Question

The element cesium (Cs) freezes at \(28.4^{\circ} \mathrm{C}\), and its molar enthalpy of fusion is \(\Delta H_{\text {fus }}=2.09 \mathrm{~kJ} / \mathrm{mol}\). (a) When molten cesium solidifies to \(\mathrm{Cs}(s)\) at its normal melting point, is \(\Delta S\) positive or negative? (b) Calculate the value of \(\Delta S\) when \(15.0 \mathrm{~g}\) of \(\mathrm{Cs}(l)\) solidifies at \(28.4^{\circ} \mathrm{C}\).

Short answer

(a) When molten cesium solidifies to Cs(s) at its normal melting point, the ΔS is negative. (b) The value of ΔS when 15.0 g of Cs(l) solidifies at 28.4°C is approximately -0.78 J/K.

Step-by-step solution

Recall the relationship between ΔH, ΔS, and temperature

For a phase transition such as the melting or solidification of a substance, the Gibbs free energy change (ΔG) is equal to zero at the equilibrium temperature. This equilibrium temperature is the melting/solidification point of the substance. The relationship between ΔG, ΔH, and ΔS is given by the following equation: ΔG = ΔH - TΔS At the melting point, ΔG = 0, so the equation becomes: 0 = ΔH - TΔS ΔS = ΔH / T

Determine the sign of ΔS

Recall that the solidification is the reverse of the melting process. Since ΔS is positive during the melting process (entropy increases due to increased disorder when a solid becomes a liquid), it must be negative during the solidification process (since the liquid becomes a solid, there is decreased disorder). Therefore, ΔS is negative when molten cesium solidifies to Cs(s) at its normal melting point. (a) ΔS is negative.

Convert molar enthalpy of fusion to J/mol

We are given the molar enthalpy of fusion, ΔHfus, in kJ/mol, which we need to convert to J/mol: ΔHfus = 2.09 kJ/mol × (1000 J / 1 kJ) = 2090 J/mol

Convert temperature to kelvin

To perform calculations with temperature in thermodynamics, we need to convert the temperature from Celsius to Kelvin: T = 28.4°C + 273.15 = 301.55 K

Calculate ΔS for the solidification process

Now we can calculate ΔS using the relationship found in Step 1: ΔS = ΔH / T ΔS = (-2090 J/mol) / (301.55 K) ΔS = -6.93 J/(mol·K) We are given the amount of cesium as 15.0 grams. To obtain ΔS for this amount, we need to convert grams to moles using the molar mass of cesium: Molar mass of Cs = 132.91 g/mol moles of Cs = (15.0 g) / (132.91 g/mol) = 0.113 mol Now multiply the moles of Cs by the calculated ΔS per mole: ΔS_total = (0.113 mol) × (-6.93 J/(mol·K)) ΔS_total ≈ -0.78 J/K

Write the final answers

(a) When molten cesium solidifies to Cs(s) at its normal melting point, the ΔS is negative. (b) The value of ΔS when 15.0 g of Cs(l) solidifies at 28.4°C is approximately -0.78 J/K.

Key concepts

Phase Transition

Understanding the phase transition of substances like cesium takes us into the heart of thermodynamics, the study of energy transitions and physical transformations within matter. When cesium transitions from liquid to solid - what we call solidification or freezing - it undergoes a phase transition, one where the orderly arrangement of atoms increases, leading to a well-structured crystal lattice in the solid phase.

A phase transition happens at a specific temperature and pressure, often characterized by a change in internal energy and other state variables like volume and entropy. For cesium, this transition occurs at its freezing point of 28.4°C. During the process, energy is released in the form of the enthalpy of fusion, which is necessary to overcome the forces keeping cesium atoms in a liquid state. A foundational understanding of these concepts is essential to make sense of phenomena such as the freezing of water or the melting of metals.

Gibbs Free Energy

The concept of Gibbs free energy, represented by \(\Delta G\), is a cornerstone in understanding chemical reactions and phase transitions. It can tell us whether a process will occur spontaneously under constant pressure and temperature. For reactions where \(\Delta G\) is negative, the process is spontaneous, indicating that it can proceed without an external energy source.

Diving deeper, \(\Delta G\) is defined as the difference between the change in enthalpy (\(\Delta H\)) and the product of the temperature (T) and change in entropy (\(\Delta S\)): \(\Delta G = \Delta H - T\Delta S\). Describing a state of equilibrium, where a substance is perfectly balanced between its phases, \(\Delta G\) is zero, which is what happens at the melting point of cesium. Understanding this fundamental relationship not only clarifies the interplay between energy and disorder during a phase change but also has broader applications in predicting the spontaneity of chemical reactions.

Entropy

Entropy is a measure of disorder or randomness within a system, a concept that's central to the second law of thermodynamics. Generally denoted as \(\Delta S\), entropy changes offer insights into the nature of a substance's phase transition. In processes where entropy increases, such as when ice melts into water, there's a dispersal of energy and an increase in disorder. Conversely, when a liquid freezes, like cesium solidifying, entropy decreases because the resulting solid structure is more ordered than the liquid state.

The concept of entropy is critical to understand not just basic physical transformations but also in the broader scope of the universe's tendency towards disorder. Thus, when cesium solidifies, we observe a decrease in entropy because the atoms arrange themselves in an ordered fashion, signifying a progression towards a state of reduced randomness.

Solidification Process

The solidification process is the transition from the liquid phase to the solid phase, with important implications in materials science and engineering. In this process, molecules slow down and assume a more stable and less energetic structure. For cesium, which freezes at 28.4°C, solidification involves the release of the enthalpy of fusion as the substance moves to a lower energy state.

During solidification, a substance like cesium releases energy to its surroundings, preventing further temperature change until the entire mass has solidified - a manifestation of the latent heat principle. The reverse process, melting, requires the same amount of energy to be absorbed to break the solid structure. Practical applications of understanding solidification range from metal casting and fabrication to the creation of ice and preservation of food.