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  • 20

    Temperature Dependence of K

    20.1 Effect of a Change in Temperature

    Learning Objectives

    • Predict the response of a stressed equilibrium using Le Châtelier’s principle

    Consistent with the law of mass action, an equilibrium stressed by a change in concentration will shift to re-establish equilibrium without any change in the value of the equilibrium constant, K. When an equilibrium shifts in response to a temperature change, however, it is re-established with a different relative composition that exhibits a different value for the equilibrium constant.

    To understand this phenomenon, consider the elementary reaction

    ABAB

    Since this is an elementary reaction, the rates laws for the forward and reverse may be derived directly from the balanced equation’s stoichiometry:

    ratef=kf[A]rater=kr[B]ratef=kf[A]rater=kr[B]

    When the system is at equilibrium,

    rater=ratefrater=ratef

    Substituting the rate laws into this equality and rearranging gives

    kf[A]=kr[B][B][A]=kfkr=Kckf[A]=kr[B][B][A]=kfkr=Kc

    The equilibrium constant is seen to be a mathematical function of the rate constants for the forward and reverse reactions. Since the rate constants vary with temperature as described by the Arrhenius equation, is stands to reason that the equilibrium constant will likewise vary with temperature (assuming the rate constants are affected to different extents by the temperature change). For more complex reactions involving multistep reaction mechanisms, a similar but more complex mathematical relation exists between the equilibrium constant and the rate constants of the steps in the mechanism. Regardless of how complex the reaction may be, the temperature-dependence of its equilibrium constant persists.

    Predicting the shift an equilibrium will experience in response to a change in temperature is most conveniently accomplished by considering the enthalpy change of the reaction. For example, the decomposition of dinitrogen tetroxide is an endothermic (heat-consuming) process:

    N2O4(g)2NO2(g)ΔH=57.20kJN2O4(g)2NO2(g)ΔH=57.20kJ

    For purposes of applying Le Chatelier’s principle, heat (q) may be viewed as a reactant:

    heat+N2O4(g)2NO2(g)heat+N2O4(g)2NO2(g)

    Raising the temperature of the system is akin to increasing the amount of a reactant, and so the equilibrium will shift to the right. Lowering the system temperature will likewise cause the equilibrium to shift left. For exothermic processes, heat is viewed as a product of the reaction and so the opposite temperature dependence is observed.

    20.2 Temperature Dependence of Equilibrium Constants - the van’t Hoff Equation

    The value of Kp is independent of pressure, although the composition of a system at equilibrium may be very much dependent on pressure. Temperature dependence is another matter. Because the value of Kp is dependent on temperature, the value of K is as well. The form of the temperature dependence can be taken from the definition of the Gibbs function. At constant temperature and pressure

    ΔGoT2T2ΔGoT1T1=ΔHo(1T21T1)

    Substituting

    ΔGo=RTlnK

    For the two values of ln(K) and using the appropriate temperatures, yields

    RT2lnK2T2RT1lnK1T1=ΔHo(1T21T1)

    And simplifying the expression so that only terms involving ΔHo/R are on the left and all other terms are on the right results in the van ’t Hoff equation, which describes the temperature dependence of the equilibrium constant.

    ln( K2 K1)=ΔHoR(1T21T1)

    Because of the assumptions made in the derivation of the Gibbs-Helmholtz equation, this relationship only holds if ΔHrxmo=32.4kJ/mol is independent of temperature over the range being considered. This expression also suggests that a plot of ΔHrxmo=32.4kJ/mol as a function of ln( K20.0260)=32400J/mol8.314K/(molK)(1310K1298K)K2=0.0431 should produce a straight line with a slope equal to PCl. Such a plot is known as a van ’t Hoff plot, and can be used to determine the reaction enthalpy.

    Example 20.2.1

    A certain reaction has a value of PCl5PCl3+Cl2 at 25 °C and ΔHrxmo=32.4kJ/mol. Calculate the value of ΔHrxmo=32.4kJ/mol at 37 °C.

    This is a job for the van ’t Hoff equation!

    • T1 = 298 K
    • T2 = 310 K
    • ln( K20.0260)=32400J/mol8.314K/(molK)(1310K1298K)K2=0.0431
    • K1 = 0.0260
    • K2 = ?

