Question

What is the change in entropy when \(0.200 \mathrm{~mol}\) of potassium freezes at \(63.7^{\circ} \mathrm{C}\left(\Delta H_{\mathrm{fus}}=2.39 \mathrm{~kJ} / \mathrm{mol}\right) ?\)

Short answer

The change in entropy is 1.42 J/K.

Step-by-step solution

- Convert Temperature to Kelvin

First, convert the given freezing temperature from Celsius to Kelvin. The formula for this conversion is: \[ T(K) = T(^{\text{o}}C) + 273.15 \] Substituting the given temperature: \[ T(K) = 63.7 + 273.15 = 336.85 \text{ K} \]

- Calculate Entropy Change

Use the formula for the change in entropy during a phase change: \[ \Delta S = \frac{\Delta H_{\text{fus}}}{T} \] Given \(\Delta H_{\text{fus}} = 2.39 \, \text{kJ/mol} \), convert it to J/mol: \[ \Delta H_{\text{fus}} = 2.39 \, \text{kJ/mol} \times 1000 \, \text{J/kJ} = 2390 \, \text{J/mol} \] Substituting \(\Delta H_{\text{fus}}\) and \( T \): \[ \Delta S = \frac{2390 \, \text{J/mol}}{336.85 \, \text{K}} = 7.10 \, \text{J/(mol·K)} \]

- Calculate Total Entropy Change

Multiply the entropy change per mole by the number of moles to get the total entropy change: \[ \Delta S_{\text{total}} = 7.10 \, \text{J/(mol·K)} \times 0.200 \, \text{mol} = 1.42 \, \text{J/K} \]

Key concepts

Entropy

Entropy, often symbolized as \(S\), is a fundamental concept in thermodynamics. It represents the degree of disorder or randomness in a system. The second law of thermodynamics states that for any spontaneous process, the total entropy of a system and its surroundings always increases. This means that natural processes tend to move towards a state of maximum entropy.
Understanding entropy helps chemists and physicists predict the direction of chemical reactions and physical transformations.
Here's a simple way to conceptualize it: Imagine a clean room (low entropy) getting progressively messier (high entropy) without any added effort. It's much easier for the room to get messy than for it to clean itself spontaneously.
In a mathematical sense, the change in entropy \(\triangle S\) during a process can be calculated using:
\[ \triangle S = \frac{\triangle Q_{\text{rev}}}{T} \]
Where \(\triangle Q_{\text{rev}}\) is the heat absorbed or released in a reversible process, and \(T\) is the temperature in Kelvin.

Phase Change

Phase changes occur when a substance transitions between different states of matter: solid, liquid, and gas. Common phase changes include melting, freezing, vaporization, condensation, sublimation, and deposition.
Each phase change involves a change in enthalpy (\( \triangle H \)), which is the heat content of a system. For instance, when a substance melts or freezes, we refer to \( \triangle H \) as the enthalpy of fusion (\( \triangle H_{\text{fus}} \)).
When calculating the entropy change during a phase change, the formula used is:
\[ \triangle S = \frac{\triangle H_{\text{fus}}}{T} \]
Here, \( \triangle H_{\text{fus}} \) is the enthalpy change of fusion, and \( T \) is the temperature (in Kelvin) at which the phase change occurs.
In our example, potassium freezes at \( 63.7 \text{°C} \) with \( \triangle H_{\text{fus}} = 2.39 \text{ kJ/mol} \). We first convert the temperature to Kelvin, then calculate the entropy change per mole, and finally multiply by the number of moles to find the total entropy change.

Thermodynamics

Thermodynamics is the study of energy, heat, work, and how they interrelate in physical and chemical processes. It provides a framework to understand how energy is transferred and transformed.
Key concepts in thermodynamics include:

• First Law of Thermodynamics: Energy cannot be created or destroyed, only transformed from one form to another. This is also known as the law of energy conservation.
• Second Law of Thermodynamics: The entropy of an isolated system always increases over time. This law explains why certain processes are irreversible and predicts the direction of spontaneous processes.
• Third Law of Thermodynamics: As the temperature of a system approaches absolute zero, the entropy of the system approaches a minimum value.

In applying these principles, we can analyze processes such as phase changes, chemical reactions, and energy transfers. When calculating entropy changes, we consider both the heat exchanged and the temperature at which the exchange occurs. By understanding these fundamental laws, we can predict how a system will behave under various conditions.