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 specific heats of \(117.2 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\) and \(72.3 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K},\) respectively. How much heat is evolved 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 heat evolved is approximately 4.8 kJ.

Step 1: Calculate moles of CCl2F2

First, determine the number of moles of \(\mathrm{CCl}_{2}\mathrm{F}_{2}\) from its mass. The molar mass of \(\mathrm{CCl}_{2}\mathrm{F}_{2}\) is approximately \(120.91 \mathrm{g/mol}\). Therefore, the number of moles is calculated as follows: \[\text{moles} = \frac{20.0 \mathrm{g}}{120.91 \mathrm{g/mol}} \approx 0.1655 \mathrm{mol}.\]

Step 2: Calculate heat released while cooling gas

Using the specific heat capacity of the gas, calculate the heat lost while cooling from \(+40^{\circ} \mathrm{C}\) to the boiling point, which is \(-29.8^{\circ} \mathrm{C}\). Use the formula \(q = m \cdot C \cdot \Delta T\) where \( q \) is the heat, \( m \) is moles, \( C \) is specific heat capacity and \(\Delta T\) is change in temperature. \[ q = 0.1655 \mathrm{mol} \times 117.2 \mathrm{J/mol\cdot K} \times (273.15 + 40 - (-29.8))\mathrm{K} \] \[ = 0.1655 \times 117.2 \times (313.15 - 243.35) \] \[ = 0.1655 \times 117.2 \times 69.8 = 1351.856 \mathrm{J}.\]

Step 3: Calculate heat released during condensation

Next, calculate the heat released during the phase change from gas to liquid at the boiling point (\(-29.8^{\circ} \mathrm{C}\)). Use the latent heat of vaporization: \[ q = n \cdot \Delta H_{vap} = 0.1655 \mathrm{mol} \times 20110 \mathrm{J/mol} = 3328.505 \mathrm{J}.\]

Step 4: Calculate heat released while cooling liquid

Finally, calculate the heat released while cooling the liquid from the boiling point to \(-40^{\circ} \mathrm{C}\) using the liquid's specific heat capacity. \[ q = 0.1655 \mathrm{mol} \times 72.3 \mathrm{J/mol\cdot K} \times (-40 - (-29.8)) \mathrm{K} \] \[ = 0.1655 \times 72.3 \times (-10.2) = -122.1794 \mathrm{J}.\] As heat is released, we consider this value as positive when measuring evolved heat.

Step 5: Total heat evolved calculation

Add all the heat values calculated from each step to find the total heat evolved. Note: Heat absorbed during cooling the liquid should have been inadvertently added as a negative value due to temperature reduction, so recalibrate appropriately. Total heat released is now the sum of steps 2, 3, 4. \[ q_{total} = 1351.856 + 3328.505 + 122.1794 = 4802.54 \mathrm{J} \approx 4.8 \mathrm{kJ}.\]

Key Concepts

Phase Change

A phase change occurs when a substance transitions from one state of matter to another, such as from gas to liquid or liquid to solid. In the context of our exercise, we focus on the transition of \(\mathrm{CCl}_2\mathrm{F}_2\) from a gaseous state to a liquid state. This process occurs at the substance's boiling point, which for \(\mathrm{CCl}_2\mathrm{F}_2\), is \(-29.8^{\circ} \mathrm{C}\).
During this change, the temperature remains constant even though heat is being released or absorbed, known as the latent heat. Understanding phase changes is crucial in thermodynamics and helps in calculating energy exchanges in systems.
To calculate the energy released during this phase transition, the enthalpy of vaporization is used, which is defined as the amount of energy required to convert one mole of a liquid into a gas (or vice versa) at a given pressure.
By applying the equation \( q = n \cdot \Delta H_{vap} \), where \( q \) is the heat released or absorbed, \( n \) is the number of moles, and \( \Delta H_{vap} \) is the latent heat of vaporization, you can calculate the energy associated with the phase change.

Specific Heat Capacity

Specific heat capacity is a property that describes how much energy is needed to raise one mole (or unit mass) of a substance by one degree Kelvin (or Celsius). In the provided exercise, the specific heat capacity of both gaseous and liquid forms of \(\mathrm{CCl}_2\mathrm{F}_2\) are given and are instrumental in calculating the thermal energy involved in temperature changes.
The gas form has a specific heat capacity of \(117.2 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\) and the liquid form \(72.3 \mathrm{J}/ \mathrm{mol} \cdot \mathrm{K}\). These values indicate that less energy is required to change the temperature of the liquid form compared to the gaseous form.
To compute energy changes due to temperature variations without a phase change, use the formula \( q = m \cdot C \cdot \Delta T \), where \( m \) is the number of moles, \( C \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. This formula is key in determining the heat exchanged when the substance is cooled or heated.

Latent Heat

Latent heat refers to the heat required to convert a substance from one phase to another at a constant temperature and pressure, without changing its temperature during the process. In the case of \(\mathrm{CCl}_2\mathrm{F}_2\), the enthalpy of vaporization, a type of latent heat, is \(20.11 \mathrm{kJ} / \mathrm{mol}\), which quantifies the energy change during the transition from gas to liquid.
The latent heat concept is crucial because it highlights that energy changes can occur without temperature change due to phase transitions. This is especially relevant in calculations involving heating and cooling processes in air conditioning systems where the medium might change states.
Calculating the total heat evolved in exercises like this involves combining the latent heat with the sensible heat (calculated using the specific heat capacities) to find the overall energy exchange in the system.

Moles Calculation

Calculating the number of moles is foundational in chemistry and essential for quantifying reaction extents and energy changes. In this exercise, we determined the moles of \(\mathrm{CCl}_2\mathrm{F}_2\) to be used in subsequent energy calculations.
First, the molar mass of \(\mathrm{CCl}_2\mathrm{F}_2\) was noted as approximately \(120.91 \mathrm{g/mol}\). With a provided mass of \(20.0 \mathrm{g}\), the number of moles was calculated using the formula: \[ \text{moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{20.0 \ \mathrm{g}}{120.91 \ \mathrm{g/mol}} \approx 0.1655 \ \mathrm{mol}. \]
Understanding moles is crucial as it forms the basis of stoichiometry in chemical reactions and links the mass of substances to the number of molecules and atoms. This enables precise calculations of energy transformations and reactions.