A-level Chemistry/WJEC/Module 3/Entropy

Entropy

Entropy is often considered to be the measure of disorder in a system.

An Example

The mass spectrum of bromine includes fragments with masses of 158, 160 and 162. this is because bromine has two isotopes with roughly equal abundance, ⁷⁹Br and ⁸¹Br. The 158 fragment is ⁷⁹Br₂⁺ and the 162 fragment is ⁸¹Br₂⁺. These two fragments are of equal abundance.

The 160 fragment is ⁷⁹Br-⁸¹Br⁺. It is twice as abundant as the ⁷⁹Br₂⁺ and ⁸¹Br₂⁺ fragments. Why? There are two ways to make a ⁷⁹Br-⁸¹Br⁺ fragment - start with a ⁷⁹Br atom and add a ⁸¹Br atom, or start with a ⁸¹Br atom and add a ⁷⁹Br atom. Both "microstates" produce the same outcome - a fragment with a mass of 160 - but there is twice the chance of making this mixed fragment.

Microstates

Systems with high entropy have many "microstates" that can produce the same observed state. The particles of a gas can exist in a Vast number of microstates without any change of pressure or temperature. In a solid, there is less freedom and hence lower entropy.

It is not an equation you will need to know, but entropy can be defined as

S = R ln Ω

where R is the ideal gas constant, 8.31 J mol^-1 K^-1, and Ω is the number of microstates with the same energy as the system.

If you toss a coin 4 times, there are five possible outcomes: 4 heads, 4 tails, and mixes of heads and tails. 4 heads and 4 tails are least likely because there is only one sequence (microstate) that results in 4 heads and one sequence for 4 tails. There are 6 different ways to get 2 heads and 2 tails, however. The entropy of the result "2 tails and 2 heads" is more than the entropy of 4 heads, or of 4 tails.

In Chemistry

The entropy of a chemical system is a measure of its disorder or chaos. More precisely, it is a measure of the dispersion of energy. A solid has low entropy (low chaos, orderly) because the molecules are locked into a rigid structure. Their energy is not dispersed freely. A gas has high entropy (high chaos, disorderly) because the molecules are free to move about randomly. The energy of the system is dispersed over a large area with unlimited possibilities of the location of each molecule.

As temperature decreases, so does the importance of entropy. Theoretically, at absolute zero (0 K, or -273 °C), the entropy of the system would be zero. This is because the solid would be perfectly crystallised so that its energy is not dispersed at all.

The Second Law of Thermodynamics tells us that the entropy of the universe always increases. An analogy: If you have tiled a floor, the entropy of the system is low. The tiles are very orderly, with each tile fitting into a very specific location. However, the tiles have to be carefully placed in position; A pile of loose tiles is very disorderly. The tiles can be in any position and still be a random pile of tiles. The entropy is naturally larger, even if the tiles are often higher piled on top of one another and therefore have higher energy. Tiles will not spontaneously cover a floor neatly, because high entropy is much more likely than low entropy.

Paving stones have to be carefully positioned in a pattern. On its own, a pile of stones does not cover a surface efficiently.

Entropy Changes

When analysing the entropy change of a chemical reaction, you would need specific numbers. As a guideline, you can estimate the entropy change based on some basic rules:

• Melting and boiling increases entropy
• Freezing and condensing decreases entropy
• Dissolving a solid or liquid solute increases entropy
• Dissolving a gaseous solute decreases entropy (gas has high entropy, which is lost if it is part of a solution)
• Forming precipitates decreases entropy
• Mixed gases (e.g. air) have higher entropy than pure gases (e.g. pure N₂)

If you do happen to know the absolute entropy of substances in a reaction (by looking it up in a chart), you can calculate the change in entropy. Entropy is symbolised with S. The change in entropy is ΔS. As with enthalpy, the Plimsoll symbol (ΔS^o) represents STP. The change in entropy is the absolute entropy of the products minus the absolute entropy of the reactants.

