Question

Consider the reaction $$2 \mathrm{O}(g) \longrightarrow \mathrm{O}_{2}(g)$$ a. Predict the signs of \(\Delta H\) and \(\Delta S .\) b. Would the reaction be more spontaneous at high or low temperatures?

Short answer

a. In the reaction, \(\Delta H < 0\) since the reaction is exothermic and \(\Delta S < 0\) as the disorder is decreasing. b. The reaction would be more spontaneous at low temperatures, as a smaller T∆S value keeps ∆G more negative.

Step-by-step solution

Analyze the reaction

The reaction given is: \[2 \mathrm{O}(g) \longrightarrow \mathrm{O}_{2}(g)\] We can see that two oxygen atoms are combining to form an oxygen molecule.

Predict the change in enthalpy (∆H)

In this reaction, two oxygen atoms are coming together and forming a covalent bond in the oxygen molecule. As a bond is formed, energy is released, which means the reaction is exothermic. Therefore, the change in enthalpy (∆H) will be negative. \[\Delta H < 0\]

Predict the change in entropy (∆S)

Entropy, represented by ∆S, is the measure of disorder or randomness in a system. In the given reaction, two gaseous oxygen atoms are coming together to form a single gaseous oxygen molecule. The number of particles in the system is decreasing, which means the disorder is decreasing. Therefore, the change in entropy (∆S) will be negative. \[\Delta S < 0\] #b. Would the reaction be more spontaneous at high or low temperatures?#

Understand the relation between spontaneity and Gibbs free energy (∆G)

A reaction is considered spontaneous if the change in Gibbs free energy (∆G) is negative. The relation between Gibbs free energy (∆G), enthalpy (∆H), and entropy (∆S) is given by the equation: \[\Delta G = \Delta H - T\Delta S\] Where T represents the temperature in Kelvin.

Analyze the effect of temperature on spontaneity

In our reaction, we found that ∆H and ∆S are both negative. If the temperature is high, the term T∆S will have a larger positive value due to the negative sign of ∆S. When this term is subtracted from the negative ∆H, it may cause the value of ∆G to become positive or less negative, decreasing the spontaneity of the reaction. On the other hand, if the temperature is low, the term T∆S will have a smaller positive value. When this term is subtracted from the negative ∆H, the value of ∆G remains more negative, which makes the reaction more spontaneous. Therefore, the reaction would be more spontaneous at low temperatures.

Key concepts

Enthalpy (ΔH)

Enthalpy, symbolized as ΔH, measures the total heat content of a system and is indicative of the energy change occurring during a reaction. When a chemical reaction occurs, bonds between atoms are broken and new bonds are formed. The key to understanding ΔH is to look at these bond energies: the energy taken to break bonds versus the energy released when new bonds form.

For instance, in the formation of an oxygen molecule ( O₂) from oxygen atoms, energy is released because a strong double bond is being formed. This energy release implies an exothermic reaction, characterized by a negative ΔH. Essentially, a negative ΔH signals that the system is releasing heat to its surroundings, which is a piece of the puzzle to predict whether a reaction will occur spontaneously.

Entropy (ΔS)

Entropy, symbolized as ΔS, is a measure of the disorder or randomness in a system. The greater the disorder, the higher the entropy. As per the second law of thermodynamics, in an isolated system, entropy tends to increase over time, leading towards equilibrium.

Examining our oxygen reaction, the initial state has two independent oxygen atoms, but the final state consists of these atoms bonded together as O₂. Here, the system transitions from a more dispersed (high entropy) state to a less dispersed (low entropy) one, resulting in a negative ΔS. Entropy is crucial for understanding spontaneity since an increase in entropy often favors spontaneous processes, but it is not the sole determining factor.

Gibbs Free Energy (ΔG)

Gibbs free energy, denoted as ΔG, is the parameter that determines spontaneity in a chemical reaction. A negative ΔG indicates that a reaction is spontaneous, meaning it may occur without the need for additional energy input. The equation ΔG = ΔH - TΔS brings together enthalpy, entropy, and temperature to predict the free energy change and thus the reaction's spontaneity.

The interaction between ΔH and ΔS is intriguing: a negative ΔH (exothermic reaction) tends to favor spontaneity, while a negative ΔS (decrease in disorder) does not. However, temperature plays a pivotal role in tipping the balance, as shown in the case of oxygen combining to form O₂. Here, both ΔH and ΔS are negative, making the sign of ΔG and spontaneity temperature-dependent.

Spontaneous Reaction

A spontaneous reaction is one that occurs without the addition of external energy once it is initiated. It’s important to note that 'spontaneous' does not necessarily mean quick; it means thermodynamically favored. The spontaneity of a reaction is determined by the values of ΔG, ΔH, and ΔS, and the environmental temperature.

In our oxygen reaction, the combination of ΔH, ΔS, and temperature can predict whether the process is a spontaneous one. Because both ΔH and ΔS are negative, an interesting question arises: under what conditions will this reaction proceed on its own? A lower temperature improves the chances of spontaneity since it makes the ΔG more negative, aligning with our intuitive understanding of the term 'spontaneous'.

Reaction Temperature

The temperature at which a reaction occurs is a vital component in determining the behavior of ΔG and, by extension, a reaction's spontaneity. As mentioned, the equation ΔG = ΔH - TΔS reveals how temperature ( T) influences the balance between enthalpy and entropy.

Consider our oxygen reaction. At high temperatures, the negative entropy change (ΔS) has a greater negative effect when multiplied by a high temperature, potentially making ΔG less negative or even positive. Conversely, at low temperatures, the effect of ΔS is minimized, allowing the negative ΔH to dominate. Thus, a lower temperature makes the reaction more likely to be spontaneous, providing us a clear connection between reaction temperature and the inherent tendency of a chemical process to proceed.