Problem 100 - Chemistry: The Central Science

Question

Most liquids follow Trouton's rule, which states that the molar entropy of vaporization lies in the range of \(88 \pm 5 \mathrm{~J} / \mathrm{mol}-\mathrm{K}\). The normal boiling points and enthalpies of vaporization of several organic liquids are as follows: $$ \begin{array}{lrl} \hline \text { Substance } & \begin{array}{l} \text { Normal Boiling } \\ \text { Point }\left({ }^{\circ} \mathrm{C}\right) \end{array} & \begin{array}{l} \Delta H_{\text {vap }} \\ \text { (kJ/mol) } \end{array} \\ \hline \text { Acetone, }\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CO} & 56.1 & 29.1 \\ \text { Dimethyl ether, }\left(\mathrm{CH}_{3}\right)_{2} \mathrm{O} & -24.8 & 21.5 \\ \text { Ethanol } \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} & 78.4 & 38.6 \\ \text { Octane, } \mathrm{C}_{8} \mathrm{H}_{18} & 125.6 & 34.4 \\ \text { Pyridine, } \mathrm{C}_{5} \mathrm{H}_{5} \mathrm{~N} & 115.3 & 35.1 \\\ \hline \end{array} $$ (a) Calculate \(\Delta \mathrm{S}_{\mathrm{vap}}\) for each of the liquids. Do all of the liquids obey Trouton's rule? (b) With reference to intermolecular forces (Section 11.2), can you explain any exceptions to the rule? (c) Would you expect water to obey Trouton's rule? By using data in Appendix \(\mathrm{B}\), check the accuracy of your conclusion. (d) Chlorobenzene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{Cl}\right)\) boils at \(131.8^{\circ} \mathrm{C}\). Use Trouton's rule to estimate \(\Delta H_{\text {vap }}\) for this substance.

Step-by-Step Solution

Answer

The calculated molar entropy of vaporization for Acetone, Dimethyl ether, Ethanol, Octane, and Pyridine are \(88.35\), \(86.55\), \(109.72\), \(86.31\), and \(90.36 \mathrm{J/mol~K}\) respectively. Only ethanol does not obey Trouton's rule due to its hydrogen bonding capability. Water also does not obey Trouton's rule, as it has molar entropy of vaporization \(109.08 \mathrm{J/mol~K}\). The estimated enthalpy of vaporization for chlorobenzene is \(35.64\) kJ/mol using Trouton's rule.

Step 1: Calculate the molar entropy of vaporization for each liquid

Recall that the molar entropy of vaporization can be calculated using the formula: \[\Delta S_{vap}=\frac{\Delta H_{vap}}{T_{b}} \] Where \(T_b\) is the normal boiling point in Kelvin and \(\Delta H_{vap}\) is the enthalpy of vaporization. First, convert the boiling point from Celsius to Kelvin by adding 273.15. Then, calculate the molar entropy of vaporization by dividing the enthalpy of vaporization by the boiling point in Kelvin. Acetone: \(T_{b} = 56.1 + 273.15 = 329.25 K\) \(\Delta S_{vap}=\frac{29.1 \times 10^3}{329.25} = 88.35 \mathrm{J/mol~K}\) Similarly, perform calculations for other liquids: Dimethyl ether: \(T_{b} = -24.8 + 273.15 = 248.35 K\) \(\Delta S_{vap}=\frac{21.5 \times 10^3}{248.35} = 86.55 \mathrm{J/mol~K}\) Ethanol: \(T_{b} = 78.4 + 273.15 = 351.55 K\) \(\Delta S_{vap}=\frac{38.6 \times 10^3}{351.55} = 109.72 \mathrm{J/mol~K}\) Octane: \(T_{b} = 125.6 + 273.15 = 398.75 K\) \(\Delta S_{vap}=\frac{34.4 \times 10^3}{398.75} = 86.31 \mathrm{J/mol~K}\) Pyridine: \(T_{b} = 115.3 + 273.15 = 388.45 K\) \(\Delta S_{vap}=\frac{35.1 \times 10^3}{388.45} = 90.36 \mathrm{J/mol~K}\)

Step 2: Check if liquids obey Trouton's rule

Now we need to check if the calculated molar entropy of vaporization lies within the range of \(88\pm5\) J/mol-K. Acetone: \(88.35 \rightarrow\) Yes Dimethyl ether: \(86.55 \rightarrow\) Yes Ethanol: \(109.72 \rightarrow\) No Octane: \(86.31 \rightarrow\) Yes Pyridine: \(90.36 \rightarrow\) Yes All the liquids, except ethanol, obey Trouton's rule.

Step 3: Explain exceptions

Ethanol is exceptional because of its capability to form hydrogen bonds, making it harder to break the intermolecular forces during vaporization. This additional energy requirement is responsible for the higher molar entropy of vaporization compared to the range dictated by Trouton's rule.

Step 4: Predict if water obeys Trouton's rule

Since water can also form hydrogen bonds, we can predict that it will not obey Trouton's rule. To check the accuracy of our conclusion, we will use the given data in Appendix B.