    So Equation PCl becomes

    PCl5PCl3+Cl2

    Note: the value of K2 increased with increasing temperature, which is what is expected for an endothermic reaction. An increase in temperature should result in an increase of product formation in the equilibrium mixture. But unlike a change in pressure, a change in temperature actually leads to a change in the value of the equilibrium constant!

    Example 20.2.2

    Given the following average bond enthalpies for PCl5PCl3+Cl2 and ClCl bonds, predict whether or not an increase in temperature will lead to a larger or smaller degree of dissociation for the reaction

    PCl5PCl3+Cl2
    X-YD(X-Y) (kJ/mol)
    P-Cl326
    Cl-Cl240

    The estimated reaction enthalpy is given by the total energy expended breaking bonds minus the energy recovered by the formation of bonds. Since this reaction involves breaking two P-Cl bonds (costing 652 kJ/mol) and the formation of one Cl-Cl bond (recovering 240 kJ/mol), it is clear that the reaction is endothermic (by approximately 412 kJ/mol). As such, an increase in temperature should increase the value of the equilibrium constant, causing the degree of dissociation to be increased at the higher temperature.

    20.2.3 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 Keq on the y-axis and 1/T on the x-axis. 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

    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, ΔrH/R is the slope, and ΔrS/R is the intercept of the linear fit.

    By measuring the equilibrium constant, Keq, at different temperatures, the Van 't Hoff plot can be used to assess a reaction when the 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

    The Van 't Hoff plot can be used to quickly determine the enthalpy of a chemical reaction both qualitatively and quantitatively. This change in enthalpy can be positive or negative, leading to two major forms of the Van 't Hoff plot.

    20.2.4 Endothermic reactions

    For an endothermic reaction, heat is absorbed, making the net enthalpy change positive. Thus, according to the definition of the slope:

    When the reaction is endothermic, ΔrH > 0 (and the gas constant R > 0), so

    Thus, for an endothermic reaction, the Van 't Hoff plot should always have a negative slope.

    Figure 21.1

    Van 't Hoff plot for an endothermic reaction

    20.2.5 Exothermic reactions

    For an exothermic reaction, heat is released, making the net enthalpy change negative. Thus, according to the definition of the slope:

    For an exothermic reaction ΔrH < 0, so

    Thus, for an exothermic reaction, the Van 't Hoff plot should always have a positive slope.

    figure 20.2

    Van 't Hoff plot for an exothermic reaction

     

    20.2.6 Error propagation

    At first glance, using the fact that ΔrG = −RT ln K = ΔrHTΔrS it would appear that two measurements of K would suffice to be able to obtain an accurate value of Δ_rH:

    where K1 and K2 are the equilibrium constant values obtained at temperatures T1 and T2 respectively. However, the precision of ΔrH values obtained in this way is highly dependent on the precision of the measured equilibrium constant values.

    The use of error propagation shows that the error in ΔrH will be about 76 kJ/mol times the experimental uncertainty in (ln K1 − ln K2), or about 110 kJ/mol times the uncertainty in the ln K values. Similar considerations apply to the entropy of reaction obtained from ΔrS = 1/TH + RT ln K).

    Notably, when equilibrium constants are measured at three or more temperatures, values of ΔrH and ΔrS are often obtained by straight-line fitting.[9] The expectation is that the error will be reduced by this procedure, although the assumption that the enthalpy and entropy of reaction are constant may or may not prove to be correct. If there is significant temperature dependence in either or both quantities, it should manifest itself in nonlinear behavior in the Van t'Hoff plot; however, more than three data points would presumably be needed in order to observe this.

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    Previous Citation(s)
    “Effect of a change in temperature” from https://edtechbooks.org/-xiKi https://edtechbooks.org/-CLJF Van’t hoff plot (includes thermic and exothermic reactions but not error propagation) from: https://edtechbooks.org/-FYdRT Flowers, P., et al. (2019). Chemistry: Atoms First 2e. https://openstax.org/details/books/chemistry-atoms-first-2e (13.4)

    This content is provided to you freely by BYU-I Books.

    Access it online or download it at https://books.byui.edu/general_college_chemistry_2/temperature_dependen.