Question

Use the following data to estimate the standard molar entropy of gaseous benzene at \(298.15 \mathrm{K} ;\) that is, \(S^{\circ}\left[\mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{g}, 1 \mathrm{atm})\right] .\) For \(\mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{s}, 1 \mathrm{atm})\) at its melting point of \(5.53^{\circ} \mathrm{C}, S^{\circ}\) is \(128.82 \mathrm{Jmol}^{-1} \mathrm{K}^{-1}\). The enthalpy of fusion is \(9.866 \mathrm{kJ} \mathrm{mol}^{-1} .\) From the melting point to 298.15 K, the average heat capacity of liquid benzene is \(134.0 \mathrm{JK}^{-1} \mathrm{mol}^{-1} .\) The enthalpy of vaporization of \(\mathrm{C}_{6} \mathrm{H}_{6}(1)\) at \(298.15 \mathrm{K}\) is \(33.85 \mathrm{kJ} \mathrm{mol}^{-1},\) and in the vapor- ization, \(\mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{g})\) is produced at a pressure of 95.13 Torr. Imagine that this vapor could be compressed to 1 atm pressure without condensing and while behaving as an ideal gas. Calculate \(S^{\circ}\left[\mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{g}, 1 \text { atm) }] .[ \text { Hint: Refer to }\right.\) Exercise \(88,\) and note the following: For infinitesimal quantities, \(d S=d q / d T ;\) for the compression of an ideal gas, \(d q=-d w ;\) and for pressure-volume work, \(d w=-P d V\).

Short answer

The total entropy calculated by adding entropy changes (in Jmol-1K-1) due to fusion, heating, vaporization and compression to the initial entropy given will give the final molar entropy of gaseous benzene at 298.15 K and 1 atm pressure.

Step-by-step solution

Calculate entropy change during fusion

First, entropy change associated with the process when benzene changes from solid to liquid at its melting point can be calculated using the formula: \[ \Delta S_{\text{fusion}} = \dfrac{\Delta H_{\text{fusion}}}{T_{\text{m}}} \] where \(\Delta H_{\text{fusion}} = 9.866 \, \text{kJ/mol}\) (enthalpy of fusion), and \(T_{\text{m}} = 5.53^{\circ}C = 278.68 \, K\). Substituting these into the formula yields \(\Delta S_{\text{fusion}}\).

Calculate entropy change during heating of liquid

Then, entropy change during heating the liquid benzene from melting point to 298.15 K is calculated using the formula: \[ \Delta S_{\text{heating}} = \int_{T_{\text{m}}}^{T_{\text{b}}} \dfrac{C_{p}}{T} \, dT \] where \(C_{p}\) is the heat capacity at constant pressure which is \(134.0 \, \text{JK}^{-1} \text{mol}^{-1}\), \(T_{\text{m}}\) is the melting temperature (278.68 K), and \(T_{\text{b}}\) is the given temperature (298.15 K). On simplification the integral yields an expression for \(\Delta S_{\text{heating}}\).

Calculate entropy change during vaporization

Next, entropy change during vaporization of benzene at 298.15 K is calculated by the formula: \[ \Delta S_{\text{vaporization}} = \dfrac{\Delta H_{\text{vaporization}}}{T_{\text{b}}} \] where \(\Delta H_{\text{vaporization}} = 33.85 \, \text{kJ/mol}\) and \(T_{\text{b}} = 298.15 \, K\). Substituting these into the formula will yield \(\Delta S_{\text{vaporization}}\).

Calculate entropy change during pressure change of gas

Entropy change during compression process of an ideal gas is calculated by the formula: \[ \Delta S_{\text{compression}} = R \ln \left( \dfrac{P_{2}}{P_{1}} \right) \] where \(R\) is universal gas constant (8.314 Jmol-1K-1), \(P_{1}\) is initial pressure (95.13 Torr = 0.125 atm) and \(P_{2}\) is final pressure (1 atm). Substituting these into the formula yields \(\Delta S_{\text{compression}}\).

Calculate total entropy change

Total entropy change is given by the sum of all individual entropy changes: \[ S^{\circ}[\text{C}_{6} \text{H}_{6}(\text{g}, 1 \text{atm})] = S^{\circ}[\text{C}_{6} \text{H}_{6}(\text{s}, 1 \text{atm})] + \Delta S_{\text{fusion}} + \Delta S_{\text{heating}} + \Delta S_{\text{vaporization}} + \Delta S_{\text{compression}} \] by substituting known values, the value for \( S^{\circ}[\text{C}_{6} \text{H}_{6}(\text{g}, 1 \text{atm})] \) is calculated.

