Question

Fill in the blanks: (a) \(\Delta H^{\circ}\) and \(\Delta G^{\circ}\) become equal at _____ \(K\). (b) \(\Delta G^{\circ}\) and \(\Delta G\) are equal when \(Q=\) _____. (c) \(S^{\circ}\) for steam is _____ than \(S^{\circ}\) for water.

Short answer

Also, compare the entropies of steam and water. Answer: ΔH° and ΔG° become equal when the temperature T = ΔH° / ΔS° K. The condition when ΔG° and ΔG are equal is when Q = 1. Comparing the entropies, S° for steam is greater than S° for water since steam has a higher degree of randomness and a higher number of microstates.

Step-by-step solution

a) Find the temperature when ΔH° and ΔG° become equal

Recall that the relationship between ΔH°, ΔG°, and T is given by the equation: ΔG° = ΔH° - TΔS° When ΔG° = ΔH°, the temperature at which this occurs can be found by making ΔG° equal to ΔH° and solving for T: ΔH° = ΔH° - TΔS° T = ΔH° / ΔS° Since we are given no values for ΔH° and ΔS°, we can just write the general relationship: T = ΔH° / ΔS° So, ΔH° and ΔG° become equal when T = ΔH° / ΔS° K.

b) Find the condition when ΔG° and ΔG are equal

The relationship between ΔG°, ΔG, R (gas constant), T, and Q is given by the equation: ΔG = ΔG° + RTlnQ When ΔG° = ΔG, the condition can be found by making ΔG° equal to ΔG and solving for Q: ΔG° = ΔG° + RTlnQ 0 = RTlnQ lnQ = 0 Taking the exponent of both sides, we get: Q = 1 So, ΔG° and ΔG are equal when Q = 1.

c) Comparing S° for steam and S° for water

Steam is the gas phase of water, while water refers to the liquid phase. The entropy (S°) of a substance is the measure of the number of possible microstates or the degree of randomness associated with the substance. In the gas phase, particles have more randomness and a higher number of microstates as they have a higher volume to move around and there are weaker intermolecular forces. Therefore, S° for steam is greater than S° for water. So, S° for steam is greater than S° for water.

Key concepts

Gibbs Free Energy (ΔG)

Gibbs Free Energy, represented by the symbol ΔG, is a fundamental concept in thermodynamics that plays a crucial role in predicting the direction of chemical reactions and determining the spontaneity of processes. Essentially, it serves as the driving force behind chemical changes. To fully grasp this concept, imagine you're looking at a ball at the top of a hill. Just as you would predict the ball to roll down the hill, chemists use ΔG to predict whether a reaction will 'roll forward' spontaneously. A negative value for ΔG indicates a spontaneous process, while a positive value suggests that the reaction is non-spontaneous and requires energy input. There's a noteworthy formula that connects ΔG to enthalpy (ΔH) and entropy (S°), expressed as: ΔG = ΔH - TΔS. The temperature (T) in this equation is measured in Kelvin (K), and it is the pivot point that balances the trade-off between ΔH and ΔS. Whenever ΔG equals ΔH, this indicates that the entropy term must be zero or the process is at a temperature that nullifies the TΔS term, leading to the two values being equal. This is crucial in understanding the dependency of reaction spontaneity on temperature.

Enthalpy (ΔH)

Enthalpy, symbolized as ΔH, can be related to the heat content of a system. It is one of the state functions used in chemistry to describe the thermodynamics of chemical reactions. Enthalpy changes can tell us whether a reaction releases heat (exothermic reaction, negative ΔH) or absorbs heat (endothermic reaction, positive ΔH). Picture a cozy campfire. When wood burns, energy is released as heat and light, which is a classic example of an exothermic reaction. On the other hand, when you dissolve certain salts in water and the solution cools down, that's an endothermic reaction absorbing heat from the surroundings. The interplay between ΔH and entropy (S°) is highlighted in the context of Gibbs Free Energy. It’s necessary to consider both enthalpy and entropy to determine the overall spontaneity of a reaction. For instance, it's entirely possible for a reaction with a positive (endothermic) ΔH to still be spontaneous if the increase in entropy (S°) is significant enough, especially at higher temperatures.

Entropy (S°)

Entropy is denoted by S° and it epitomizes disorder or randomness within a thermodynamic system. Think of entropy as a measure of how many different ways the particles within a system can be arranged without changing the overall energy—a classic move from order to disorder. For an intuitive understanding, consider a deck of cards. When the cards are neatly arranged into suits, they are in a state of low entropy. However, once shuffled, the order is lost, and the entropy increases. In the context of steam versus water (as addressed in the original exercise), steam has a higher entropy because it's a gas and its molecules are spread out and have more freedom to move, compared to water which is a liquid with molecules that are closer together. Notably, the second law of thermodynamics states that the total entropy of a closed system can never decrease over time. This implies that the universe tends towards a state of maximum entropy. During phase transitions, such as boiling water to create steam, entropy increases since there's a change from a more ordered liquid state to a less ordered gaseous state.
So, entropy doesn't just measure disorder—it predicts the direction and spontaneity of chemical processes, and in tandem with enthalpy, forms the backbone of our understanding of the Gibbs Free Energy formula.