Original Article

Phys Chem Ind J, Volume: 12( 2)

# The Alternative Formulation of Van't Hoff Equation

- *Correspondence:
- Irfan H Lone , Department of Chemistry, Government Degree College, Kupwara-193222, Kashmir, India,
**Tel:**7051248200;**E-mail:**[email protected]

**Received:** May 01, 2017; **Accepted:** June 07, 2017; **Published:** June 09, 2017

**Citation:** Lone IH. The Alternative Formulation of Van't Hoff Equation. Phys Chem Ind J. 2017;12(2):110.

### Abstract

An alternative formulation (named total entropy equation) of Van't Hoff's equation has been presented. An explanation of the effect of temperature on exothermic and endothermic processes is provided in terms of this alternative formulation. The treatment can be useful in directly highlighting the central role that the total entropy change plays in determining the direction of the process. As a chemical example the calculations of total entropy change are presented for the Haber-Bosch process. Finally, the correlation between Van't Hoff's equation and Total entropy equation is demonstrated by showing that it is possible to derive the former from the later and vice versa.

### Keywords

*Van't Hoff equation; Entropy; Exothermic; Endothermic; Gibbs-Helmholtz equation*

### Introduction

The Van’t Hoff equation [1]:

(1)

is usually derived [2-6] by making use of the Van’t Hoff isotherm:

(2)

and the Gibbs-Helmholtz [7-9] equation:

(3)

More specifically, to obtain equation (1) one substitutes equation (2) for Δ_{r}G° into the left-hand side of equation (3) and
integrates. The Van’t Hoff equation is used to know the variation of an equilibrium constant K with temperature [2,3]. This
effect of temperature on the equilibrium constant is usually understood through its effect on the standard Gibbs energy of
reaction Δ_{r}G° [10,11]. The standard state of a substance refers to 1 bar pressure. In other words, any change in K with T is
perceived as a change in the value of the standard Gibbs energy of reaction Δ_{r}G° [3]. Furthermore, the dependence of K on
temperature is determined by the sign of Δ_{r}H° and the Van’t Hoff equation provides the theoretical basis of this dependence
[12]. Alternatively, the effect of temperature on equillibria has sometimes correctly, but only qualitatively, been explained in
terms of the relative importance of the magnitudes of the entropy changes of the system and its immediate surroundings [2].
This is done by employing the following relation:

(4)

In order to provide a mathematical expression corresponding to this alternative explanation the Van’t Hoff equation may be formulated in terms of entropy. This we do in the following sections. The presented equation, it is hoped, is a useful expression that highlights in a more direct manner the role that total entropy change (System+Surroundings) plays in the process.

**The Equation**

If one multiplies both sides of equation (4) by the negative sign, one gets

If one identifies the first term on the right hand side of above equation with the entropy change of the surroundings [13] and denotes it by then, since , one finds [14]:

(5)

Here ΔS°_{total} is the total entropy change that the reaction generates and T denotes the absolute temperature. ΔS_{total} is
sometimes also called ΔS_{universe} or ΔS_{isolated} but we won’t use such designations here. For us the universe will consist of the
system and its immediate surroundings as is usually the case in a typical chemistry laboratory. Although equation (5) does
give some indication of the role that ΔS°_{total} plays in determining the favourable path of a process, it is useful to appreciate the role of ΔS°_{total} fully. If instead of equation (2) one substitutes equation (5) into the left hand side of equation (3) one
obtains:

(6)

Equation (6) can be directly integrated between the two specified temperatures T_{i} and T_{f}, say, to give the following general
result:

(7)

Here T_{i} is the initial temperature while T_{f} is the final temperature. In order to carry out above integration further one must
know how the enthalpy of the system varies with the temperature [15-18]. If one uses the approximation that the standard
reaction enthalpy is independent of temperature, a valid approximation for at least small ranges of temperature [2], then
equation (7) can be easily integrated to give the following special result:

(8)

Equations (7) and (8) represent an alternative formulation of Van't Hoff's equation in terms of entropy.

