Problem 23 - Chemistry and Chemical Reactivity

Question

The freezing point of mercury is \(-38.8^{\circ} \mathrm{C} .\) What quantity of heat energy, in joules, is released to the surroundings if \(1.00 \mathrm{mL}\) of mercury is cooled from \(23.0^{\circ} \mathrm{C}\) to \(-38.8^{\circ} \mathrm{C}\) and then frozen to a solid? (The density of liquid mercury is \(13.6 \mathrm{g} / \mathrm{cm}^{3} .\) Its specific heat capacity is \(0.140 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}\) and its heat of fusion is \(11.4 \mathrm{J} / \mathrm{g} .\) )

Step-by-Step Solution

Answer

The total heat energy released is 36.97 J.

Step 1: Calculate Mass of Mercury

First, determine the mass of mercury using its density and volume. Since the density of liquid mercury is \(13.6\, \text{g/cm}^3\) and the volume is \(1.00\, \text{mL}\), we have: \[ m = \text{Density} \times \text{Volume} = 13.6\, \text{g/cm}^3 \times 1.00\, \text{cm}^3 = 13.6\, \text{g}. \]

Step 2: Calculate Heat Released During Cooling

Calculate the heat released when mercury is cooled from \(23.0^{\circ} \text{C}\) to \(-38.8^{\circ} \text{C}\). Use the formula: \[ q = m \times c \times \Delta T, \] where \(m = 13.6\, \text{g}\), \(c = 0.140\, \text{J/g} \cdot \text{K}\), and \(\Delta T = -38.8^{\circ} \text{C} - 23.0^{\circ} \text{C} = -61.8\, \text{K}.\)\[ q = 13.6\, \text{g} \times 0.140\, \text{J/g} \cdot \text{K} \times (-61.8\, \text{K}) = -118.07\, \text{J}.\]

Step 3: Calculate Heat Released During Freezing

When mercury freezes, it releases latent heat. The heat of fusion of mercury is \(11.4\, \text{J/g}\). Therefore, the heat released during freezing is: \[ q = m \times \text{Heat of Fusion} = 13.6\, \text{g} \times 11.4\, \text{J/g} = 155.04\, \text{J}.\]

Step 4: Calculate Total Heat Energy Released

Add the heat released from cooling and the heat released from freezing to find the total heat energy released: \[ q_{\text{total}} = q_{\text{cooling}} + q_{\text{freezing}} = -118.07\, \text{J} + 155.04\, \text{J} = 36.97\, \text{J}.\]

Key Concepts

Freezing Point

The freezing point is the temperature at which a liquid turns into a solid. For mercury, this is environments this means that when mercury reaches -38.8°C, it will start forming solid crystals. This is important for heat energy calculations because once mercury reaches this temperature, additional heat is released as it transitions from liquid to solid. The release of heat during this phase change does not change the temperature; instead, it is used to arrange the mercury particles into a solid structure. This is known as latent heat release.

• Freezing occurs at the freezing point.
• Temperature remains constant during the phase change.
• Energy is utilized to form a solid structure.

Understanding the freezing point is crucial to calculate the total amount of heat energy involved, because it marks the temperature at which the calculation for latent heat needs to begin. In the exercise, noticing the freezing point allows you to split your examination of heat energy loss between cooling the mercury and the freezing process itself.

Specific Heat Capacity

Specific heat capacity is a measure of the amount of heat required to raise the temperature of a given mass of substance by one degree Celsius (or Kelvin). For mercury, this is low at 0.140 J/g·K, signifying that it takes relatively little heat to change its temperature. This is crucial in exercises where temperature changes over a range are involved because it helps quantify how much heat is needed for temperature adjustments before a phase change occurs.

• Defined as heat required per unit mass per degree change.
• Low value means temperature changes easily with little heat.
• Essential for calculating heat in non-phase changes.

In the problem, the specific heat capacity is vital for calculating heat released when mercury goes from 23°C to its freezing point. This helps in determining the first stage of our total energy released, just before the freezing begins.

Heat of Fusion

Heat of fusion is the energy required to change a substance from solid to liquid or vice versa without a change in temperature. For mercury, this amount is 11.4 J/g. During freezing, this energy is released into the surroundings as mercury's particles arrange themselves into a solid without changing the temperature further.

• Involves energy needed for solid-liquid phase change.
• The process occurs isothermally (at constant temperature).
• Helps calculate energy released during phase transitions.

In the exercise, once mercury is cooled to its freezing point, it doesn't cool further until it freezes completely. The heat of fusion accounts for the additional energy released as the mercury solidifies, forming a crucial part of the total energy release calculation. Adding this value to the energy calculated from cooling provides a comprehensive picture of the process, ensuring that the energy required for both cooling and phase transition is accounted for.