Problem 36 - Chemistry and Chemical Reactivity

Question

If your air conditioner is more than several years old, it may use the chlorofluorocarbon \(\mathrm{CCl}_{2} \mathrm{F}_{2}\) as the heat transfer fluid. The normal boiling point of \(\mathrm{CCl}_{2} \mathrm{F}_{2}\) is \(-29.8^{\circ} \mathrm{C},\) and the enthalpy of vaporization is \(20.11 \mathrm{kJ} / \mathrm{mol} .\) The gas and the liquid have molar heat capacities of \(117.2 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\) and \(72.3 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K},\) respec- tively. How much energy is evolved as heat when \(20.0 \mathrm{g}\) of \(\mathrm{CCl}_{2} \mathrm{F}_{2}\) is cooled from \(+40^{\circ} \mathrm{C}\) to \(-40^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The total energy evolved is \(-4787.4\,\text{J}.\)

Step 1: Calculate the moles of CCl2F2

First, we determine the number of moles of \( \mathrm{CCl}_{2} \mathrm{F}_{2} \) using its molar mass. The molar mass of \( \mathrm{CCl}_{2} \mathrm{F}_{2} \) is calculated as follows: \[ \text{Molar mass} = (12.01 + 2 \times 35.45 + 2 \times 18.998) \ \text{g/mol} = 120.91 \ \text{g/mol}. \] Now, calculate the moles for the given mass (20.0 g): \[ \text{Moles} = \frac{20.0 \ \text{g}}{120.91 \ \text{g/mol}} = 0.165 \ \text{mol}. \]

Step 2: Calculate cooling in the gas phase

Cool the gas from \(40^{\circ} \mathrm{C} \) to the boiling point \(-29.8^{\circ} \mathrm{C}\). Use the formula for heat transfer in the gas phase: \[ q = nC_{p,\text{gas}} \Delta T. \] Here, \( C_{p,\text{gas}} = 117.2 \ \mathrm{J/mol \cdot K} \), \( n = 0.165 \ \text{mol} \), and \( \Delta T = -29.8 - 40 = -69.8^{\circ} \mathrm{C} = -69.8 \ \text{K} \). \[ q = 0.165 \times 117.2 \times (-69.8) = -1344.9 \ \text{J}. \]

Step 3: Phase Change (Condensation)

Now convert the gas to liquid at the boiling point. Use the enthalpy of vaporization: \( \Delta H_{\text{vap}} = 20.11 \ \mathrm{kJ/mol} \). The energy change for this phase transition is: \[ q = -n \Delta H_{\text{vap}} = -0.165 \times 20.11 \times 1000 = -3320.8 \ \text{J}. \]

Step 4: Cooling in the liquid phase

Cool the liquid from the boiling point \(-29.8^{\circ} \mathrm{C}\) to \(-40^{\circ} \mathrm{C}\). Use the formula for heat transfer in the liquid phase: \[ q = nC_{p,\text{liquid}} \Delta T. \] Here, \( C_{p,\text{liquid}} = 72.3 \ \mathrm{J/mol \cdot K} \), \( n = 0.165 \ \text{mol} \), and \( \Delta T = -40 - (-29.8) = -10.2 \ \text{K} \). \[ q = 0.165 \times 72.3 \times (-10.2) = -121.7 \ \text{J}. \]

Step 5: Sum all energy changes

Sum all calculated energy changes from each step to find the total energy evolved: \[ q_{\text{total}} = q_{\text{gas}} + q_{\text{phase}} + q_{\text{liquid}} = -1344.9 - 3320.8 - 121.7 = -4787.4 \ \text{J}. \]

Key Concepts

Phase Change

Phase change refers to the transition of a substance from one state of matter to another, such as solid to liquid, liquid to gas, or vice versa. In this problem, we specifically deal with the transition from gas to liquid, known as condensation.
A phase change is significant because it involves a large amount of energy absorption or release. During condensation, energy is released as the molecules decrease their kinetic energy to form intermolecular bonds.
This is why we use enthalpy of vaporization to calculate the energy needed or released during these transitions. Understanding phase changes is crucial in thermodynamics, as they are involved in processes like refrigeration and heating.

Enthalpy of Vaporization

Enthalpy of vaporization (\( \Delta H_{\text{vap}} \)) is the amount of heat required to convert one mole of a liquid into a gas at constant temperature and pressure. For the reverse process—condensation—it is the heat released when a gas turns back into a liquid.
In the exercise, \( \Delta H_{\text{vap}} \) for \(\text{CCl}_{2}\text{F}_{2}\) is given as 20.11 kJ/mol. This indicates that when the gas condenses, \(20.11 kJ/mol\) is released.
This concept is essential when trying to understand how much energy is absorbed or released during a phase transition, impacting systems like air conditioners, which rely on substantial energy changes to operate efficiently.

Heat Capacity

Heat capacity (\(C_p\)) is the amount of heat needed to raise the temperature of a given quantity of a substance by 1 Kelvin. It is specific to the state of matter of the substance, as seen with \(\text{CCl}_{2}\text{F}_{2}\) in this problem, which has different heat capacities for its liquid and gas states.

• \(C_{p,\text{gas}} = 117.2 \text{ J/mol} \cdot \text{K}\) while \(C_{p,\text{liquid}} = 72.3 \text{ J/mol} \cdot \text{K}\).

These values influence the amount of energy released or absorbed during temperature changes within the gaseous and liquid states of the material.
Understanding heat capacity helps predict how much a substance's temperature will change when it absorbs or releases a given amount of energy, crucial for designing and operating thermodynamic systems effectively.