Problem 24 - Chemistry The Central Science In Si Units

Question

The element sodium (Na) melts at \(97.8^{\circ} \mathrm{C}\), and its molar enthalpy of fusion is \(\Delta H_{\text {fus }}=2.60 \mathrm{~kJ} / \mathrm{mol}\). (a) When molten sodium solidifies to \(\mathrm{Na}(s),\) is \(\Delta S\) positive or negative? (b) Calculate the value of \(\Delta S\) when \(50.0 \mathrm{~g}\) of \(\mathrm{Na}(I)\) solidifies at \(97.8^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

(a) ΔS is negative. (b) ΔS ≈ -15.2 J/K.

Step 1: Determine the Sign of ΔS for Solidification

When a substance solidifies, it changes from a more disordered state (liquid) to a more ordered state (solid). This transition results in a decrease in entropy (ΔS). Therefore, the sign of ΔS is negative.

Step 2: Calculate Moles of Sodium

To calculate ΔS, we need to know the number of moles of sodium. The molar mass of sodium (Na) is approximately 23.0 g/mol. Thus, for 50.0 g of Na:\[\text{Moles of Na} = \frac{50.0 \, \text{g}}{23.0 \, \text{g/mol}} \approx 2.17 \, \text{mol}\]

Step 3: Calculate ΔS Using the Enthalpy of Fusion

The entropy change (ΔS) can be calculated using the formula: \[\Delta S = -\frac{\Delta H_{\text{fus}}}{T}\]where ΔH_fus = 2.60 kJ/mol and T must be in Kelvin: \[T = 97.8 + 273.15 = 370.95 \text{ K}\]Insert the values:\[\Delta S = -\frac{2.60 \, \text{kJ/mol}}{370.95 \, \text{K}} \approx -0.00701 \, \text{kJ/(mol·K)}\]For 2.17 mol of Na:\[\Delta S = -0.00701 \, \text{kJ/(mol·K)} \times 2.17 \, \text{mol} \approx -0.0152 \, \text{kJ/K}\]Convert kJ to J:\[\Delta S \approx -15.2 \, \text{J/K}\]

Key Concepts

Understanding Entropy Change

Entropy is a measure of the disorder or randomness of a system. In chemistry, we often discuss changes in entropy during phase transitions. When sodium (Na) undergoes a change from a liquid to a solid, it transitions from a more disordered to a less disordered state. This transition results in a decrease in entropy. Entropy change, denoted as \( \Delta S \), can be positive or negative.

• Positive \( \Delta S \) implies an increase in disorder
• Negative \( \Delta S \) implies a decrease in disorder

In the case of sodium solidifying, \( \Delta S \) is negative because the atoms arrange into an orderly solid structure, decreasing randomness.

Exploring the Solidification Process

The solidification process occurs when a liquid turns into a solid upon cooling. During solidification, energy is released as the liquid's molecules form a more ordered array, typically in a crystalline structure. This release of energy, often observed as the molar enthalpy of fusion, involves the removal of heat from the system.

A simple analogy is freezing water to form ice. Just as water molecules slow down and settle into a structured state, sodium atoms do likewise when moving from a liquid to a solid form. The solidification of sodium releases energy in the form of heat, illustrating the concept of molar enthalpy of fusion.

• The energy change during solidification is negative.
• The process decreases the entropy of the system.

This is because the structured solid state is less random than that of the liquid.

Sodium (Na) Melting Point

The melting point of sodium is a fundamental physical property that's critical for understanding its phase transitions. Sodium melts at \(97.8^{\circ} \text{C}\), a temperature where it transitions between solid and liquid states.

Temperature is a key factor in determining the state of a substance. At temperatures below the melting point, sodium remains solid. On the other hand, at temperatures above the melting point, it becomes liquid. The melting point is also the freezing point, highlighting the equilibrium between the liquid and solid states.

• At the melting point, both phases coexist in equilibrium.
• Molar enthalpy of fusion helps us quantify the energy involved during the melting or freezing process.

Understanding the melting point of sodium is essential for calculations involving phase change, such as determining entropy change during solidification.