Question

The specific heat capacity of copper metal is \(0.385 \mathrm{J} / \mathrm{g} \cdot \mathrm{K} .\) How much energy is required to heat 168 g of copper from \(-12.2^{\circ} \mathrm{C}\) to \(+25.6^{\circ} \mathrm{C} ?\)

Short answer

2444.74 J

Step-by-step solution

Identify Known Values

We have the specific heat capacity of copper, which is \(c = 0.385 \, \text{J/g} \cdot \text{K}\), the mass \(m = 168 \, \text{g}\), the initial temperature \(T_{i} = -12.2^{\circ} \text{C}\), and the final temperature \(T_{f} = 25.6^{\circ} \text{C}\).

Calculate Temperature Change

The change in temperature \( \Delta T \) is calculated using the formula:\[\Delta T = T_{f} - T_{i}\]Substitute the values:\[\Delta T = 25.6^{\circ} \text{C} - (-12.2^{\circ} \text{C}) = 25.6 + 12.2 = 37.8 \, \text{K}\]

Utilize the Energy Formula

To find the total energy required, use the specific heat formula:\[Q = m \cdot c \cdot \Delta T\]where \(Q\) is the energy, \(m\) is the mass, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change.

Perform the Calculation

Substitute the known values into the formula:\[Q = 168 \, \text{g} \cdot 0.385 \, \text{J/g} \cdot \text{K} \cdot 37.8 \, \text{K}\]Calculate the result:\[Q = 168 \times 0.385 \times 37.8 \]\[ Q = 2444.736 \, \text{J}\]

Finalize with Conclusion

Therefore, the total energy required to heat 168 g of copper from \(-12.2^{\circ} \mathrm{C}\) to \(+25.6^{\circ} \mathrm{C}\) is approximately \(2444.74 \, \text{J}\).

Key concepts

Energy Calculation

Energy calculation is fundamental to understanding how much energy is needed to change the temperature of a substance. When dealing with problems involving heating or cooling, we use a specific formula that considers the mass, specific heat capacity, and temperature change of the material.

The formula for calculating energy is:

• \[ Q = m \cdot c \cdot \Delta T \]

Where:

• \( Q \) is the energy in Joules (J)
• \( m \) is the mass of the substance in grams (g)
• \( c \) is the specific heat capacity of the substance (J/g·K)
• \( \Delta T \) is the temperature change in Kelvin (K)

This equation helps calculate the amount of heat energy required when heating a given mass of material by a certain temperature difference. Understanding and effectively utilizing this formula is key to solving thermostat problems like the one concerning copper we examined.

Temperature Change

Temperature change often requires a straightforward calculation to establish how much heat we need to add or subtract. To find the change in temperature, denoted as \( \Delta T \), you subtract the initial temperature from the final temperature.

The equation used is:

• \[ \Delta T = T_f - T_i \]

Where:

• \( T_f \) is the final temperature
• \( T_i \) is the initial temperature

In the case of our copper example, it's important to effectively calculate \( \Delta T \), as this difference determines the amount of energy we need to use or release for temperature change. With correct values, calculating \( \Delta T \) ensures that you input the right numbers into your energy calculation formula.

Copper Specific Heat

Copper's specific heat capacity is a unique property important when performing energy calculations. Specific heat capacity is a measure of how much energy is required to increase the temperature of one gram of a substance by one degree Kelvin (or Celsius).

For copper, this value is known to be:

• \( c = 0.385 \, \text{J/g} \cdot \text{K} \)

Compared to other substances, copper has a relatively low specific heat capacity, meaning it doesn't require much energy to change its temperature.

This property makes it an excellent material for applications where quick heat exchange is important, such as in electrical components and cookware. Being aware of copper's specific heat capacity allows for precise energy calculations needed for heating or cooling processes, ensuring efficiency in practical applications.