Question

An ice cube tray contains enough water at \(22.0{ }^{\circ} \mathrm{C}\) to make 18 ice cubes that each have a mass of \(30.0 \mathrm{~g} .\) The tray is placed in a freezer that uses \(\mathrm{CF}_{2} \mathrm{Cl}_{2}\) as a refrigerant. The heat of vaporization of \(\mathrm{CF}_{2} \mathrm{Cl}_{2}\) is \(158 \mathrm{~J} / \mathrm{g} .\) What mass of \(\mathrm{CF}_{2} \mathrm{Cl}_{2}\) must be vaporized in the refrigeration cycle to convert all the water at \(22.0^{\circ} \mathrm{C}\) to ice at \(-5.0^{\circ} \mathrm{C}\) ? The heat capacities for \(\mathrm{H}_{2} \mathrm{O}(s)\) and \(\mathrm{H}_{2} \mathrm{O}(I)\) are \(2.03 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\) and \(4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\), respectively, and the enthalpy of fusion for ice is \(6.02 \mathrm{~kJ} / \mathrm{mol}\).

Short answer

The total energy required can be calculated using the following steps: 1. Calculate the energy for cooling the water from 22.0 °C to 0 °C: \(Q_{cooling} = (540 \mathrm{~g}) (4.18 \mathrm{~J / g \cdot ^{\circ}C})(0 - 22)\) 2. Calculate the energy for converting liquid water to ice: \(Q_{fusion} = (30 \mathrm{~mol})(6.02 \mathrm{~kJ/mol}) \times 1000 \mathrm{~J/kJ}\) 3. Calculate the energy for cooling the ice from 0 °C to -5.0 °C: \(Q_{cooling\_ice} = (540 \mathrm{~g}) (2.03 \mathrm{~J / g \cdot ^{\circ}C})(-5)\) 4. Calculate the total energy: \(Q_{total} = Q_{cooling} + Q_{fusion} + Q_{cooling\_ice}\) 5. Calculate the mass of CF2Cl2 needed to provide the total energy: \(m = \frac{Q_{total}}{158 \mathrm{~J/g}}\) Use the Q_total value found in step 4 to calculate the mass of CF2Cl2 required for the refrigeration process.

Step-by-step solution

Calculate the energy for cooling the water from 22.0 °C to 0 °C

First, we have to calculate the energy required to lower the temperature of the water from 22.0 °C to 0 °C. The formula to calculate the energy is: \(Q = mcΔT\) Where: - Q is the energy required (Joules) - m is the mass of the water (grams) - c is the specific heat capacity of liquid water (4.18 J/g· °C) - ΔT is the temperature change (from 22.0 °C to 0 °C) Total mass of the water is given as 18 ice cubes each having a mass of 30.0 g, so the total mass is 18 * 30.0 g = 540 g. Now we can substitute the values and find the energy for cooling the water: \(Q_{cooling} = (540 \mathrm{~g}) (4.18 \mathrm{~J / g \cdot ^{\circ}C})(0 - 22)\)

Calculate the energy for converting liquid water to ice

Next, we need to calculate the energy for converting water (at 0 °C) to ice. We will use the enthalpy of fusion value: Enthalpy of fusion = 6.02 kJ/mol The molar mass of water is around 18 g/mol. We have 540 g of water which is equivalent to \(\frac{540g}{18g/mol} = 30 mol\). Now we can calculate the energy required for converting water to ice: \(Q_{fusion} = (30 \mathrm{~mol})(6.02 \mathrm{~kJ/mol})\) Convert the energy to Joules: \(Q_{fusion} = (30 \mathrm{~mol})(6.02 \mathrm{~kJ/mol}) \times 1000 \mathrm{~J/kJ}\)

Calculate the energy for cooling the ice from 0 °C to -5.0 °C

In this step, we will calculate the energy needed to lower the temperature of the ice from 0 °C to -5.0 °C. Using the formula \(Q = mcΔT\), we can calculate the energy required: Here, c is the specific heat capacity of ice (2.03 J/g· °C). \(Q_{cooling\_ice} = (540 \mathrm{~g}) (2.03 \mathrm{~J / g \cdot ^{\circ}C})(-5)\)

Calculate the total energy

Now, combining the energy required from steps 1, 2, and 3, we can calculate the total energy: \(Q_{total} = Q_{cooling} + Q_{fusion} + Q_{cooling\_ice}\)

Calculate the mass of CF2Cl2 needed to provide the total energy

We are given the heat of vaporization of CF2Cl2 as 158 J/g. Let m be the mass of CF2Cl2 required to provide the total energy (Q_total). Then: \(Q_{total} = (m \mathrm{~g})(158 \mathrm{~J/g})\) Now we can calculate the mass of CF2Cl2 needed: \(m = \frac{Q_{total}}{158 \mathrm{~J/g}}\) Calculate m using the Q_total value found in Step 4 to get the final answer.

Key concepts

Thermal Energy Transfer

Thermal energy transfer is a crucial concept when studying how heat moves from one object to another. In our problem, water is transformed into ice and changes temperature, requiring thermal energy transfer at each stage. This occurs through processes like heating, cooling, or phase changes.
When water in the tray starts at 22.0 °C, thermal energy is extracted to lower its temperature to 0 °C, then phase change occurs as it freezes, and finally, more energy is removed to cool the ice to -5.0 °C. The direction of heat transfer is important here: heat always moves from the water to the colder environment of the freezer until reaching the desired temperature of the ice. Each step of this process involves specific calculations, highlighting how thermal energy is transferred through different phases and temperatures.

Specific Heat Capacity

The concept of specific heat capacity is foundational in understanding how much energy is needed to change the temperature of a substance. Specific heat capacity (\(c\)) is defined as the amount of heat required to change the temperature of one gram of a substance by one degree Celsius.
For water as a liquid, this value is \(4.18 \, \text{J/g}\cdot^{\circ}\text{C}\). This high specific heat capacity explains why water can absorb a lot of heat before showing an increase in temperature. Conversely, for ice, this capacity is reduced to \(2.03 \text{J/g}\cdot^{\circ}\text{C}\).
Understanding these values helps us calculate the energy required to change the state of water and ice, as shown when determining the heat needed for each temperature change in the exercise.

Phase Change

Phase change is when a substance transitions from one state of matter (e.g., liquid) to another (e.g., solid). In this task, the process is the freezing of liquid water into solid ice.
To compute the energy needed, we use the enthalpy of fusion, which measures the heat necessary to change from liquid to solid without altering the temperature. In our problem, this is quantified as \(6.02 \, \text{kJ/mol}\).
We first calculate how many moles of water are present, as this helps us determine the total energy needed to complete the phase change. For the 540 g of water turning into ice using an enthalpy of fusion, it's clear that a significant amount of energy is involved in this conversion, exemplifying how phase changes require specialized calculations separate from simple temperature changes.

Molar Calculations

Molar calculations involve converting the mass of a substance into the number of moles, using the molar mass. This computation is critical when applying thermal energy equations that are expressed in unit per mole, such as the enthalpy of fusion or heat of vaporization.
In the exercise, we work with water which has a molar mass of approximately \(18 \text{ g/mol}\). By converting 540 g of water into moles, we use this factor to engage with enthalpy of fusion values, enabling us to compute the total energy required for phase changes.
Furthermore, this skill is utilized when calculating the mass of \(\text{CF}_2\text{Cl}_2\) that must vaporize to provide the necessary cooling in the refrigeration cycle, demonstrating the interrelation of mole calculations in energy-related computations.