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 \(\mathrm{Na}(\mathrm{s})\), is \(\Delta S\) positive or negative? (b) Calculate the value of \(\Delta S\) when \(50.0 \mathrm{~g}\) of \(\mathrm{Na}(l)\) solidifies at \(97.8^{\circ} \mathrm{C}\).

Short answer

(a) When molten sodium solidifies to Na(s), the ΔS is negative. (b) The value of ΔS when 50.0 g of Na(l) solidifies at 97.8°C is -0.0152 kJ/K.

Step-by-step solution

Determine the sign of ΔS

At the melting point, the process of solidification is in equilibrium, which means ΔG = 0. According to the equation ΔG = ΔH - TΔS, at equilibrium, ΔH = TΔS. Since fusion is the process of melting, the enthalpy change for the reverse process, solidification, will have the opposite sign. In this case, ΔH_fus = 2.60 kJ/mol, so ΔH_solidification = -2.60 kJ/mol. We know that ΔH_solidification is negative and T (temperature in Kelvin) is positive. Therefore, to satisfy the equilibrium condition, ΔS must be negative. The answer to part (a) is that ΔS is negative during the solidification of sodium.

Convert temperature to Kelvin

In order to calculate ΔS, we need to convert the given temperature in Celsius to Kelvin. To do this, simply add 273.15 to the Celsius value: T = 97.8°C + 273.15 = 370.95 K

Calculate ΔS

Now we can use the equilibrium condition ΔH = TΔS to calculate the change in entropy during the solidification process. Rearrange the formula to find ΔS: ΔS = ΔH / T ΔS = (-2.60 kJ/mol) / (370.95 K) ΔS = -0.0070 kJ/(mol·K)

Calculate the moles of Na

To find the change in entropy for 50.0 g of Na, we need to calculate the number of moles in that mass. The molar mass of sodium is 22.99 g/mol. Divide the mass by the molar mass: moles_Na = (50.0 g) / (22.99 g/mol) = 2.174 mol

Calculate total ΔS

Now we can calculate the total change in entropy for the given mass of Na by multiplying the moles by the change in entropy per mole: ΔS_total = moles_Na × ΔS ΔS_total = (2.174 mol) × (-0.0070 kJ/(mol·K)) ΔS_total = -0.0152 kJ/K The answer to part (b) is that the change in entropy when 50.0 g of Na solidifies at 97.8°C is -0.0152 kJ/K.

Key concepts

Enthalpy

Enthalpy is a key concept in thermodynamics, representing the total heat content of a system under constant pressure. It is denoted by the symbol \( \Delta H \). In chemical reactions, enthalpy change indicates whether the process absorbs or releases heat. For example, during a phase change such as melting, the molar enthalpy of fusion \( \Delta H_{\text{fus}} \) quantifies the amount of energy required to convert a solid into a liquid at its melting point. In the reverse process, solidification, this energy change is negative, indicating that heat is released as the liquid cools into a solid.

Entropy

Entropy \( \Delta S \) is a measure of the disorder or randomness in a system. In thermodynamics, it helps determine the feasibility of a process when combined with enthalpy and temperature. The equation \( \Delta G = \Delta H - T\Delta S \) shows Gibbs free energy \( \Delta G \) reflects a system's changes, with equilibrium achieved when \( \Delta G = 0 \). During phase change like solidification, entropy decreases, reflected by a negative \( \Delta S \), since liquid solidifying into a solid represents order with molecules in a more structured state. Entropy is key in understanding natural processes' spontaneity.

Molar Mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). It plays a crucial role in calculations involving chemical reactions, allowing conversion between mass and moles. For sodium, the molar mass is 22.99 g/mol. To calculate quantities in reactions, such as the entropy change when mass of sodium solidifies, you first find the moles by dividing mass by molar mass. This conversion is essential in linking macroscopic quantities like grams to the microscopic scale of moles, facilitating correct application of equations like \( \Delta S = \Delta H / T \).

Phase Change

A phase change involves a substance transitioning between different states of matter. Common examples include melting, freezing, and vaporization. During a phase change, energy is either absorbed or released by the substance, but the temperature remains constant until the change is complete. This energy facet is closely tied to enthalpy. For instance, a substance must absorb energy for melting (fusion), and the same amount of energy is released when it solidifies. Understanding phase changes helps explain behavior like the latent heat concept—energy absorbed or released during a phase change without temperature variation. These principles are crucial in approaches to energy efficiency and material design.