Problem 7 - Chemistry and Chemical Reactivity

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 \(168 \mathrm{g}\) of copper from \(-12.2^{\circ} \mathrm{C}\) to \(+25.6^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

2444.8 J

Step 1: Identify Given Values

We need to find how much energy is required to heat copper. The given values are: the mass of copper is 168 g, the specific heat capacity is 0.385 \(\text{J/g} \cdot \text{K}\), the initial temperature is \(-12.2^{\circ}C\) and the final temperature is \(25.6^{\circ}C\).

Step 2: Calculate Temperature Change

Determine the change in temperature by subtracting the initial temperature from the final temperature: \[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 25.6 - (-12.2) = 37.8^{\circ}C \]

Step 3: Apply the Formula

Use the formula for calculating the energy required: \[ q = m \cdot c \cdot \Delta T \]where \( q \) is the energy in joules, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change. Substitute in the known values:\[ q = 168 \cdot 0.385 \cdot 37.8 \]

Step 4: Calculate the Energy

Perform the calculations:\[ q = 168 \cdot 0.385 \cdot 37.8 = 2444.796 \text{ J} \]

Key Concepts

Energy Calculation

Energy calculation is a fundamental concept in physics and chemistry, especially when dealing with heat transfer. To calculate the energy required to change the temperature of a substance, we use the formula:
\[ q = m \cdot c \cdot \Delta T \]
Where:

• \( q \) is the energy in joules.
• \( m \) is the mass of the substance in grams.
• \( c \) is the specific heat capacity.
• \( \Delta T \) is the change in temperature.

Energy calculation helps in understanding how much energy is needed to increase or decrease the temperature of a substance. By using this formula, you can determine how different factors, such as mass and specific heat capacity, affect the energy required. In real-world scenarios, accurate energy calculations are crucial for designing heating systems and managing energy consumption.

Temperature Change

Understanding temperature change is key when performing calculations involving heat. Temperature change, denoted as \( \Delta T \), is the difference between the final temperature and the initial temperature. In this exercise:
\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \]
For copper being heated from \(-12.2^{\circ}C\) to \(25.6^{\circ}C\), the temperature change is calculated as:
\[ \Delta T = 25.6 - (-12.2) = 37.8^{\circ}C \]
This temperature change signifies how much the temperature has increased or decreased. It is crucial not only in academic exercises but also in practical applications like cooking or metallurgical processes, where precise control of temperature changes is often necessary to achieve desired results.

Copper Heating

Copper heating illustrates the application of specific heat capacity in real-world scenarios. Copper has a specific heat capacity of \(0.385 \text{J/g} \cdot \text{K}\), meaning it requires 0.385 joules to raise the temperature of 1 gram of copper by 1 degree Celsius (or Kelvin).
When heating copper, several factors need to be considered:

• The mass of the copper: In this case, it's \(168 \text{g}\).
• The specific heat of copper dictates how quickly it heats up or cools down compared to other substances.

By understanding copper's specific heat, you can effectively calculate and manage the energy required for heating or cooling processes, ensuring efficient energy use in industrial applications like electrical wiring or in culinary arts where copper is often used for its excellent thermal conductivity.