Problem 12 - Chemistry and Chemical Reactivity

Question

After absorbing \(1.850 \mathrm{kJ}\) of heat, the temperature of a \(0.500-\mathrm{kg}\) block of copper is \(37^{\circ} \mathrm{C} .\) What was its initial temperature?

Step-by-Step Solution

Answer

The initial temperature was approximately \( 27.38^{\circ} \mathrm{C} \).

Step 1: Identify Given Values

First, note down the values given in the exercise. The heat absorbed is \( Q = 1.850 \, \mathrm{kJ} \), the mass of the copper block is \( m = 0.500 \, \mathrm{kg} \), and the final temperature of the block is \( T_f = 37^{\circ} \mathrm{C} \).

Step 2: Convert Heat to Joules

Convert the heat absorbed into joules since the specific heat capacity will use these units. Given \( Q = 1.850 \, \mathrm{kJ} \), convert it: \( Q = 1.850 \times 1000 = 1850 \, \mathrm{J} \).

Step 3: Use the Specific Heat Capacity of Copper

Look up the specific heat capacity of copper, which is \( c = 0.385 \, \mathrm{J/g^{\circ}C} \). Convert this to \( \mathrm{J/kg^{\circ}C} \) by multiplying by 1000: \( c = 0.385 \times 1000 = 385 \, \mathrm{J/kg^{\circ}C} \).

Step 4: Apply the Heat Transfer Formula

Use the formula for heat transfer: \( Q = mc\Delta T \), where \( \Delta T = T_f - T_i \) is the change in temperature. Rearrange the formula to solve for the initial temperature \( T_i \): \( T_i = T_f - \frac{Q}{mc} \).

Step 5: Calculate the Initial Temperature

Substitute the known values into the formula \( T_i = 37 - \frac{1850}{0.500 \times 385} \). Compute the value to find \( T_i \).

Step 6: Simplify the Calculation

First calculate \( mc = 0.500 \times 385 = 192.5 \, \mathrm{J/^{\circ}C} \). Then, \( \frac{1850}{192.5} \approx 9.62 \, ^{\circ}C \). Finally, \( T_i = 37 - 9.62 = 27.38 \, ^{\circ}C \).

Key Concepts

Understanding Heat Transfer

Heat transfer is the process by which heat energy moves from one body or substance to another. In the context of the exercise, heat is absorbed by the copper block, resulting in a change in temperature. There are three primary modes of heat transfer: conduction, convection, and radiation. Here, the focus is on conduction, as the heat is directly transferred to the copper block.

• Conduction occurs when heat travels through a material without the movement of the material itself.
• Copper, being a metal, is an excellent conductor of heat, meaning it readily absorbs heat energy.

Heat transfer is quantitatively described by the formula: \[ Q = mc \Delta T \] where:

• \( Q \) is the amount of heat transferred, measured in joules (J).
• \( m \) is the mass of the substance, in kilograms (kg).
• \( c \) is the specific heat capacity, indicating how much energy is needed to change the temperature of 1 kg of the substance by 1°C.
• \( \Delta T \) represents the change in temperature.

This formula is pivotal for determining how heat affects substances and is a fundamental aspect of thermodynamics.

The Role of Specific Heat Capacity

Specific heat capacity is a property of a substance that describes how much heat is necessary to change its temperature by 1 degree Celsius. In this exercise, copper has a specific heat capacity of 385 J/kg°C.

• A high specific heat capacity means more energy is needed to change the temperature.
• A low specific heat implies that even a small amount of energy can cause a significant temperature change.

Copper’s specific heat capacity determines how much heat energy is absorbed to raise the temperature of the copper block. Given its moderate specific heat, copper heats and cools relatively quickly compared to other materials with higher specific heats.
Understanding specific heat capacity is critical when calculating how substances will react to heat gain or loss, which is essential in engineering and many scientific applications.

Analyzing Temperature Change

Temperature change (\( \Delta T \)) occurs as a result of heat being absorbed or lost by a substance. The change in temperature for our copper block was calculated using the heat transfer formula and the values provided in the exercise.

• The final temperature (\( T_f \)) was given as 37°C.
• The equation for temperature change is: \( \Delta T = T_f - T_i \) where \( T_i \) is the initial temperature.

By rearranging the heat transfer formula, the initial temperature (\( T_i \)) can be found, demonstrating how heat energy directly influences the temperature of a material. In practical applications, knowing how to calculate temperature change is valuable in areas such as climate science, cooking, and material science.

Key Copper Properties

Copper is an essential metal known for its excellent thermal and electrical conductivity. Its properties make it ideal for various applications, including electrical wiring and heat exchangers.

• Copper’s conductivity allows for efficient heat and electrical energy transfer, making it indispensable in many industries.
• Its moderate specific heat capacity (385 J/kg°C) means it both heats and cools efficiently without enormous energy consumption.
• Copper is also durable and resistant to corrosion, maintaining its properties over long periods.

Understanding copper's properties helps in designing systems that either distribute heat effectively or require minimal energy loss. In contexts like this exercise, these properties explain why copper was chosen and how it responds to heat influx.