Animated Solution for Chemistry - Chemical Thermodynamics: For a dimerisation reaction $2A(g)⟶A_2(g)$, at $298\text{ K}$, $\Delta U^{\ominus }=-20{\text{ kJ mol}}^{-1}$, $\Delta S^{\ominus }=-30{\text{ JK}}^{-1}{\text{ mol}}^{-1}$, then the $\Delta G^{\ominus }$ will be ......... $\text{J}$.

Imagine you are observing a microscopic dance floor where individual gas molecules of $A$ are floating around freely. Suddenly, two of these molecules collide and decide to stick together, forming a single, larger molecule known as a dimer, $A_2$. This process is called a dimerisation reaction, represented chemically as $2A(g)⟶A_2(g)$.

In the realm of chemical thermodynamics, every reaction is accompanied by changes in energy and disorder. For this specific dimerisation at a temperature of $298\text{ K}$, we are given two crucial pieces of information: the change in standard internal energy, $\Delta U^{\ominus }=-20{\text{ kJ mol}}^{-1}$, and the change in standard entropy, $\Delta S^{\ominus }=-30{\text{ J K}}^{-1}{\text{ mol}}^{-1}$. Our ultimate quest is to determine the standard Gibbs free energy change, $\Delta G^{\ominus }$, which will tell us if this molecular dance happens spontaneously.

Before we can calculate the Gibbs free energy, we need to find the enthalpy change, $\Delta H^{\ominus }$. Enthalpy is essentially the total heat content of the system, and it is intimately connected to the internal energy through the work done by expanding or contracting gases. The bridge connecting them is the classic equation:

Here, $\Delta n_g$ represents the change in the number of moles of gas during the reaction. Let's look at our balanced equation: we start with $2$ moles of gaseous reactants and end up with $1$ mole of gaseous product. Therefore, $\Delta n_g=1-2=-1$. This negative value makes perfect physical sense; the system is contracting as two molecules merge into one.

Now, we must be incredibly careful with our units. A common trap is mixing kilojoules with Joules. Let's convert our internal energy to Joules: $\Delta U^{\ominus }=-20000{\text{ J mol}}^{-1}$. Substituting our values into the equation, we get:

Let's take a moment to appreciate the entropy change, $\Delta S^{\ominus }=-30{\text{ J K}}^{-1}{\text{ mol}}^{-1}$. Entropy is a measure of disorder or randomness. Since two freely moving gas molecules are combining to form one, the system is becoming more ordered. Hence, the entropy decreases, which is perfectly reflected by the negative sign of $\Delta S^{\ominus }$.

Gibbs Free Energy

Now we have all the pieces of the puzzle. To determine if the reaction is spontaneous, we turn to the master equation of chemical thermodynamics, the Gibbs Helmholtz equation:

This beautiful equation balances the two driving forces of nature: the tendency to achieve a lower energy state (negative $\Delta H$) and the tendency to achieve maximum disorder (positive $\Delta S$).

Spontaneity

Let's carefully substitute our calculated enthalpy, the given temperature, and the entropy into the equation. Watch out for the double negative!

Rounding to two decimal places, we get our final answer: $-13537.57\text{ J}$.

The negative sign of $\Delta G^{\ominus }$ is the grand finale. It tells us that at $298\text{ K}$, the drive towards a lower energy state (the exothermic enthalpy) overpowers the decrease in disorder. Therefore, the dimerisation of molecule $A$ is a spontaneous process!