The Van 't Hoff equation in chemical thermodynamics relates the change in the equilibrium constant, K_{eq}, of a chemical equilibrium to the change in temperature, T, given the standard enthalpy change, ΔH^{o}, for the process. It was proposed by Dutch chemist Jacobus Henricus van 't Hoff in 1884.^{[1]}
The Van 't Hoff equation has been widely utilized to explore the changes in state functions in a thermodynamic system. The Van 't Hoff plot, which is derived from this equation, is especially effective in estimating the change in enthalpy, or total energy, and entropy, or amount of disorder, of a chemical reaction.
Contents

Equation 1

Under standard conditions 1.1

Development from thermodynamics 1.2

Van 't Hoff isotherm 1.3

Van 't Hoff plot 2

Endothermic reactions 2.1

Exothermic reactions 2.2

Temperature dependence of the equilibrium constant 2.3

Applications of the Van 't Hoff plot 3

Van 't Hoff analysis 3.1

Mechanistic studies 3.2

Temperature dependence 3.3

See also 4

References 5
Equation
Under standard conditions
Under standard conditions, the Van 't Hoff equation is^{[2]}^{[3]}

\frac{d \ln K_{eq}}{dT} = \frac{\Delta H^\ominus}{RT^2},

where R is the gas constant. This equation is exact at any one temperature. In practice, the equation is often integrated between two temperatures under the assumption that the reaction enthalpy ΔH is constant. Since in reality ΔH as well as ΔS do vary with temperature for most processes,^{[4]} the integrated equation is only approximate.
A major use of the integrated equation is to estimate a new equilibrium constant at a new absolute temperature assuming a constant standard enthalpy change over the temperature range.
To obtain the integrated equation, it is convenient to first rewrite the van't Hoff equation as^{[2]}

\frac{d \ln K_{eq}}{d {\frac}} = \frac{\Delta H^\ominus}{R}.
The definite integral between temperatures T_{1} and T_{2} is then

\ln \left( {\frac} \right) = \frac{R}\left( {\frac{1}  \frac{1}} \right).
In this equation K_{1} is the equilibrium constant at absolute temperature T_{1}, and K_{2} is the equilibrium constant at absolute temperature T_{2}.
Development from thermodynamics
From the definition of Gibbs free energy

\Delta G^\ominus = \Delta H^\ominus  T\Delta S^\ominus
where S is the entropy of the system, and from the Gibbs free energy isotherm equation, ^{[5]} and

\Delta G^\ominus = RT \ln K_{eq}
These equations are combined to obtain

\ln K_{eq} =  \frac{RT}+ \frac{R}.
Differentiation of this expression with respect to the variable (1/T) yields the Van 't Hoff equation.
Provided that ΔH^{o} and ΔS^{o} are constant, the preceding equation gives ln K as a linear function of 1/T, and is known as the linear form of the Van't Hoff equation. Therefore, when the range in temperature is small enough that the standard enthalpy and entropy changes are essentially constant, a plot of the natural logarithm of the equilibrium constant versus the reciprocal temperature gives a straight line. The slope of the line may be multiplied by the gas constant R to obtain the standard enthalpy change of the reaction, and the intercept may be multiplied by R to obtain the standard entropy change.
Van 't Hoff isotherm
The Gibbs free energy can change with the change of the temperature and pressure of the thermodynamic system. The Van 't Hoff isotherm can be used to determine the Gibbs free energy for nonstandard state reactions at a constant temperature:^{[6]}
\left(\frac {dG}{d\xi}\right)_{T,p} = \Delta_rG + RT \ln Q_r~
where \Delta_rG is the Gibbs free energy for the reaction, and Q_r~ is the reaction quotient. When a reaction is at equilibrium, Q_r~=K_{eq}. The Van 't Hoff isotherm can help estimate the equilibrium reaction shift. When \Delta_rG<0, the reaction moves in the forward direction. When \Delta_rG>0, the reaction moves in the backwards directions. See Chemical equilibrium.
Van 't Hoff plot
For a reversible reaction, the equilibrium constant can be measured at a variety of temperatures. This data can be plotted on a graph with \ln K_{eq} on the Yaxis and 1/T on the Xaxis. The data should have a linear relationship, the equation for which can be found by fitting the data using the linear form of the Van 't Hoff equation