Δ_rS = ΣS(products) - ΣS(reactants)

General Information

American scientist Josiah Willard Gibbs (1839-1903) created the theory of available energy, known as Gibbs Free Energy, in 1873. The theory relates the energy changes within the chemical reaction and how they depend upon the following quantities: enthalpy, temperature, reagents concentration and entropy of the system. In other words, these quantities will determine whether the reaction is favourable (exergonic) or not (endergonic).

The free energy change of a reaction (ΔG) can tell us whether or not a reaction occurs spontaneously. Reactions that occur spontaneously have a negative ΔG value. When ΔG is positive, the reaction does not occur spontaneously. When a system forms an equilibrium, then ΔG is close to zero. The ΔG of a reaction is the free energy of the products minus the free energy of the reactants, making it is independent of the reaction pathway. However, the value of ΔG provides no information on the rate of a reaction.

Gibbs Free Energy Equation

The signs of entropy change (Δ_rS) or enthalpy change (Δ_rH) cannot, on their own, determine if a process is spontaneous. For instance, an exothermic reaction becomes spontaneous under certain conditions, and the reverse endothermic reaction can also become spontaneous under different conditions. The water/ice equilibrium is an example to illustrate such conditions.

H₂O(l) → H₂O(s) Δ_rH = -6.02 kJ mol^-1, Δ_rS = -22.04 J mol^-1 K^-1 (an exothermic reaction, forming new bonds to make the solid but also losing entropy as the molecules have less freedom in the solid state; spontaneous when θ < 0 °C)

H₂O(s) → H₂O(l) Δ_rH = +6.02 kJ mol^-1, Δ_rS = +22.04 J mol^-1 K^-1 (an endothermic reaction; spontaneous when θ > 0 °C)"

In both reactions, one cannot use enthalpy alone or entropy alone to determine the direction of a spontaneous reaction. Gibbs free energy (Δ_rG) combines enthalpy with the effect of entropy (which is proportional to temperature):

The most commonly used equation for Gibbs free energy is:

Δ_rG = Δ_rH - TΔ_rS

Where ΔH is change in enthalpy, T is the absolute temperature of the system (in kelvin, K), ΔS is change in entropy of the system. Take care when using this formula, because Δ_rS is usually measured in J mol^-1 K^-1 while Δ_rG and Δ_rH are measured in kJ mol^-1. It is simplest to convert Δ_rS to kJ e.g. Δ_rS = +22.04 J mol^-1 K^-1 = +0.02204 kJ mol^-1 K^-1.

Numerical Meaning of ΔG

The more negative the Gibbs free energy, the more favourable the reaction, and the larger the value of the equilibrium constant, K.

• If ΔG < 0 (negative), then the reaction will proceed spontaneously, meaning the reaction is favourable. K > 1.
• If ΔG = 0 (equal to zero), then the reaction is at equilibrium. K = 1
• If ΔG > 0 (positive), then the reaction will not proceed spontaneously, meaning the reaction is unfavourable. K < 1.

When a reaction becomes spontaneous, the effect of entropy balances the effect of enthalpy and Δ_rG = 0

T = Δ_rH / Δ_rS

In the example of water above, T = +6.02 / +0.02204 = 273 K i.e. θ = 0 °C. Notice that the values for the reverse reaction give the same value: T = -6.02 / -0.02204 = 273 K.

Is this the temperature where the reaction becomes spontaneous, or where it stops being spontaneous? If T = 0 K then Δ_rG = Δ_rH. For the reaction H₂O(l) → H₂O(s), Δ_rH = -6.02 kJ mol^-1, meaning that this reaction is spontaneous at 0 K and stops being spontaneous when T reaches 273 K.

Free energy can also be found if the free energies of formation are known:

Δ_rG = ΣΔ_fG(products) - ΣΔ_fG(reactants)

References

Reece, Jane (2011). Biology. Pearson. ISBN 978-0-321-55823-7. {{cite book}}: Text "coauthors+ Lisa A. Urry, Michael L. Cain, Steven A. Wasserman, Peter V. Minorsky, Robert B. Jackson" ignored (help)

Seader, J. D., and Ernest J. Henley. Separation Process Principles. Hoboken, NJ: Wiley, 2006. Print.