Key concepts

Entropy Change

Entropy change, often represented by the symbol \( \Delta S \), is a measure of the dispersal of energy in a system. In the context of physical transformations, such as melting or boiling, it quantifies how much disorder is introduced when a substance transitions between phases. The general formula to calculate the entropy change during a phase transition or heating process is:

• \( \Delta S = \frac{\Delta Q}{T} \), where \( \Delta Q \) is the heat added to the system and \( T \) is the temperature.

It's vital to remember that entropy increases when a system absorbs heat because the atoms and molecules become more disordered.
For instance, when solid benzene melts or vaporizes, there's a significant increase in entropy due to the separation and increased random motion of the molecules.
​ Understanding entropy change can help you predict the feasibility of reactions and transformations. The Second Law of Thermodynamics states that the total entropy of an isolated system can never decrease over time, which suggests that phase transitions such as melting or evaporation naturally drive toward higher entropy states.

Vaporization

Vaporization is the transition of a substance from the liquid phase to the gas phase. This process requires energy, known as the enthalpy of vaporization (\( \Delta H_{\text{vaporization}} \)), to overcome intermolecular forces. To evaluate the change in the system's entropy during vaporization, the formula used is:

• \( \Delta S_{\text{vaporization}} = \frac{\Delta H_{\text{vaporization}}}{T} \), where \( T \) is the temperature.

It is important to note that vaporization always leads to a positive entropy change.
This is because the molecules are moving freely in the gaseous state, leading to higher disorder compared to the liquid state.
In the case of benzene, the entropy increase reflects the dramatic shift in the degree of freedom and arrangement of molecules.
In practical applications, understanding vaporization and its associated entropy changes can inform processes like distillation, where controlling phase transitions is essential for separation techniques.

Fusion

Fusion refers to the melting of a solid into a liquid, and is characterized by the enthalpy of fusion (\( \Delta H_{\text{fusion}} \)). During fusion, heat is absorbed by the substance to break the lattice structure of the solid, resulting in an increase in entropy. The formula for the entropy change during fusion is:

• \( \Delta S_{\text{fusion}} = \frac{\Delta H_{\text{fusion}}}{T_m} \), where \( T_m \) is the melting temperature.

This relationship shows that the entropy change is directly proportional to the enthalpy and inversely proportional to the temperature.
Therefore, higher temperatures at melting points generally result in less entropy change for the same amount of absorbed heat.
For benzene, transitioning from a solid to a liquid phase brings a notable increase in disorder as the structured solid becomes a fluid liquid.
Understanding fusion helps in predicting how a material will behave under changes in temperature and can be crucial in designing processes that require precise control over phase changes, like metal forging or ice sculpting.

Ideal Gas Compression

Ideal gas compression involves reducing the volume of a gas while ideally not affecting its temperature. During this process, the gas molecules are forced closer together, often leading to decreased entropy. The formula for calculating the entropy change during gas compression is:

• \( \Delta S_{\text{compression}} = R \ln \left( \frac{P_2}{P_1} \right) \), where \( R \) is the universal gas constant, \( P_1 \) is the initial pressure, and \( P_2 \) is the final pressure.

This equation indicates that entropy decreases during compression as pressure increases, reflecting the greater ordering of gas molecules in a smaller volume.
In the benzene gas example, this kind of entropy change is of particular interest because the gas was transitioned from a lower pressure to a standard condition of 1 atm.
Comprehending ideal gas compression and its effect on entropy is critical in numerous applications, such as optimizing the efficiency of engines and designing air conditioning systems.

Heat Capacity

Heat capacity (\( C_p \)) is a measure of the amount of heat required to raise the temperature of a substance by one Kelvin. It is crucial when calculating the entropy change associated with heating the substance. The entropy change during heating a substance can be calculated using:

• \( \Delta S_{\text{heating}} = \int_{T_1}^{T_2} \frac{C_p}{T} \, dT \)

This formula helps determine how the distribution of energy changes as a substance is heated. A high heat capacity means a material requires more heat energy to change its temperature, resulting in smaller temperature changes for a given amount of heat added.
In the benzene example, knowing the heat capacity of liquid benzene allows one to compute the entropy change as benzene is heated from its melting point to the desired state.
Understanding heat capacity and its role in entropy changes is invaluable in processes involving heating and cooling, such as in climate science and material production.