Question

An ice cube tray contains enough water at \(22.0^{\circ} \mathrm{C}\) to make 18 ice cubes that each has 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 \(\mathrm{H}_{2} \mathrm{O}(l)\) 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 mass of CF2Cl2 that must be vaporized in the refrigeration cycle to convert all the water at 22.0°C to ice at -5.0°C is 1,494.6 g.

Step-by-step solution

Calculate the total mass of the water

We know that the ice cube tray can make 18 ice cubes, each with a mass of 30.0 g. So, the total mass of the water can be calculated as: Total mass of water = Number of ice cubes × Mass of each ice cube Total mass of water = 18 × 30.0 g = 540.0 g

Calculate the heat needed to cool the water from 22.0°C to 0°C

To calculate the heat needed to cool down the water to 0°C, we will use the formula: Heat (Q) = Mass (m) × Specific Heat Capacity (c) × Temperature difference (ΔT) Given that the specific heat capacity of water (H2O(l)) is 4.18 J/(g·°C): Q1 = 540.0 g × 4.18 J/(g·°C) × (0 - 22.0)°C Q1 = 540.0 g × 4.18 J/(g·°C) × -22.0°C Q1 = -50,173.6 J; we take the absolute value, as we are considering only the magnitude of heat exchange: Q1 = 50,173.6 J

Calculate the heat needed to convert the water at 0°C into ice at 0°C

For freezing the water at 0°C to ice at 0°C, we will use the formula: Q2 = mass × (ΔHf / molar mass of H2O) where ΔHf is the enthalpy of fusion, and the molar mass of H2O is 18.015 g/mol. In this case, ΔHf = 6.02 kJ/mol. Q2 = 540.0 g × (6.02 kJ/mol) / 18.015 g/mol Q2 = 540.0 g × (6020 J/mol) / 18.015 g/mol Q2 = 180,569.1 J

Calculate the heat needed to cool the ice from 0°C to -5.0°C

To calculate the heat needed to cool down the ice from 0°C to -5.0°C, we will use the formula: Heat (Q) = Mass (m) × Specific Heat Capacity (c) × Temperature difference (ΔT) Given that the specific heat capacity of ice (H2O(s)) is 2.03 J/(g·°C): Q3 = 540.0 g × 2.03 J/(g·°C) × (0 - (-5.0))°C Q3 = 540.0 g × 2.03 J/(g·°C) × 5.0°C Q3 = 5,490.7 J

Calculate the mass of refrigerant needed

Now, we will sum up the heats calculated in steps 2, 3, and 4, and divide the total heat by the heat of vaporization of CF2Cl2 to find the mass of refrigerant needed: Total heat (Q_total) = Q1 + Q2 + Q3 Q_total = 50,173.6 J + 180,569.1 J + 5,490.7 J Q_total = 236,233.4 J Next, we can find the mass of CF2Cl2 required using the heat of vaporization (Lv = 158 J/g): Mass of CF2Cl2 = Q_total / Lv Mass of CF2Cl2 = 236,233.4 J / 158 J/g Mass of CF2Cl2 = 1,494.6 g Therefore, 1,494.6 grams of CF2Cl2 must be vaporized in the refrigeration cycle to convert all the water at 22.0°C to ice at -5.0°C.

Key concepts

Specific Heat Capacity

Specific heat capacity is a term used in thermodynamics that describes how much heat energy is needed to change the temperature of a given quantity of a substance by one degree Celsius (or Kelvin). It is an intrinsic property of materials, meaning that it is specific to each kind of substance and depends on its physical state. Understandably, this tells us how resistant a substance is to changing its temperature when heat is added or removed.
In the context of our problem involving water and ice, the specific heat capacities are given for water in its liquid state \(H_2O(l)\) and in its solid state \(H_2O(s)\). For liquid water, the specific heat capacity is 4.18 J/(g·°C), which is higher than that of ice, which is 2.03 J/(g·°C).
These values indicate that it takes more energy to raise the temperature of water compared to ice. This is why, in our calculations, different amounts of energy are required to cool or warm the same mass of water depending on its state.

• Liquid water has a higher specific heat capacity than ice, meaning it retains heat more efficiently.
• Ice requires less energy to change its temperature as it has a lower specific heat capacity.

These characteristics are particularly important during phase changes and temperature adjustments between liquid and solid forms.

Enthalpy of Fusion

The enthalpy of fusion is another fundamental concept in thermodynamics. It represents the amount of energy required to change a substance from solid to liquid at its melting point without changing its temperature. In simpler terms, it's the energy needed to "melt" a solid into a liquid.
For water turning into ice (or vice versa), the enthalpy of fusion is a key player. In our exercise, the enthalpy of fusion for ice is provided as 6.02 kJ/mol. This means that when one mole of ice melts or forms, 6.02 kJ of energy is either absorbed or released, depending on the direction of the process.
In our calculations:

• The energy needed to freeze water at 0°C into ice at 0°C was calculated using the given enthalpy of fusion value.
• The molar mass of water (18.015 g/mol) was used to determine the energy exchange per gram.

This value helps us understand the energy dynamics during the phase transition from liquid to solid and is crucial for calculations involving melting or freezing processes.

Heat of Vaporization

The heat of vaporization, also known as enthalpy of vaporization, is the amount of heat required to convert a unit mass of a liquid into a gas at constant temperature and pressure. This concept is essential when dealing with processes where a liquid becomes a vapor, as energy must be provided to overcome molecular attractions.
For this exercise, the heat of vaporization of CF2Cl2 (used as a refrigerant) is given as 158 J/g. This value indicates how much energy is required to vaporize one gram of this substance.
This property comes into play when the refrigerant in the freezer vaporizes to absorb the heat required for the phase transition of water from liquid to ice. This process allows us to calculate the required mass of CF2Cl2 to achieve the desired temperature change because:

• More energy would need more refrigerant to reach the effective cooling needed.
• The total energy calculated to cool the water and freeze it is divided by the vaporization energy to find out how much refrigerant must evaporate.

Analyzing the heat of vaporization helps us ensure efficient cooling operations and energy use within refrigeration cycles.