The freezing point of mercury is -38.8 C . What quantity of energy, in joules, is released to the surroundings if 1.00 mL of mercury is cooled from 23.0 C to -38.8 C and then frozen to a solid? (the density of liquid mercury is 13.6 g/cm3. Its specific heat capacity is 0.140 J/g*K and its heat of fusion is 11.4 J/g.
The Correct Answer and Explanation is :
To calculate the energy released to the surroundings, we need to consider two separate processes:
- Cooling the mercury from 23.0°C to its freezing point (-38.8°C).
- Freezing the mercury at -38.8°C.
We can use the following formulas:
- Energy for cooling (Q₁):
[
Q₁ = m \cdot c \cdot \Delta T
]
where:
- ( m ) is the mass of mercury,
- ( c ) is the specific heat capacity of mercury,
- ( \Delta T ) is the change in temperature (final temperature – initial temperature).
- Energy for freezing (Q₂):
[
Q₂ = m \cdot H_f
]
where:
- ( H_f ) is the heat of fusion of mercury.
Step 1: Calculate the mass of mercury
The density of liquid mercury is given as 13.6 g/cm³. The volume of mercury is 1.00 mL, which is equivalent to 1.00 cm³. The mass ( m ) is given by:
[
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 the energy for cooling (Q₁)
The temperature change ( \Delta T ) is:
[
\Delta T = 23.0°C – (-38.8°C) = 23.0°C + 38.8°C = 61.8°C
]
Now, use the formula for cooling:
[
Q₁ = 13.6 \, \text{g} \cdot 0.140 \, \text{J/g·K} \cdot 61.8 \, \text{K}
]
[
Q₁ = 118.7 \, \text{J}
]
Step 3: Calculate the energy for freezing (Q₂)
Now, calculate the energy required to freeze the mercury at -38.8°C:
[
Q₂ = 13.6 \, \text{g} \cdot 11.4 \, \text{J/g} = 154.0 \, \text{J}
]
Step 4: Total energy released
The total energy released to the surroundings is the sum of the energy for cooling and the energy for freezing:
[
Q_{\text{total}} = Q₁ + Q₂ = 118.7 \, \text{J} + 154.0 \, \text{J} = 272.7 \, \text{J}
]
Thus, the total energy released is 272.7 joules.
Explanation:
The process involves two stages: first, cooling the mercury from room temperature (23°C) to its freezing point (-38.8°C), which releases energy as the mercury loses heat. Then, as mercury freezes at this temperature, additional energy is released due to the phase change from liquid to solid, governed by its heat of fusion. The total energy released is the sum of both processes, as the mercury first cools and then freezes, with both steps contributing to the heat released to the surroundings.