\ln K_{eq} =  \frac{RT}+ \frac{R}.
This graph is called the Van 't Hoff plot and is widely used to estimate the enthalpy and entropy of a chemical reaction. From this plot, \frac{\Delta H}{R} is the slope and \frac{\Delta S}{R} is the intercept of the linear fit.
By measuring the equilibrium constant, K_{eq}, at different temperatures, the Van 't Hoff plot can be used to assess a reaction when temperature changes.^{[7]}^{[8]} Knowing the slope and intercept from the Van 't Hoff plot, the enthalpy and entropy of a reaction can be easily obtained using
\Delta H =  R * slope,
\Delta S = R * intercept.
The Van 't Hoff plot can be used to quickly determine the enthalpy of a chemical reaction both qualitatively and quantitatively. Change in enthalpy can be positive or negative, leading to two major forms of the Van 't Hoff plot.
Endothermic reactions
Endothermic Reaction Van 't Hoff Plot
For an endothermic reaction, heat is absorbed, making the net enthalpy change positive. Thus, according to the definition of the slope:
Slope= \frac{\Delta H}{R}
for an endothermic reaction,
\Delta H > 0 and R is the gas constant
So
Slope= \frac{\Delta H}{R} < 0
Thus, for an endothermic reaction, the Van 't Hoff plot should always have a negative slope.
Exothermic reactions
Exothermic Reaction Van 't Hoff Plot
For an exothermic reaction, heat is released, making the net enthalpy change negative. Thus, according to the definition of the slope:
Slope= \frac{\Delta H}{R}
from an exothermic reaction,
\Delta H < 0 and R is the gas constant
So
Slope= \frac{\Delta H}{R} > 0
Thus, for an exothermic reaction, the Van 't Hoff plot should always have a positive slope.
Temperature dependence of the equilibrium constant
By analyzing the Van 't Hoff plots for endothermic and exothermic reactions, it is possible to determine how the equilibrium constant is affected by changes in temperature. On a Van 't Hoff plot, temperature is plotted as 1/T. Because of this, the left side of the plot represents higher temperatures. Looking at a plot of an endothermic reaction, it can be determined that as temperature increases, the equilibrium constant also increases. This leads to a greater amount of product formation as temperature increases.
Conversely, it can be seen on the exothermic reaction plot that as temperature increases, the equilibrium constant decreases. This leads to less product formation as temperature increases.
Applications of the Van 't Hoff plot
Van 't Hoff analysis
Van 't Hoff analysis
In biological research, the Van 't Hoff plot is also called Van 't Hoff analysis.^{[9]} It's most effective in determining the favored product in a reaction.
Assume two products B and C form in a reaction:
\mathrm{a\ A + d\ D \longrightarrow b\ B}
\mathrm{a\ A + d\ D \longrightarrow c\ C}
In this case, K_{eq} can be defined as ratio of B to C rather than the equilibrium constant.
When \frac {B}{C} >1 , B is the favored product, and the data on the Van 't Hoff plot will be in the positive region.
When \frac {B}{C} <1 , C is the favored product, and the data on the Van 't Hoff plot will be in the negative region.
Using this information, a Van 't Hoff analysis can help determine the most suitable temperature for a favored product.
Recently, a Van 't Hoff analysis was used to determine whether water preferentially forms a hydrogen bond with the Cterminus or the Nterminus of the amino acid proline.^{[10]} The equilibrium constant for each reaction was found at a variety of temperatures, and a Van 't Hoff plot was created. This analysis showed that enthalpically, the water preferred to hydrogen bond to the Cterminus, but entropically it was more favorable to hydrogen bond with the Nterminus. Specifically, they found that Cterminus hydrogen bonding was favored by 4.26.4 kJ/mol. The Nterminus hydrogen bonding was favored by 3143 J/(mol K).
This data alone could not conclude which site water will preferentially hydrogen bond to, so additional experiments were used. It was determined that at lower temperatures, the enthalpically favored species, the water hydrogen bonded to the Cterminus, was preferred. At higher temperatures, the entropically favored species, the water hydrogen bonded to the Nterminus, was preferred.
Mechanistic studies
Van 't Hoff plot in Mechanism study
A chemical reaction may undergo different reaction mechanisms under different temperatures.^{[11]}
In this case, a Van 't Hoff plot with two or more linear fits may be exploited. Each linear fit has a different slope and intercept which indicates different changes in enthalpy and entropy for each distinct mechanisms. The Van 't Hoff plot can be used to find the enthalpy and entropy change for each mechanism and the favored mechanism under different temperatures.
\Delta H_1 =  R * slope_1,
\Delta S_1 = R * interception_1;
\Delta H_2 =  R * slope_2,
\Delta S_2 = R * interception_2;
In the example figure, the reaction undergoes mechanism 1 at high temperature and mechanism 2 at low temperature.
Temperature dependence
Temperature dependent Van 't Hoff plot
The Van 't Hoff plot is linear based on the assumption that the enthalpy and entropy are almost constant with temperature changes. However, in some cases the enthalpy and entropy do change dramatically with temperature. In this case, an additional term, \frac {1}{T^2} , can be added to the Van 't Hoff equations. A polynomial fit can then be used to analyze the data.:^{[12]}
ln K_{eq} = a+ \frac {b}{T} + \frac {c}{T^2} ,
where
\Delta H =  R * (b+2\frac {c}{T}),
\Delta S = R * (a\frac {c}{T^2}).
Thus, the enthalpy and entropy of a reaction can still be determined at specific temperatures even when a temperature dependence exists.
See also
References

^ Biography on Nobel prize website. Nobelprize.org (19110301). Retrieved on 2013118.

^ ^{a} ^{b} Atkins, Peter; De Paula, Julio (10 March 2006). Physical Chemistry (8th ed.). W.H. Freeman and Company. p. 212.

^ Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0356037363

^ Craig, Norman (1996). "Entropy Diagrams". J. Chem. Educ. (73): 710.

^ Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, ISBN 0198551487

^ Monk, Paul (2004). Physical Chemistry: Understanding our Chemical World. Wiley. p. 162.

^ Kim, Tae Woo (2012). "Dynamic [2]Catenation of Pd(II) Selfassembled Macrocycles in Water". Chem. Lett. (41): 70.

^ Ichikawa, Takayuki (2010). "Thermodynamic properties of metal amides determined by ammonia pressurecomposition isotherms". J. Chem. Thermodynamics (42): 140.

^ "Van't Hoff Analysis". Protein Analysis and Design Group.

^ Prell, James; Williams E (2010). "Entropy Drives an Attached Water Molecule from the C to NTerminus on Protonated Proline". J. Am. Chem. Soc. 132 (42): 14733.

^ Chatake, Toshiyuki (2010). Cryst. Growth Des. 10: 1090.

^ David, Victor (28 April 2011). "Deviation from van't Hoff dependence in RPLC induced by tautomeric interconversion observed for four compounds". Journal of Separation Science 34 (12): 1423.
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