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 find the quantity of energy released to the surroundings, we need to calculate the energy released in two stages: first, when mercury is cooled from 23.0°C to its freezing point (-38.8°C), and then when it freezes into a solid at that temperature.
Step 1: Energy released during cooling
The energy released when a substance is cooled is given by the formula:
[
q = m \cdot c \cdot \Delta T
]
Where:
- (q) is the heat energy released (in joules),
- (m) is the mass of mercury (in grams),
- (c) is the specific heat capacity of mercury (in J/g·K),
- (\Delta T) is the change in temperature (in °C or K).
Find the mass of mercury
The volume of mercury is given as 1.00 mL, and the density of mercury is 13.6 g/cm³. Since the density of mercury is 13.6 g/mL, the mass (m) is:
[
m = \text{density} \times \text{volume} = 13.6 \, \text{g/mL} \times 1.00 \, \text{mL} = 13.6 \, \text{g}
]
Change in temperature
The temperature change (\Delta T) is the difference between the initial temperature (23.0°C) and the freezing point (-38.8°C):
[
\Delta T = 23.0°C – (-38.8°C) = 23.0 + 38.8 = 61.8 \, \text{°C}
]
Calculate the heat released during cooling
Now, we can calculate the energy released during cooling:
[
q = m \cdot c \cdot \Delta T = 13.6 \, \text{g} \cdot 0.140 \, \text{J/g·K} \cdot 61.8 \, \text{K}
]
[
q = 13.6 \cdot 0.140 \cdot 61.8 = 118.8 \, \text{J}
]
Step 2: Energy released during freezing
The energy released during freezing is calculated using the heat of fusion, given by:
[
q = m \cdot \Delta H_f
]
Where:
- (\Delta H_f) is the heat of fusion of mercury (11.4 J/g).
[
q = 13.6 \, \text{g} \cdot 11.4 \, \text{J/g} = 154.4 \, \text{J}
]
Total energy released
The total energy released to the surroundings is the sum of the energy released during cooling and the energy released during freezing:
[
q_{\text{total}} = 118.8 \, \text{J} + 154.4 \, \text{J} = 273.2 \, \text{J}
]
Final Answer
The total energy released to the surroundings is 273.2 J.
Explanation
To calculate the total energy released when mercury is cooled and frozen, we used two key concepts:
- Heat transfer during cooling: The heat energy released when a substance is cooled is determined by its mass, specific heat capacity, and the change in temperature.
- Latent heat during freezing: The energy released during phase change (from liquid to solid) is determined by the substance’s heat of fusion, which represents the amount of energy required to freeze 1 gram of a substance.
These calculations account for both the cooling of the mercury to its freezing point and its subsequent phase change to a solid. The total energy released is the sum of these two processes.