Question

Thenormal boiling point of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) is \(64.7^{\circ} \mathrm{C}\), and its molar enthalpy of vaporization is \(\Delta H_{\mathrm{vap}}=\) \(71.8 \mathrm{~kJ} / \mathrm{mol} .\) (a) When \(\mathrm{CH}_{3} \mathrm{OH}(l)\) boils at its normal boiling point, does its entropy increase or decrease? (b) Calculate the value of \(\Delta S\) when \(1.00\) mol of \(\mathrm{CH}_{3} \mathrm{OH}(t)\) is vaporized at \(64.7^{\circ} \mathrm{C}\).

Short answer

(a) When methanol boils at its normal boiling point, its entropy increases as it transitions from a more ordered liquid phase to a less ordered gaseous phase. (b) The entropy change, \(\Delta S\), when 1.00 mol of methanol is vaporized at \(64.7^{\circ}\mathrm{C}\) (or \(337.85\mathrm{K}\)) is \(212.69\, \mathrm{J/(mol\cdot K)}\).

Step-by-step solution

Determine the phase transition taking place at the normal boiling point

At the normal boiling point of methanol, the substance is transforming from the liquid phase to the gaseous phase (boiling).

Explain the relation between entropy and phase transitions

During a phase transition, when a substance transitions from a more ordered phase to a less ordered phase, its entropy increases. Since liquid to gas transition represents an increase in disorder, the entropy will increase in this case.

Apply the formula relating enthalpy change and entropy change for phase transitions

The formula that relates enthalpy change, temperature, and entropy change during phase transitions is: \[\Delta S = \frac{\Delta H}{T}\] where \(\Delta S\) is the entropy change, \(\Delta H\) is the enthalpy change, and \(T\) is the temperature. Note that the temperature should be in Kelvin for this formula.

Convert the temperature from Celsius to Kelvin

Since the normal boiling point is given in Celsius, we must convert it to Kelvin: \[64.7^{\circ}\mathrm{C} + 273.15 = 337.85\mathrm{K}\]

Calculate the entropy change

Now, we can plug in the given values into the formula: \[\Delta S = \frac{71.8 \, \mathrm{kJ/mol}}{337.85\,\mathrm{K}}\] First, convert kJ to J by multiplying by 1000: \[\Delta S = \frac{71.8 \times 10^3 \, \mathrm{J/mol}}{337.85\,\mathrm{K}}\] Now perform the division: \[\Delta S = 212.69\, \mathrm{J/(mol\cdot K)}\] The entropy change when 1.00 mol of methanol is vaporized at its normal boiling point is 212.69 J/mol·K.

Key concepts

Normal Boiling Point

The normal boiling point of a substance is a fundamental concept in thermodynamics. It refers to the temperature at which a liquid changes to a gas at 1 atmospheric pressure. This is the point where the vapor pressure of the liquid equals the surrounding atmospheric pressure.
The boiling occurs at this specific temperature because molecules gain sufficient energy to overcome intermolecular forces and escape into the vapor phase.

• For methanol, the normal boiling point is 64.7°C. At this point, the liquid methanol gains enough energy to become gaseous.
• This transition doesn’t require a change in temperature once the boiling point is reached, because energy continuously goes into completing the phase change.

Therefore, the normal boiling point is crucial because it indicates the specific energy state required for a substance to transform from liquid to gas at a given pressure.

Enthalpy of Vaporization

The enthalpy of vaporization is the amount of energy required to turn a liquid into a gas at its boiling point. This reflects the energy needed to break bonds or intermolecular forces within the liquid.
This is a vital concept because it helps predict how much energy substances will require to change phases, which is essential in processes like distillation and energy calculations.

• For methanol, the molar enthalpy of vaporization is \(71.8 \, \mathrm{kJ/mol}\).
• This energy input is necessary to separate the methanol molecules so they can transition to a gaseous state.

An understanding of enthalpy of vaporization lets us see the energetic demands of phase changes and its coordination with temperature control in practical applications.

Entropy Change

When substances change phases, such as from liquid to gas, there is a change in entropy. Entropy is a measure of disorder or randomness in a system. In the case of boiling, molecules move from a more ordered liquid state to a less ordered gaseous state. This transition increases the system's entropy.
The formula used to calculate the entropy change for vaporization is:\[\Delta S = \frac{\Delta H}{T}\]where \(\Delta S\) is entropy change, \(\Delta H\) is the enthalpy of vaporization, and \(T\) is the temperature in Kelvin.

• In the example of methanol, the entropy change when 1.00 mol vaporizes at \(64.7^{\circ}\mathrm{C}\) (converted to 337.85 K) is calculated as:
• \(\Delta S = \frac{71.8 \times 10^3 \, \mathrm{J/mol}}{337.85\,\mathrm{K}} = 212.69\, \mathrm{J/(mol\cdot K)}\).

This increase in entropy highlights the natural tendency towards greater disorder and is a key factor in understanding the spontaneity of phase transitions.