Gibbs Free Energy As Predictor Of Chemical Reaction Spontaneity

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Gibbs Free Energy As Predictor Of Chemical Reaction Spontaneity

When deviations from generally accepted rules occur, smart humans want to know why.

Generally, an *exothermic reaction* involves an increase in *disorder*. So, an exothermic reaction that involves a decrease in disorder (increase in order) is a deviation from expectation.

Generally an *endothermic reaction* involves a decrease in *disorder*. So, an endothermic reaction that involves an increase in disorder (decrease in order) is a deviation from expectation.

Theoretical physicist and chemist J. Willard Gibbs (A.D. 1839 - 1903) was one of the people who wanted to know why the deviations stated above exist. The answer he developed, introduced a quantity called *free energy* (now called *Gibbs Free Energy* in his honor).

(a) State Rudolf Clausius' mathematical definition of *entropy*

(b)Relate change in *Gibbs Free Energy* to Change in *enthalpy*, temperature and change in *entropy*

(c) Relate change in *Gibbs free energy* to *chemical equilibrium constant*

(d) Relate change in *Gibbs free energy* to *standard cell potentials* through the *Nernst equation*

)e) Explain the spontaneity of the reaction of the combustion of 2 moles of hydrogen gas despite the decrease in entropy.

(f) State Ludwig Boltzmann's mathematical definition of *entropy*.

**The strings**:

S_{7}P_{2}A_{22} (Identity - Chemical Properties)
**The math**:

Pj Problem of Interest is of type *Identity*. A *predictor* must *identify* that which it predicts. The identification of the chemical behavior (chemical properties) of a chemical reaction in the context of change in enthalpy, temperature and change in entropy is the reason the change in Gibbs free Energy is a plausible predictor of chemical reaction spontaneity. It is in this sense that the Pj Problem of Interest is of type *Identity*.

(a) Rudolph Clausius, was one of the primary pioneers of classical thermodynamics. The others were Sadi Carnot, James Prescott Joule, Benoit Claperyon, James Clerk Maxwell and William Thomson (Lord Kelvin). Clausius coined the term *entropy* which is a measure of the *disorder* of a system. The letter S is used to denote entropy.

Clausius mathematical definition of entropy:

S = Q/T --------(1).

Where S is entropy of system; Q is heat content of system and T is temperature (degree Kelvin). Clausius assumed thermodynamic equilibrium as a result T is constant and the following equation holds:

ΔS = ΔQ/T -------(2)

Equation (2) says that given thermodynamic equilibrium, change in entropy equals change in heat content divided by temperature (degree Kelvin). Equation (2) implies that ΔS increases (positive) when ΔQ increases (positive) and ΔS decreases (negative) when ΔQ decreases (negative)

Suppose the energy ΔU is pumped into the system, some of this energy will be absorbed by the system and some will be used by the system to do work (for example, the push of a piston by hot gas in a car engine).

Let Δw represent this work. Then the heat absorbed by the system is given by:

ΔU-Δw

Therefore, equation (2) can be restated as:

ΔS = (ΔU-Δw)/T ----(3)

Now suppose:

ΔU is expressed as ΔH (change in enthalpy)

and Δw is epressed as ΔG (*change in Gibbs Free Energy*).

Then equation (3) can be written as follows:

ΔS = (ΔH-ΔG)/T ----(4)

So, ΔG = ΔH - TΔS ------(5)

Where ΔG is change in *Gibbs free energy*; ΔH, is the change in enthalpy; ΔS is change entropy; and T is temperature (degree Kelvin).

The states of ΔH, ΔS and ΔG relative to reaction spontaneity is shown in table 12.1.

(c) ΔG = - 2.30RT(logK_{eq}) --------(6)

Where ΔG is change in Gibbs free energy, R is the universal gas constant, T is temperature (degrees Kelvin) and K_{eq} is the equilbrium constant.

(d) The Nernst equation is as follows:

E = E^{o} - (1/nF)(2.30RT(logK_{eq})) -------(7)

Where E is the voltage of a *voltaic cell* at other than standard conditions; E^{o} is
the voltage of the *voltaic cell* at standard conditions; R is the universal gas constant, T is absolute temprature (degrees Kelvin); K_{eq} is equilibrium constant; *n* is the number of electrons transfered in the balanced equation of the reaction taking place in the cell; and F is the value of the conversion factor from volts to joules per mole (96,485).

At equilibrium E = 0.

So, nFE^{o} = 2.30RT(logK_{eq})

So, substituting nFE^{o} in equation (6), we have:

ΔG = -nFE^{o} -------(8)
*Gibbs free energy* is related to *standard cell potentials* and *equilibrium constants*. So, any of them can be used as a predictor of reaction spontaneity.

A reaction tends to be spontaneous if K_{eq} is significantly larger than 1, if E^{o} is positive , and if ΔG is less than zero.

(e) The chemical reaction for the combustion of 2 moles of hydrogen gas is as follows:

2H_{2(g)} + O_{2(g)} -------> 2H_{2}O_{(g)}.

This reaction indicates a decrease in entropy since two gas molecules are formed from three gas molecules.

However, ΔG is negative, because of the large negative value of the change in enthalpy (ΔH) relative to + TΔS. Hence the reaction is spontaneous.

(f) Ludwig Boltzmann was one of the primary pioneers of statistical mechanics applied to molecular theory.

Boltzmann mathematical definition of entropy:

S = klnW ----(9)

Equation (9) was established by Ludwig Boltzmann in the context of *statistical mechanics* applied to molecular theory.

The point of molecular theory is that a system is a *macrostate* made up of *microstates*. The temperature of the macrostate is the average kinetic energy of the motions of the particles that constitute the *microstates*.

Statistical mechanics is concerned with the uncertainty involved in the position of a particle at any given time. In other words, the probability of a particle being in a given microstate. Equation (9) assumes the same pobability for all the particles of the macrostate and expresses *entropy* as the product of the logarithm of *W, the multiplicity, or total number of microstates available to the system* and a constant *k* (1.380658 x 10^{-23}), known as the *Boltzmann constant*. This *multiplicity, W * is not the total number of particles, rather it is the total number of microstates that the particles could occupy, without changing the nature of the *macrostate*.

The linear proportionality between *entropy* and the *multiplicity* of a macrostate as established by Boltzmann's equation, is a reason for the conception of *entropy* as a measure of *disorder* and as a measure of energy dispersal. Increased *entropy* implies increased *multiplicity* of the system. This *multiplicity* is a tendency towards *randomness* which can be *disorderly*. *Randomness* is also a tendency towards increased kinetic energy and increased collisions of the particles of the macrostate. These collisions do *disperse* energy.

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