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Part 3: ΔS Entropy Changes and ΔG Free Energy Changes

3.3b Entropy Changes & Feasibility of a Chemical Reaction

 In Part 3.3a the total entropy change for the 'system' and its' surroundings' was considered and now applied to whether a chemical reaction is feasible i.e the criteria is a positive entropy is required. The entropy changes for thermal decomposition of limestone (a very endothermic reaction) and the burning of hydrogen in oxygen (a very exothermic reaction) are discussed in detail and in each case the feasibility of the reaction at different temperatures is discussed.

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3.3b Entropy Changes and feasibility of a Chemical Change

ΔS^θ_tot must be >0 for a chemical change to be feasible.

Example 3.3b1 Thermal decomposition of calcium carbonate (limestone)

• Energy is distributed not just in translational KE, but also in rotation, vibration and also distributed in electronic energy levels (if input great enough, bond breaks).

• All four forms of energy are quantised and the quanta ‘gap’ differences increases from trans. KE ==> electronic.

• Entropy (S) and energy distribution: The energy is distributed amongst the energy levels in the particles to maximise their entropy.

• Entropy is a measure of both the way the particles are arranged AND the ways the quanta of energy can be arranged.

• We can apply ΔS^θ_sys/surr/tot ideas to chemical changes to test feasibility of a reaction:

  • ΔS^θ_tot = ΔS^θ_sys +  ΔS^θ_surr

  • ΔS^θ_tot must be >=0 for a chemical change to be feasible.

  • For example: CaCO_3(s) ==> CaO_(s) + CO_2(g) 

    • ΔS^θ_sys = ΣS^θ_products – ΣS^θ_reactants 

    • ΔS^θ_sys = S^θ_CaO(s) + S^θ_CO2(g) – S^θ_CaCO3(s) 

    • ΔS^θ_surr is –ΔH^θ/T(K) and ΔH is very endothermic (very +ve),

  • Now ΔS^θ_sys is approximately constant with temperature and at room temperature the ΔS^θ_surr term is too negative for ΔS^θ_tot to be plus overall.

  • But, as the temperature is raised, the ΔS^θ_surr term becomes less negative and eventually at about 800^oC ΔS^θ_tot becomes plus overall (and ΔG^θ becomes negative), so the decomposition is now chemically, and 'commercially' feasible in a lime kiln.

CaCO_3(s) ==> CaO_(s) + CO_2(g)  ΔH^θ = +179 kJ mol^–1  (very endothermic)

This important industrial reaction for converting limestone (calcium carbonate) to lime (calcium oxide) has to be performed at high temperatures in a specially designed limekiln – which these days, basically consists of a huge rotating angled ceramic lined steel tube in which a mixture of limestone plus coal/coke/oil/gas? is fed in at one end and lime collected at the lower end. The mixture is ignited and excess air blasted through to burn the coal/coke and maintain a high operating temperature.

ΔS^θ_sys = ΣS^θ_products – ΣS^θ_reactants

ΔS^θ_sys = S^θ_CaO(s) + S^θ_CO2(g) – S^θ_CaCO3(s) = (40.0) + (214.0) – (92.9) = +161.0 J mol^–1 K^–1

ΔS^θ_surr is –ΔH^θ/T = –(179000/T)

ΔS^θ_tot = ΔS^θ_sys +  ΔS^θ_surr

ΔS^θ_tot = (+161) + (–179000/T) = 161 – 179000/T

If we then substitute various values of T (in Kelvin) you can calculate when the reaction becomes feasible.

For T = 298K (room temperature)

ΔS^θ_tot = 161 – 179000/298 = –439.7 J mol^–1 K^–1, no good, negative entropy change

For T = 500K (fairly high temperature for an industrial process)

ΔS^θ_tot = 161 – 179000/500 = –197.0, still no good

For T = 1200K (limekiln temperature)

ΔS^θ_tot = 161 – 179000/1200 = +11.8 J mol^–1 K^–1, definitely feasible, overall positive entropy change

Now assuming ΔS^θ_sys is approximately constant with temperature change and at room temperature the ΔS^θ_surr term is too negative for ΔS^θ_tot to be plus overall. But, as the temperature is raised, the ΔS^θ_surr term becomes less negative and eventually at about 800–900^oC ΔS^θ_tot becomes plus overall, so the decomposition is now chemically, and 'commercially' feasible in a lime kiln.

You can approach the problem in another more efficient way by solving the total entropy expression for T at the point when the total entropy change is zero. At this point calcium carbonate, calcium oxide and carbon dioxide are at equilibrium.

ΔS^θ_tot–equilib = 0 = 161 – 179000/T, 179000/T = 161, T = 179000/161 = 1112 K

This means that 1112 K is the minimum temperature to get an economic yield. Well at first sight anyway. In fact because the carbon dioxide is swept away in the flue gases so an equilibrium is never truly attained so limestone continues to decompose even at lower temperatures.

Lime is actually formed at temperatures above 900^oC (1173K) and a typical modern limekiln operating temperature range is 950–980^oC (from web). These calculations are approximate above 298K because it assumed that enthalpy and entropy values do not change with temperature. This is not true BUT the above calculations exemplify the sort of calculation you can do to calculate at what temperature becomes feasible.

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In the next few examples I haven't bothered listing the entropy values, I've just slotted them into the entropy equations and the reaction enthalpy values are by the relevant chemical equation.

Example 3.3b2 The entropy change in forming water from hydrogen and oxygen

(i) Calculate the entropy changes ΔS for the combustion of hydrogen to form water vapour.