### Results and Discussion

Equation (8) is an important result as it can be used to explain the behavior of exothermic and endothermic processes towards
temperature directly. According to equation (8), for exothermic processes (the processes that are accompanied by the
evolution of energy as heat as the reactants get converted to products) the increase of temperature leads to a net decrease in
total entropy. Hence the equilibrium constant for an exothermic process decreases with increase of temperature as can be
easily seen from a comparison of equations (2) and (5). On the other hand, a decrease in temperature leads to a net increase in
total entropy and the equilibrium constant increases, also obvious from a comparison of (2) and (5). This must be so because
ΔrH°<0 for an exothermic system. Now if we apply equation (8) to endothermic processes (the processes that are
accompanied by the absorption of energy as heat as the reactants get converted to products) we find that the increase of
temperature leads to a net increase in total entropy. Hence for endothermic processes the equilibrium constant increases with
increase in temperature. On the other hand decrease in temperature leads to a net decrease in total entropy and K_{eq} decreases
with a decrease in temperature, again obvious from a comparison of equations (2) and (5). This must be so because Δ_{r}H°>0
for an endothermic system.

**An Application: The Haber-Bosch Process**

The Haber-Bosch process named after its inventors, the German chemists Fritz Haber and Carl Bosch, who developed it in
the first half of the 20th century [19], is an artificial nitrogen fixation process and is the main industrial procedure for the production of ammonia today [20]. This important process consumes over one percent of humanity's energy production but is
responsible for feeding roughly one-third of its population and has been of somewhat greater fundamental importance to the
modern world than the invention of the airplane, nuclear energy, space flight, or television [21]. The Haber-Bosch process is
an exothermic process that converts atmospheric nitrogen (N_{2}) to ammonia (NH_{3}) by reaction with hydrogen (H_{2}) using a
metal catalyst under high temperature and pressure and is usually depicted by the following chemical equation:

N_{2}(g)+3H_{2}(g) ≡ 2NH_{3}(g)

Now for the Haber-Bosch equilibrium process the value of Δ_{r}H°=-92.2 KJmol^{-1} at 298 K and the value of K_{eq} is then 6.1 ×
105. Suppose the value of Δ_{r}H° is independent of temperature up to about 500°C and the temperature is increased above room
temperature in steps of 50°C. One can then calculate the variation of ΔS°_{total} with temperature by using equation (8). These
values of ΔS°_{total} at different temperatures above room temperature [with ΔS°_{total}=110.40 JK^{-1}mol^{-1} at 298 K, available from
the already existing chemical literature, for example see reference-2 or calculated by using equation (5)] are shown in table
below.

As is clear from above **Table 1**, ΔS° _{total}decreases as T increases for the above Haber-Bosch exothermic process. This
means that the process becomes unfavourable at higher temperatures.

Temperature/K | 298 | 348 | 398 | 448 | 498 |

ΔS°total/JK^{-1}mol^{-1} |
110.40 | 65.95 | 32.67 | 6.28 | -13.84 |

**Table 1: **Values of ∆S°total at different temperatures for the Haber-Bosch process.

**The Correlation between Van't Hoff Equation and Total Entropy Equation**

Now using equation (8) one can easily obtain the Van't Hoff equation or Van't Hoff's law. If one uses equation (5) one can also write (8) as

(9)

Now one substitutes equation (2) into equation (9) to get the desired result

(10)

Alternatively, if one takes equation (10) and works backwards by using equation (2) in it one gets the total entropy equation,equation (8).

### Conclusion

An alternative thermodynamic interpretation of the behavior of exothermic and endothermic processes towards temperature is in terms of the total entropy change that the process causes. It is possible to present a mathematical expression, named as the total entropy equation, underlying this explanation. The presented equation makes it very direct to appreciate the role of total entropy change in the processes.

### Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be considered as a potential conflict of interest.

### Acknowledgements

The author would like to thank Prof. Carl O. Trindle (University of Virginia) for many useful suggestions for the improvement of the article.

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