Step 5: Verify the conclusion using given data

From Appendix B, we can find \(\Delta H_{vap}\) for water as \(40.7\) kJ/mol and the normal boiling point as \(100^\circ C\) or \(373.15 K\). Calculate the molar entropy of vaporization for water: \[\Delta S_{vap}=\frac{40.7 \times 10^3}{373.15} = 109.08 \mathrm{J/mol~K}\] As expected, since the value exceeds the range of \(88 \pm 5\) J/mol-K, water does not obey Trouton's rule.

Step 6: Estimate enthalpy of vaporization for chlorobenzene using Trouton's rule

Chlorobenzene boils at \(131.8^\circ C\), convert to Kelvin: \[T_{b} = 131.8 + 273.15 = 404.95 K\] Using Trouton's rule, we assume that \(\Delta S_{vap} = 88\ \mathrm{J/mol~K}\). Now, we can estimate the enthalpy of vaporization using the formula: \[\Delta H_{vap} = \Delta S_{vap} \times T_{b}\] \[\Delta H_{vap} = 88 \times 404.95 = 35636.60 \mathrm{J/mol}\] Converting to kJ/mol: \[\Delta H_{vap} = \frac{35636.60}{1000} = 35.64 \mathrm{kJ/mol}\] So, the estimated enthalpy of vaporization for chlorobenzene is \(35.64\) kJ/mol.

Key Concepts

Trouton's Rule

Trouton's Rule is a fascinating observation in thermodynamics that applies to many liquids. It states that the molar entropy of vaporization, which measures how much disorder is introduced when a liquid turns into gas, is typically around 88 J/mol-K. This holds true as long as the conditions are not affected strongly by intermolecular forces, such as hydrogen bonding.

This rule is quite useful for estimating the enthalpy of vaporization for various substances, especially when it's difficult to measure directly. Its approximate nature means it works best for non-polar and non-hydrogen-bonded liquids. Trouton's Rule is also a great indicator for identifying outliers in molecular behavior. For example, substances like ethanol and water deviate from this rule because of their strong hydrogen bonding.

• The standard range is 88 ± 5 J/mol-K.
• Works well for non-polar substances without hydrogen bonding.
• Used to estimate other thermodynamic properties.

Entropy of Vaporization

Entropy of Vaporization (\(\Delta S_{vap}\) ) is a crucial thermodynamic concept that helps us understand how energy is distributed when a liquid turns into vapor. It essentially measures the increase in disorder or randomness as molecules transition from the condensed state to a more disorganized gaseous state.

This can be calculated by dividing the enthalpy of vaporization by the absolute temperature (in Kelvin) at which the phase change occurs:\[\Delta S_{vap} = \frac{\Delta H_{vap}}{T_b}\]The unit for entropy is typically J/mol-K.

Every substance has a characteristic entropy of vaporization. Most substances follow Trouton's rule, but there are exceptions. For molecules with strong intermolecular forces, such as hydrogen bonds, the additional energy required disrupts Trouton's typical range for \(\Delta S_{vap}\)

• Calculates disorder from liquid to vapor phase.
• Often aligns with Trouton's range unless influenced by strong forces.
• Expressed in J/mol-K.

Hydrogen Bonding

Hydrogen Bonding is a powerful type of dipole-dipole attraction that plays a crucial role in the physical properties of substances. This type of bonding occurs when there is a strong attraction between a hydrogen atom, which is covalently bonded to an electronegative atom (such as oxygen or nitrogen), and another electronegative atom nearby.

This bond is much stronger than a typical dipole interaction but weaker than covalent or ionic bonds. It significantly increases the enthalpy of vaporization and entropy of vaporization for substances like water and ethanol because breaking these intermolecular forces requires more energy. This additional energy also means these liquids often do not follow Trouton's Rule.

Hydrogen bonds lead to distinctive properties, such as the high boiling point of water, ice's lower density than liquid water, and providing secondary and tertiary structures to proteins and nucleic acids.

• Occurs between molecules with hydrogen attached to strong electronegative atoms.
• Increases both \(\Delta H_{vap}\) and \(\Delta S_{vap}\) .
• Leads to deviations from Trouton's Rule.

Enthalpy of Vaporization

The Enthalpy of Vaporization (\(\Delta H_{vap}\) ) denotes the amount of energy needed to convert a liquid into vapor at a constant temperature and pressure. This encompasses more than just the energy needed to overcome cohesive forces; it's an essential factor in understanding phase changes in thermodynamics.

To determine \(\Delta H_{vap}\) for a given substance, we reference data from experiments or use quantitative rules like Trouton's Rule. Knowing the enthalpy of vaporization helps in the design of chemical processes and industrial applications, as it tells us how much energy is consumed during phase changes.

For substances without strong hydrogen bonding or other strong intermolecular forces, Trouton's Rule can be used to estimate \(\Delta H_{vap}\) from the molar entropy of vaporization. Substances with strong intermolecular forces, such as hydrogen bonds, however, require more detailed consideration due to additional energy needed to break these bonds.

• Measures energy needed for liquid to vapor phase change.
• Critical for process design and analysis.
• Affected by strength of intermolecular forces.