Question

Calculate the amount of heat liberated (in kJ) from 366 \(\mathrm{g}\) of mercury when it cools from \(77.0^{\circ} \mathrm{C}\) to \(12.0^{\circ} \mathrm{C}\).

Short answer

The heat liberated is 3.33 kJ.

Step-by-step solution

Identify Known Values

We are given the following information:- Mass of mercury: \( m = 366 \, \text{g} \)- Initial temperature: \( T_i = 77.0^{\circ} \mathrm{C} \)- Final temperature: \( T_f = 12.0^{\circ} \mathrm{C} \)- Specific heat capacity of mercury: \( c = 0.140 \frac{\text{J}}{\text{g} \cdot {\text{°C}}} \)We need to calculate the amount of heat \( q \) liberated in kilojoules.

Calculate Temperature Change

Calculate the change in temperature \( \Delta T \):\[\Delta T = T_f - T_i = 12.0^{\circ} \mathrm{C} - 77.0^{\circ} \mathrm{C} = -65.0^{\circ} \mathrm{C}\]The temperature change is \(-65.0^{\circ} \mathrm{C}\).

Apply Heat Transfer Formula

Use the formula for heat transfer:\[q = m \cdot c \cdot \Delta T\]Substitute the known values:\[q = 366 \, \text{g} \times 0.140 \frac{\text{J}}{\text{g} \cdot {\text{°C}}} \times (-65.0^{\circ} \mathrm{C})\]

Calculate the Heat Liberated in Joules

Calculate the value:\[q = 366 \times 0.140 \times -65.0 = -3331.8 \, \text{J}\]Since the heat is being liberated, the value is negative. The magnitude of heat liberated is \(3331.8 \, \text{J}\).

Convert Joules to Kilojoules

Convert the heat from joules to kilojoules by dividing by 1000:\[q = \frac{3331.8}{1000} = 3.3318 \, \text{kJ} \]Thus, the heat liberated is approximately \(3.33 \, \text{kJ} \).

Key concepts

Specific Heat Capacity

Specific heat capacity is an important concept in thermodynamics. It refers to the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. In our case, mercury has a specific heat capacity of 0.140 J/g°C. This relatively low value indicates that mercury requires less heat to change its temperature compared to substances with higher specific heat capacities.

Understanding specific heat capacity helps us comprehend how different materials absorb and release heat. For example, water has a much higher specific heat capacity than mercury, which means it can store more heat energy. This is why substances like water take longer to heat up or cool down.

When performing calculations, the specific heat capacity allows us to quantify the heat change given the mass and temperature difference of a substance. It is crucial for calculations like the one provided in the exercise.

Heat Transfer Calculation

Heat transfer calculations are essential to determine the thermal energy change in a substance. They involve multiplying the mass of the substance by its specific heat capacity and the change in temperature.

In the exercise, we are tasked to find the heat liberated from mercury as it cools. The heat transfer formula used is:

• \( q = m \cdot c \cdot \Delta T \)

Where:

• \( q \) = amount of heat (Joules)
• \( m \) = mass (grams)
• \( c \) = specific heat capacity (J/g°C)
• \( \Delta T \) = change in temperature (°C)

By substituting the given values, we can easily calculate how much thermal energy is released or absorbed by a substance. Positive values of \( q \) indicate heat is absorbed, while negative values signify heat is released. This is particularly useful for determining energy efficiency and energy savings in various applications.

Temperature Change

Temperature change is a simple yet vital concept in understanding how heat affects substances. It measures the difference between the initial and final temperatures of a substance and is denoted by \( \Delta T \).

In the exercise, mercury starts at 77°C and cools down to 12°C. This results in a temperature change calculated as:

• \( \Delta T = T_f - T_i = 12.0^{\circ} \mathrm{C} - 77.0^{\circ} \mathrm{C} = -65.0^{\circ} \mathrm{C} \)

The negative sign of \( \Delta T \) denotes a decrease in temperature, which corresponds to a release of energy as the substance cools.

Temperature change is particularly important in everyday applications such as climate control in buildings, cooking, and refrigeration. When combined with specific heat capacity and mass, it allows us to calculate the total heat exchanged during the heating or cooling process.