H_2(g) + ¹/₂O_2(g) ==> H₂O_(g)  ΔH_sys = –242 kJ mol^–1

ΔS^θ_sys–298K = S^θ_298K(products) – S^θ_298K(reactants) J mol^–1 K^–1

Watch the signs, all absolute S values are +ve, but ΔS can be +ve or –ve.

ΔS^θ_sys–298K = S^θ_298K(H₂O_(g)) – S^θ_298K(H_2(g)) – S^θ_298K(O_2(g))/2

ΔS^θ_sys = 188.7 –130.6 –205/2 = –44.4 J mol^–1 K^–1

This shows what you might expect in terms of an entropy change, you have gone from 1.5 moles of gas particles down to just 1.0 moles of gas particles – therefore less gas particles means less microstates or ways of arranging the particles. BUT we know this reaction is very feasible with the help of a lit splint! POP! Therefore to complete the 'feasibility calculation' we must calculate the total entropy change.

ΔS^θ_tot = ΔS^θ_sys +  ΔS^θ_surr

ΔS^θ_tot = ΔS^θ_sys +  (–ΔH^θ_sys/T)

ΔS^θ_tot = –44.4 + (––242 x 1000/298) don't forget ΔH values come along in kJ

ΔS^θ_tot = –44.4 +812.1  = +767 J mol^–1 K^–1

So, despite the loss of gas particles the total entropy change is very positive! Why?

The reason for this is that the reaction is very exothermic, and the heat released is absorbed by the surroundings whose particles absorb the energy and distribute it in many available ways in the various energy levels e.g. kinetic energy – translational levels or rotational/vibrational quantum levels.

(ii) Calculate the entropy changes ΔS for the combustion of hydrogen to form liquid water.

H_2(g) + ¹/₂O_2(g) ==> H₂O_(l)  ΔH_sys = –286 kJ mol^–1

ΔS^θ_sys–298K = S^θ_298K(H₂O_(l)) – S^θ_298K(H_2(g)) – S^θ_298K(O_2(g))/2

ΔS^θ_sys = 69.9 –130.6 –205/2 = –163.2 J mol^–1 K^–1 notice the much smaller entropy for liquid water

The entropy decrease in this case is much larger in forming liquid water compared to forming gaseous water. The liquid state offers less ways of distributing the particles and there energy.

ΔS^θ_tot = ΔS^θ_sys +  ΔS^θ_surr

ΔS^θ_tot = ΔS^θ_sys +  (–ΔH^θ_sys/T)

ΔS^θ_tot = –163.2 + (––286 x 1000/298) don't forget ΔH values come along in kJ

ΔS^θ_tot = –163.2 +959.7  = +796.5 J mol^–1 K^–1

So despite, the smaller entropy of the product liquid water, the overall entropy change is even more positive! because reaction (ii) involves condensation which is an exothermic process and more energy is transferred to increase the entropy of the surroundings.

Example 3.3b2 The entropy change in forming water from hydrogen ions and hydroxide ions – neutralisation

Calculate the entropy change for the neutralisation reaction

H⁺_(aq) + OH^–_(aq) ==> H₂O_(l)  ΔH_sys = –57.1 kJ mol^–1

S^θ₂₉₈(H⁺_(aq)) = 0 by definition, S^θ₂₉₈(OH^–_(aq)) = –10.7, S^θ₂₉₈(H₂O_(l)) = 69.9

Its difficult to estimate absolute entropy values for aqueous ions, so the aqueous hydrogen ion is given an arbitrary value of zero and all the entropies of other ions are measured against it so that you can get negative entropy values here – but don't worry about it – I don't know if its required for UK A2 level courses.

ΔS^θ_sys–298K = S^θ_298K(products) – S^θ_298K(reactants) J mol^–1 K^–1

ΔS^θ_sys–298K = S^θ_298K(H₂O_(l)) – S^θ_298K(H⁺_(aq)) – S^θ_298K(OH^–_(aq))

ΔS^θ_sys = 69.9 –0  ––10.7 = +80.6 J mol^–1 K^–1

Despite going from two particles to one particle, an apparent considerable decrease in the possible ways of arranging the particles there is still a large increase in in entropy of the system! Why?

Both of the ions are hydrated and the hydrogen ion is strongly associated with water molecules,

e.g. it forms H₃O⁺, H₅O₂⁺, H₇O₃⁺ ions etc. (H⁺ and 1, 2, 3 etc. associated water molecules) because of its strong polarising power and electrostatic field effect on the negative lone pair electron oxygens of water molecules. So, when water molecules are formed, all the water molecules associated with the hydrated ions are freed! and thereby raising the entropy of the system. Just to complete the calculation, though its feasibility hardly needs any introduction ...

ΔS^θ_tot = ΔS^θ_sys +  ΔS^θ_surr

ΔS^θ_tot = ΔS^θ_sys +  (–ΔH^θ_sys/T)

ΔS^θ_tot = +80.6 + (––57.1 x 1000/298) don't forget ΔH values come along in kJ

ΔS^θ_tot = +80.6 +191.6  = +272.2 J mol^–1 K^–1

A very healthy increase in entropy!

FINAL COMMENTS

summary to do

These theoretical calculations can be used for any reaction BUT there are limitations:

You can’t say the reaction will definitely spontaneously happen (go without help!) because there may be rate limits especially if the reaction has a high activation energy or a very low concentration of an essential reactant.

However you can employ a catalyst, raise reactant concentrations or raise the temperature to get the reaction going! There is usually a way of getting most, but not all, feasible reactions to actually occur.

For more details on this last point see 3.6 Kinetic stability versus thermodynamic instability for a detailed discussion

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