Question

For ammonia ( \(\mathrm{NH}$$_3\)), the enthalpy of fusion is \(5.65 \mathrm{kJ} / \mathrm{mol}\) and the entropy of fusion is \(28.9 \mathrm{J} / \mathrm{K} \cdot\) mol. a. Will \(\mathrm{NH}_{3}(s)\) spontaneously melt at \(200 . \mathrm{K} ?\) b. What is the approximate melting point of ammonia?

Short answer

a. Yes, NH₃(s) will spontaneously melt at 200 K. b. The approximate melting point of ammonia is \(195.5\, \mathrm{K}\).

Step-by-step solution

Write down the given values

The given values are: Enthalpy of fusion, \(\Delta H = 5.65\, \mathrm{kJ/mol} = 5650\, \mathrm{J/mol}\) Entropy of fusion, \(\Delta S =28.9\, \mathrm{J/K \cdot mol}\) Temperature, \(T = 200\, \mathrm{K}\)

Calculate Gibbs free energy change, \(\Delta G\)

Use the Gibbs free energy change equation: \[\Delta G = \Delta H - T \Delta S\] Plug in the given values: \[\Delta G = (5650\, \mathrm{J/mol}) - (200\, \mathrm{K}) \cdot (28.9\, \mathrm{J/K \cdot mol})\]

Determine if the process is spontaneous

Using a calculator, we find that: \[\Delta G = 5650 - 200 \cdot 28.9 = 5650 - 5780 = -130\, \mathrm{J/mol}\] Since the value of \(\Delta G\) is negative, the process of melting ammonia at 200 K is spontaneous. So, the answer to the question (a) is: Yes, NH₃(s) will spontaneously melt at 200 K. b. Find the approximate melting point of ammonia:

Use the Gibbs free energy change equation at melting point

At the melting point, the process of melting is at equilibrium, so \(\Delta G\) is equal to 0: \[0 = \Delta H - T \Delta S\] We are given: \(\Delta H = 5650\, \mathrm{J/mol}\) \(\Delta S = 28.9\, \mathrm{J/K \cdot mol}\)

Solve for the melting point temperature \(T\)

Re-arrange the above equation to find the temperature: \[T = \frac{\Delta H}{\Delta S}\]

Calculate the melting point temperature

Plug in the given values: \[T = \frac{5650\, \mathrm{J/mol}}{28.9\, \mathrm{J/K \cdot mol}}\] Using a calculator, we get: \[T \approx195.5\, \mathrm{K}\] So, the answer to the question (b) is: The approximate melting point of ammonia is \(195.5\, \mathrm{K}\).

Key concepts

Gibbs Free Energy

Gibbs free energy, denoted as \( \Delta G \), is a thermodynamic quantity that represents the maximum amount of reversible work that can be performed by a system at constant temperature and pressure. It is given by the equation \( \Delta G = \Delta H - T\Delta S \), where \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy.

The sign of \( \Delta G \) helps determine whether a process will occur spontaneously. A negative \( \Delta G \) implies that the process is spontaneous, while a positive value indicates non-spontaneity. At equilibrium, \( \Delta G = 0 \) and the system experiences no net change as forward and reverse processes occur at equal rates.

Entropy of Fusion

The entropy of fusion, symbolized as \( \Delta S_{\text{fusion}} \), represents the change in entropy when a substance changes from a solid to a liquid at its melting point. It is a measure of the increased randomness or disorder in the system due to the phase change.

More precisely, it quantifies the energy distribution among particles as solid arrangements break down into a less ordered, more mobile liquid state. For the substance ammonia \( (\mathrm{NH}_3) \), this value indicates how much the degree of disorder increases when it melts. A higher entropy of fusion means a significant increase in disorder during the melting process. Understanding the entropy change can tell us about the energy distribution involved in phase transitions.

Spontaneous Process

A spontaneous process is one that occurs naturally without the input of external energy. It is driven by a system's move towards a state of lower Gibbs free energy, resulting in a more thermodynamically stable condition.

This concept is not to be confused with the rate of the process; even if a process is spontaneous, it may happen slowly. Spontaneity is determined by \( \Delta G \) as mentioned earlier. If \( \Delta G < 0 \), the process is spontaneous. For instance, as in the exercise, the melting of ammonia at a certain temperature is spontaneous if the calculated \( \Delta G \) is negative.

Melting Point

The melting point of a substance is the temperature at which it changes from a solid to a liquid at atmospheric pressure. It is a unique physical property of each material and can be determined by analyzing the dependencies between enthalpy, entropy, and Gibbs free energy.

At the melting point, the transition from solid to liquid is in equilibrium, and \( \Delta G = 0 \), meaning that the system sees no net change despite the ongoing processes. By setting the Gibbs free energy change equation to zero and rearranging it, \( T = \frac{\Delta H}{\Delta S} \), one can calculate the melting point temperature of the substance, such as ammonia in the exercise provided.