The quantum casino: tutorials

∆G and temperature

We can see from the expression

∆G = ∆H – T∆S_system

that the value of ∆G depends on temperature. Sometimes, changing the temperature can change the sign of ∆G. This explains why some reactions go in one direction at one temperature and in the opposite direction at a different temperature.

Let us look at the reaction

CaO(s) + CO₂(g) CaCO₃(s)

that we considered above.

∆H is –178 kJ mol^-1(–178 000 J mol^-1) and ∆S_system is –161 J K^-1 mol^-1

Note ∆S_system is calculated from the difference between the total entropy of the product(s) and the total entropy of the reactants. The data can be obtained from Table 1.

S (CaCO₃) = 93 J K^-1 mol^-1

S (CaO) = 40 J K^-1 mol^-1

S (CO₂) = 214 J K^-1 mol^-1

So ∆S_system = 92 – (40 + 214) = –161 J K^-1 mol^-1

If ∆G is negative, the reaction ‘goes’ from left to right. If it is positive, the reaction goes in the opposite direction.

Let us put in some figures.

At 298 K (room temperature)

∆G = ∆H – T∆S_system

∆G = –178 – (298 x –0.161) (remember to convert entropy in J K^-1 mol^-1 to kJ K^-1 mol^-1)

∆G = –178 + 48

∆G = –130 kJ mol^-1

∆G is negative and the reaction is feasible at this temperature. It is the reaction that happens in an absorption tube.

At 1500 K (the sort of temperature in a lime kiln)

∆G = ∆H – T∆S_system

∆G = –178 – (1500 x –0.161) (remember to convert entropy in J K^-1 mol^-1 to kJ K^-1 mol^-1)

∆G = –178 + 242

∆G = +64 kJ mol^-1

∆G is positive and the reaction is not feasible at this temperature. In fact the reverse reaction, which will have ∆G = –64 kJ mol^-1 is feasible and this is the reaction that occurs in a lime kiln

The ‘crossover point ‘is when ∆G is zero. At this point, 0 = ∆H – T∆S_system

ie ∆H = T∆S_system

Putting in the figures

–178 000 = T x –161

So T = 178 000 / 161 = 1105 K

So this temperature is a sort of ‘tipping point’ between the reaction being feasible or not.

Note. In this calculation we have assumed that ∆H does not change with temperature. This is an approximation only, but for most reactions the change with temperature is small. ∆S_system may change with temperature. For example if one of the reactants or products changes state (melts or boils) this will affect ∆S_system. In fact the reaction does not simply ‘flip’ between feasibility and non–feasibility (or going in one direction or the other). If the reaction were to take place in a closed system, an equilibrium would be set up. As a rule of thumb, reactions with ∆G more negative than –60 kJ mol^-1 are considered to go to completion while those with ∆G more positive than +60 kJ mol^-1 are considered not to occur at all. Between these temperatures, the reaction is considered to be reversible and in a closed system an equilibrium would be set up containing some of all the reactants and products concerned.

∆G and temperature