits. It is produced naturally in plants and is the main constituent of white sugar. It has the molecular formula .
For human consumption, sucrose is extracted and refined ...
or
sodium chloride
Sodium chloride , commonly known as Salt#Edible salt, edible salt, is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. It is transparent or translucent, brittle, hygroscopic, and occurs a ...
, this approach will not work since they do not have measurable vapor pressures at most temperatures. However, in such cases it is possible to measure the vapor pressure of the ''solvent'' instead. Using the
Gibbs–Duhem relation it is possible to translate the change in solvent vapor pressures with concentration into activities for the solute.
The simplest way of determining how the activity of a component depends on pressure is by measurement of densities of solution, knowing that real solutions have deviations from the additivity of (molar) volumes of pure components compared to the (molar) volume of the solution. This involves the use of
partial molar volume
In thermodynamics, a partial molar property is a quantity which describes the variation of an extensive property of a solution or mixture with changes in the molar composition of the mixture at constant temperature and pressure. It is the par ...
s, which measure the change in chemical potential with respect to pressure.
Another way to determine the activity of a species is through the manipulation of
colligative properties
In chemistry, colligative properties are those properties of solutions that depend on the ratio of the number of solute particles to the number of solvent particles in a solution, and not on the nature of the chemical species present. The numb ...
, specifically
freezing point depression. Using freezing point depression techniques, it is possible to calculate the activity of a weak acid from the relation,
:
where is the total equilibrium molality of solute determined by any colligative property measurement (in this case ), is the nominal molality obtained from titration and is the activity of the species.
There are also electrochemical methods that allow the determination of activity and its coefficient.
The value of the mean ionic activity coefficient of
ions in solution can also be estimated with the
Debye–Hückel equation, the
Davies equation
The Davies equation is an empirical extension of Debye–Hückel theory which can be used to calculate activity coefficients of electrolyte solutions at relatively high concentrations at 25 °C. The equation, originally published in 1938, wa ...
or the
Pitzer equations
Pitzer equations are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water. They were first described by physical chemist Kenneth Pitzer. The parameters of the Pitzer equations ar ...
.
Single ion activity measurability revisited
The prevailing view that single ion activities are unmeasurable, or perhaps even physically meaningless, has its roots in the work of
Edward A. Guggenheim
Edward Armand Guggenheim FRS (11 August 1901 – 9 August 1970) was an English physical chemist, noted for his contributions to thermodynamics.
Life
Guggenheim was born in Manchester 11 August 1901, the son of Armand Guggenheim and Marguerite B ...
in the late 1920s. However, chemists have not given up the idea of single ion activities. For example,
pH is defined as the negative logarithm of the hydrogen ion activity. By implication, if the prevailing view on the physical meaning and measurability of single ion activities is correct it relegates pH to the category of thermodynamically unmeasurable quantities. For this reason the
International Union of Pure and Applied Chemistry
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
(IUPAC) states that the activity-based definition of pH is a notional definition only and further states that the establishment of primary pH standards requires the application of the concept of 'primary method of measurement' tied to the Harned cell. Nevertheless, the concept of single ion activities continues to be discussed in the literature, and at least one author purports to define single ion activities in terms of purely thermodynamic quantities. The same author also proposes a method of measuring single ion activity coefficients based on purely thermodynamic processes. A different approach
has a similar objective.
Use
Chemical activities should be used to define
chemical potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
s, where the chemical potential depends on the
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
,
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 eve ...
and the activity according to the
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
:
:
where is the
gas constant and is the value of under standard conditions. Note that the choice of concentration scale affects both the activity and the standard state chemical potential, which is especially important when the reference state is the infinite dilution of a solute in a solvent. Chemical potential has units of joules per mole (J/mol), or energy per amount of matter. Chemical potential can be used to characterize the specific
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol is a thermodynamic potential that can be used to calculate the maximum amount of Work (thermodynamics), work, other than Work (thermodynamics)#Pressure–v ...
changes occurring in chemical reactions or other transformations.
