Question

How much energy in kilojoules is released when \(25.0 \mathrm{~g}\) of ethanol vapor at \(93.0{ }^{\circ} \mathrm{C}\) is cooled to \(-11.0{ }^{\circ} \mathrm{C} ?\) Ethanol has \(\mathrm{mp}=-114.1{ }^{\circ} \mathrm{C}, \mathrm{bp}=78.3^{\circ} \mathrm{C}, \Delta H_{\mathrm{vap}}=38.56 \mathrm{~kJ} / \mathrm{mol}\), and \(\Delta H_{\text {fusion }}=4.93 \mathrm{~kJ} / \mathrm{mol}\). The molar heat capacity is \(112.3\) \(\mathrm{J} /(\mathrm{K} \cdot \mathrm{mol})\) for the liquid and \(65.6 \mathrm{~J} /(\mathrm{K} \cdot \mathrm{mol})\) for the vapor.

Short answer

The total energy released is approximately 41.10 kJ.

Step-by-step solution

Convert Mass to Moles

First, we need to convert the given mass of ethanol, 25.0 g, to moles. The molar mass of ethanol (C₂H₅OH) is approximately 46.08 g/mol. The number of moles of ethanol is calculated using the formula:\[\text{moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{25.0 \text{ g}}{46.08 \text{ g/mol}} \approx 0.542 \text{ moles}.\]

Cool Vapor to Boiling Point

The ethanol vapor needs to be cooled from 93.0°C to its boiling point, 78.3°C. We use the formula for heat absorbed or released:\[ q_1 = n \cdot C_{\text{vapor}} \cdot \Delta T \]where \( n = 0.542 \text{ moles} \), \( C_{\text{vapor}} = 65.6 \text{ J/(K·mol)} \), and \( \Delta T = 93.0 - 78.3 = 14.7 \text{ °C} \):\[q_1 = 0.542 \times 65.6 \times 14.7 \approx 522.55 \text{ J}.\]

Condense Vapor to Liquid

Calculate the energy released when ethanol vapor condenses to a liquid. Use the heat of vaporization:\[ q_2 = n \times \Delta H_{\text{vap}} = 0.542 \times 38.56 \text{ kJ/mol} = 20.89 \text{ kJ}. \]

Cool Liquid to Melting Point

Next, cool the liquid ethanol from the boiling point, 78.3°C, to the melting point, -114.1°C. The formula used is:\[ q_3 = n \cdot C_{\text{liquid}} \cdot \Delta T \]where \( C_{\text{liquid}} = 112.3 \text{ J/(K·mol)} \), and \( \Delta T = 78.3 + 114.1 = 192.4 \text{ °C} \):\[ q_3 = 0.542 \times 112.3 \times 192.4 \approx 11708 \text{ J} \approx 11.71 \text{ kJ}.\]

Freeze Liquid to Solid

Calculate the energy change when ethanol freezes, using the heat of fusion:\[ q_4 = n \times \Delta H_{\text{fusion}} = 0.542 \times 4.93 \text{ kJ/mol} \approx 2.67 \text{ kJ}. \]

Cool Solid to -11.0°C

Finally, cool the solid ethanol from its melting point, -114.1°C, to the final temperature, -11.0°C. Assume liquid specific heat during this process (approximation). The formula used is the same as for cooling liquid:\[ q_5 = n \cdot C_{\text{solid}} \cdot \Delta T \]where \( C_{\text{solid}} = 112.3 \text{ J/(K·mol)} \) and \( \Delta T = -11.0 + 114.1 = 103.1 \text{ °C} \):\[q_5 = 0.542 \times 112.3 \times 103.1 \approx 6273 \text{ J} \approx 6.27 \text{ kJ}.\]

Calculate Total Energy Released

Add up all the energy changes to find the total energy released:\[ q_{\text{total}} = -(-0.522) + (-20.89) + (-11.71) + (-2.67) + (-6.27) \approx -42.03 \text{ kJ}.\] The total energy released is approximately 41.10 kJ (since it’s negative as energy is released).

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in thermochemistry. It describes how energy is exchanged between substances due to a difference in temperature. In the context of this exercise, heat transfer occurs during various steps as ethanol transitions between different phases and temperature states. Each phase change or temperature adjustment involves a calculation of energy change. ### Forms of Heat Transfer - **Conduction** is the transfer of energy through a material without the material itself moving. - **Convection** involves the movement of heat by fluid motion. - **Radiation** is the transfer of heat through electromagnetic waves. When dealing with chemical processes, the focus is primarily on conduction within the substance and how it absorbs or releases heat as it undergoes a change in temperature or state.
In this problem, the ethanol releases energy as it cools from a vapor to a solid, demonstrating heat transfer through conduction.

Phase Changes

Phase changes refer to the transitions between solid, liquid, and gas states. Each phase change involves specific energy alterations due to the differences in molecular arrangement and energy states.### Types of Phase Changes- **Melting** and **Freezing** occur between the solid and liquid states.- **Vaporization** and **Condensation** happen between the liquid and gas states.In the exercise, ethanol undergoes:- **Condensation** from vapor to liquid (energy release due to vaporization heat being eliminated).- **Freezing** of the liquid into a solid form as it is cooled past the melting point.Each phase change involves specific energy exchange, calculated using the formula:\[ q = n \times \Delta H \]where \( n \) is the moles and \( \Delta H \) is the enthalpy change (either heat of vaporization or fusion, depending on the phase change).
By properly calculating these energy changes, we understand how much energy is needed—or released—during these transitions.

Molar Heat Capacity

Molar heat capacity is the amount of energy required to change the temperature of one mole of a substance by one degree Kelvin. It is an essential factor in calculating the energy needed for temperature changes in a substance, without changing its phase.### Understanding Molar Heat Capacity- **For Liquids:** This refers to how much heat, per mole, is required to raise the temperature of a liquid by one degree.- **For Vapors:** It applies similarly to gases, factoring in the entropic changes as gases have more freedom for energy distribution.Within our exercise, different molar heat capacities for liquid and vapor phases of ethanol are given:- **Vapor:** 65.6 J/(K·mol)- **Liquid:** 112.3 J/(K·mol)These values are used in specific heat calculations to determine the energy change as the temperature of ethanol adjusts in various phases:\[ q = n \cdot C \cdot \Delta T \]where \( C \) represents the molar heat capacity, and \( \Delta T \) signifies the temperature change.
By using the appropriate molar heat capacity values, we accurately quantify how much energy is either absorbed or released during the temperature adjustments in the ethanol.