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

Around \(1491.51\,\mathrm{g}\) of \(\mathrm{CF}_{2} \mathrm{Cl}_{2}\) must be vaporized to convert all the water at \(22.0^{\circ}\mathrm{C}\) to ice at \(-5.0^{\circ}\mathrm{C}\).

Step-by-step solution

Calculate total mass of water

To find the total mass of water, multiply the mass of each ice cube (30.0 g) by the number of cubes (18): \(30.0\,\mathrm{g} \times 18 = 540\,\mathrm{g}\). Next, we need to calculate the heat required to lower the temperature of the water to its freezing point (0°C).

Calculate heat to lower water temperature to 0°C

Using the heat capacity for liquid water and the formula \(q = mcΔT\), we have: \(q_1 = (540\,\mathrm{g})(4.18\,\mathrm{J}/\mathrm{g} \cdot{ } ^{\circ}\mathrm{C})(22.0 - 0)\,\mathrm{J} = 49{,}514\,\mathrm{J}\). Now, we will calculate the heat needed to change the water from liquid to solid at 0°C using the enthalpy of fusion.

Calculate heat to change water to ice

First, we need to convert the mass of water to moles. We know that the molar mass of water is \(18.015\,\mathrm{g}/\mathrm{mol}\). So the number of moles is \(\frac{540\,\mathrm{g}}{18.015\,\mathrm{g}/\mathrm{mol}} = 30\,\mathrm{mol}\). Next, we multiply the moles by the enthalpy of fusion per mole: \(q_2 = 30\,\mathrm{mol} \times 6.02\,\mathrm{kJ}/\mathrm{mol} = 180.6\,\mathrm{k}\mathrm{J}\). Converting this to joules, we get \(180{,}600\,\mathrm{J}\). Afterward, we need to calculate the heat necessary to lower the ice temperature to -5°C.

Calculate heat to lower ice temperature to -5°C

Using the heat capacity for solid water and the formula \(q = mcΔT\), we have: \(q_3 = (540\,\mathrm{g})(2.03\,\mathrm{J}/\mathrm{g} \cdot{ } ^{\circ}\mathrm{C})(0 - (-5))\,\mathrm{J} = 5{,}481\,\mathrm{J}\). Now, by adding all the heats, we can find the total heat removed from the water.

Calculate total heat removed from water

The total heat removed from the water is: \(q_\text{total} = q_1 + q_2 + q_3 = 49{,}514\,\mathrm{J} + 180{,}600\,\mathrm{J} + 5{,}481\,\mathrm{J} = 235{,}595\,\mathrm{J}\). Finally, we will determine the mass of \(\mathrm{CF}_{2} \mathrm{Cl}_{2}\) needed.

Calculate mass of \(CF_2Cl_2\) needed

We know that the heat of vaporization for \(\mathrm{CF}_{2}\mathrm{Cl}_{2}\) is \(158 \mathrm{J} / \mathrm{g}\). To find the mass, we divide the total heat by the heat of vaporization: \(m_\text{\(CF_2\)}\) = \(\frac{235{,}595\,\mathrm{J}}{158\,\mathrm{J}/\mathrm{g}} = 1491.51\,\mathrm{g}\). Therefore, around \(1491.51\,\mathrm{g}\) of \(\mathrm{CF}_{2} \mathrm{Cl}_{2}\) must be vaporized to convert all the water at \(22.0^{\circ}\mathrm{C}\) to ice at \(-5.0^{\circ}\mathrm{C}\).

Key concepts

Heat Capacity

Heat capacity is an important concept in thermodynamics and physical chemistry. It tells us how much heat is needed to change the temperature of a substance. More specifically, it's the amount of energy required to raise the temperature of 1 gram of a substance by 1 degree Celsius. For water, which can exist in both liquid and solid states, the heat capacity changes. While the heat capacity of liquid water is 4.18 J/g°C, the heat capacity of ice is lower, at 2.03 J/g°C.

This difference demonstrates that more energy is needed to heat water than ice, owing to the additional energy stored in the structure of liquid water. Knowing the heat capacity of a substance allows us to compute the amount of heat (q) gained or lost using the formula: \(q = mcΔT\), where \(m\) is mass and \(ΔT\) is the change in temperature. This formula is fundamental in calculating thermal energy changes when heating or cooling materials.

• Liquid water has a high heat capacity, which means it resists temperature changes.
• Ice, having a lower heat capacity, requires less energy for temperature change.
• Heat capacity helps us understand and calculate how substances react to thermal energy.

Heat of Vaporization

The heat of vaporization is the energy needed to transform a given amount of a liquid into a gas. This is without changing its temperature, emphasizing the energy required to overcome intermolecular forces in a liquid. For example, the heat of vaporization for \(\mathrm{CF}_{2} \mathrm{Cl}_{2}\), a refrigerant used in this problem, is 158 J/g.

Vaporization encompasses both evaporation and boiling, where molecules have enough energy to escape the liquid's surface into the gaseous phase. This concept is crucial in refrigeration cycles, where energy absorbed during vaporization creates a cooling effect. To calculate the mass of refrigerant needed in such processes, the heat removed from the substance is divided by the heat of vaporization.

• Heat of vaporization is pivotal in cooling and refrigeration systems.
• Understanding it helps explain phase changes from liquid to gas.
• It measures the energy necessary to overcome attractive forces within a liquid.

Molar Mass

Molar mass is the weight of one mole of a substance, expressed in grams per mole. For water, the molar mass is 18.015 g/mol, allowing us to convert between mass and moles, which is crucial for finding the energy associated with phase changes. This conversion relies on Avogadro's number, defining the quantity of atoms or molecules in a mole.

By understanding molar mass, we can calculate moles from a given mass of a substance, and in turn, determine enthalpy changes using thermochemical data like the enthalpy of fusion or vaporization. For instance, in this problem, we converted 540 g of water into moles to apply the known enthalpy of fusion value, enabling us to determine the thermal dynamics during the water's phase change to ice.

• Molar mass bridges mass to quantity (moles).
• It is vital for calculating energy changes in chemical reactions and phase changes.
• It helps convert mass into moles, facilitating stoichiometric calculations.