Formulae involving activities can be simplified by considering that:
* For a chemical solution:
** the
solvent
A solvent (from the Latin language, Latin ''wikt:solvo#Latin, solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a Solution (chemistry), solution. A solvent is usually a liquid but can also be a solid, a gas ...
has an activity of unity (only a valid approximation for rather dilute solutions)
** At a low concentration, the activity of a solute can be approximated to the ratio of its concentration over the standard concentration:
Therefore, it is approximately equal to its concentration.
* For a mix of
gas at low pressure, the activity is equal to the ratio of the
partial pressure
In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal g ...
of the gas over the standard pressure:
Therefore, it is equal to the partial pressure in atmospheres (or bars), compared to a standard pressure of 1 atmosphere (or 1 bar).
* For a solid body, a uniform, single species solid has an activity of unity at standard conditions. The same thing holds for a pure liquid.
The latter follows from any definition based on Raoult's law, because if we let the solute concentration go to zero, the vapor pressure of the solvent will go to . Thus its activity will go to unity. This means that if during a reaction in dilute solution more solvent is generated (the reaction produces water for example) we can typically set its activity to unity.
Solid and liquid activities do not depend very strongly on pressure because their molar volumes are typically small.
Graphite
Graphite () is a Crystallinity, crystalline allotrope (form) of the element carbon. It consists of many stacked Layered materials, layers of graphene, typically in excess of hundreds of layers. Graphite occurs naturally and is the most stable ...
at 100 bars has an activity of only 1.01 if we choose as standard state. Only at very high pressures do we need to worry about such changes. Activity expressed in terms of pressure is called
fugacity.
Example values
Example values of activity coefficients of
sodium chloride
Sodium chloride , commonly known as Salt#Edible salt, edible salt, is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. It is transparent or translucent, brittle, hygroscopic, and occurs a ...
in aqueous solution are given in the table.
In an ideal solution, these values would all be unity. The deviations ''tend'' to become larger with increasing molality and temperature, but with some exceptions.
See also
*
Fugacity
In thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of chemical equilibrium. It is equal to the pressure of an ideal gas which has the same tempe ...
, the equivalent of activity for
partial pressure
In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal g ...
*
Chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which both the Reagent, reactants and Product (chemistry), products are present in concentrations which have no further tendency to change with time, so that there is no observable chan ...
*
Electrochemical potential
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference and identifiable chemical change. These reactions involve electrons moving via an electronically conducting phase (typi ...
*
Excess chemical potential
In thermodynamics, the excess chemical potential is defined as the difference between the chemical potential of a given species and that of an ideal gas under the same conditions (in particular, at the same pressure, temperature, and composition). ...
*
Partial molar property
In thermodynamics, a partial molar property is a quantity which describes the variation of an extensive property of a solution or mixture with changes in the molar composition of the mixture at constant temperature and pressure. It is the par ...
*
Thermodynamic equilibrium
Thermodynamic equilibrium is a notion of thermodynamics with axiomatic status referring to an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...
*
Thermal expansion
Thermal expansion is the tendency of matter to increase in length, area, or volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions).
Substances usually contract with decreasing temp ...
*
Virial expansion
The virial expansion is a model of thermodynamic equations of state. It expresses the pressure of a gas in local Thermodynamic equilibrium, equilibrium as a power series of the density. This equation may be represented in terms of the compre ...
*
Water activity
In food science, water activity (''aw'') of a food is the ratio of its vapor pressure to the vapor pressure of water at the same temperature, both taken at equilibrium. Pure water has a water activity of one. Put another way, ''aw'' is the equi ...
*
Non-random two-liquid model
The non-random two-liquid model (abbreviated NRTL model) is an activity coefficient model introduced by Renon
and John Prausnitz, Prausnitz in 1968 that correlates the activity coefficients \gamma_i of a compound with its mole fractions x_i in th ...
(NRTL model) – phase equilibrium calculations
*
UNIQUAC model – phase equilibrium calculations
References
External links
Equivalences among different forms of activity coefficients and chemical potentialsCalculate activity coefficients of common inorganic electrolytes and their mixturesAIOMFAC online-model calculator for activity coefficients of inorganic ions, water, and organic compounds in aqueous solutions and multicomponent mixtures with organic compounds.
{{Authority control
Dimensionless numbers of chemistry
Physical chemistry
Thermodynamic properties