Problem 45 - Chemistry: The Central Science in SI Units, Global Edition

Question

Acetone \(\mathrm{CH}_{3} \mathrm{COCH}_{3}\), is a common organic solvent with relatively low melting point \((178 \mathrm{~K})\) and boiling point \((329 \mathrm{~K})\). The enthalpy of fusion of acetone is \(5.72 \mathrm{~kJ} / \mathrm{mol}\), and its enthalpy of vaporization is \(29.1 \mathrm{~kJ} / \mathrm{mol}\). The specific heats of solid and liquid acetone are \(96 \mathrm{~J} / \mathrm{mol}-\mathrm{K}\) and \(125.5 \mathrm{~J} / \mathrm{mol}-\mathrm{K}\) respectively. (a) How much heat is required to convert \(23.0 \mathrm{~g}\) of acetone at \(273 \mathrm{~K}\) to the vapor phase at \(329 \mathrm{~K} ?(\mathbf{b})\) How much heat is required to convert the same amount of acetone at \(77 \mathrm{~K}\) to the vapor phase at \(329 \mathrm{~K} ?\)

Step-by-Step Solution

Answer

(a) The total heat required to convert 23.0 g of acetone at 273 K to the vapor phase at 329 K can be found by calculating and adding the heat required to heat the solid and liquid acetone, melting, and vaporization: \(q_{total} = q_{solid} + q_{fus} + q_{liquid} + q_{vap} = 0.396 \text{ mol} * 96 \frac{\text{J}}{\text{mol} \cdot \text{K}} * (178 \text{ K} - 273 \text{ K}) + 0.396 \text{ mol} * 5.72 \frac{\text{kJ}}{\text{mol}} + 0.396 \text{ mol} * 125.5 \frac{\text{J}}{\text{mol} \cdot \text{K}} * (329 \text{ K} - 178 \text{ K}) + 0.396 \text{ mol} * 29.1 \frac{\text{kJ}}{\text{mol}}\) (b) To find the heat required to convert the same amount of acetone at 77 K to the vapor phase at 329 K, calculate the heat required to heat the solid acetone from 77 K to the melting point, then follow steps 3-6 from part (a), and finally add all heat values: \(q_{total} = q_{solid} + q_{fus} + q_{liquid} + q_{vap} = 0.396 \text{ mol} * 96 \frac{\text{J}}{\text{mol} \cdot \text{K}} * (178 \text{ K} - 77 \text{ K}) + q_{fus} + q_{liquid} + q_{vap}\)

Step 1: Calculate the number of moles of acetone

First, calculate the number of moles (n) of acetone using its molar mass, 58.08 g/mol: n = (mass) / (molar mass) n = (23.0 g) / (58.08 g/mol) n = 0.396 mol

Step 2: Calculate the heat required to heat acetone from 273 K to its melting point, 178 K

We'll use the specific heat of solid acetone (C_s = 96 J/mol-K) and the temperature difference to calculate the heat needed (q_solid): q_solid = n * C_s * (T₂ - T₁) q_solid = 0.396 mol * 96 J/mol-K * (178 K - 273 K)

Step 3: Calculate the heat required for melting acetone

Use the enthalpy of fusion (ΔH_fus = 5.72 kJ/mol) to calculate the heat needed (q_fus): q_fus = n * ΔH_fus q_fus = 0.396 mol * 5.72 kJ/mol

Step 4: Calculate the heat required to heat liquid acetone from its melting point to its boiling point

We'll use the specific heat of liquid acetone (C_l = 125.5 J/mol-K) and the temperature difference to calculate the heat needed (q_liquid): q_liquid = n * C_l * (T₄ - T₃) q_liquid = 0.396 mol * 125.5 J/mol-K * (329 K - 178 K)

Step 5: Calculate the heat required for vaporization of acetone

Use the enthalpy of vaporization (ΔH_vap = 29.1 kJ/mol) to calculate the heat needed (q_vap): q_vap = n * ΔH_vap q_vap = 0.396 mol * 29.1 kJ/mol

Step 6: Calculate the total heat required

Add the heat required for heating solid and liquid acetone, melting, and vaporization: q_total = q_solid + q_fus + q_liquid + q_vap (b) Heat required to convert the same amount of acetone at 77 K to the vapor phase at 329 K

Step 1: Calculate the heat required to heat acetone from 77 K to its melting point, 178 K

We'll use the specific heat of solid acetone (C_s = 96 J/mol-K) and the temperature difference to calculate the heat needed (q_solid): q_solid = n * C_s * (T₂ - T₁) q_solid = 0.396 mol * 96 J/mol-K * (178 K - 77 K) Follow steps 3-6 from part (a) to calculate the corresponding heat values.

Step 7: Calculate the total heat required

Add the heat required for heating solid and liquid acetone, melting, and vaporization: q_total = q_solid + q_fus + q_liquid + q_vap

Key Concepts

Enthalpy of Vaporization

The enthalpy of vaporization is the energy required to convert a substance from a liquid to a vapor at a constant temperature and pressure. This process is critical as it involves breaking intermolecular forces, allowing molecules to move freely in the gaseous state. For acetone, the enthalpy of vaporization is given as 29.1 kJ/mol. This value indicates that each mole of acetone requires 29.1 kJ to overcome the molecular forces keeping it in the liquid state and to become vapor. When calculating the total heat needed for phase transitions, this value plays an essential role, especially in processes like boiling. In exercises, this step will typically follow the heating of the liquid to its boiling point using its specific heat capacity. Once at the boiling point, the enthalpy of vaporization is applied to find the energy needed to vaporize the liquid.

Enthalpy of Fusion

Enthalpy of fusion refers to the amount of energy needed to change a substance from a solid to a liquid at its melting point. This process involves overcoming the forces holding the molecules in the solid state. For acetone, the enthalpy of fusion is 5.72 kJ/mol, which implies that each mole of acetone requires this amount of energy to melt completely at its melting temperature of 178 K. In practice, this value is crucial when calculating the energy required for a phase transition from solid to liquid during exercises focusing on thermal energy changes. The enthalpy of fusion is the key energy component required immediately after the solid has been heated to its melting point, and before the liquid is heated toward its boiling point.

Specific Heat Capacity

Specific heat capacity is a measure of the heat energy required to raise the temperature of one mole of a substance by one Kelvin. It is a fundamental concept in thermochemistry and differs for each state of a substance—solid or liquid. For acetone, the specific heat is 96 J/mol-K in the solid state and 125.5 J/mol-K in the liquid state. This property is used in calculations to determine the amount of heat energy required to change the temperature of acetone without changing its phase. When heating solid acetone to its melting point, we apply its solid specific heat capacity. Similarly, when heating liquid acetone from the melting point to the boiling point, we use the liquid specific heat capacity. Understanding this concept allows for precise calculation of the energy required for temperature changes within a phase without triggering a phase transition.

Phase Transition

Phase transitions involve changes between different states of matter—solid, liquid, and gas. They occur under specific conditions of temperature and pressure and involve significant energy changes related to intermolecular forces. Key phase transitions include melting, vaporization, and sublimation. In calculations involving acetone, several transitions are considered:

• Solid to liquid (melting) at the melting point, utilizing the enthalpy of fusion.
• Liquid to gas (vaporization) at the boiling point, utilizing the enthalpy of vaporization.

Each phase transition requires a combination of heating to reach the transition temperature, followed by applying the respective enthalpy change to actually transition between phases. Understanding these transitions is crucial for accurately computing the total heat involved in processes that take acetone through multiple phases, such as the given exercise of keeping acetone from a solid form all the way to vapor.