Question

How many joules of energy are required to change \(50.0 \mathrm{~g} \mathrm{Cu}\) from \(25.0^{\circ} \mathrm{C}\) to a liquid at its melting point, \(1083^{\circ} \mathrm{C}\) ? Specific heat of \(\mathrm{Cu}=0.385 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}\) Heat of fusion for \(\mathrm{Cu}=134 \mathrm{~J} / \mathrm{g}\)

Short answer

27,045.5 J

Step-by-step solution

Determine the Heat Required to Raise the Temperature

Use the formula for heat: \( q = mc\Delta T \). Here, \( m = 50.0 \, \mathrm{g} \), \( c = 0.385 \, \mathrm{J/g^{\circ}C} \), and \( \Delta T = 1083^{\circ} \mathrm{C} - 25^{\circ} \mathrm{C} = 1058^{\circ} \mathrm{C} \). Substitute these values into the formula: \[ q = 50.0 \, \mathrm{g} \times 0.385 \, \mathrm{J/g^{\circ}C} \times 1058^{\circ} \mathrm{C} = 20,345.5 \, \mathrm{J} \]

Determine the Heat Required to Melt the Copper

Use the formula for the heat of fusion: \( q = mL \). Here, \( m = 50.0 \, \mathrm{g} \) and \( L = 134 \, \mathrm{J/g} \). Substitute these values into the formula: \[ q = 50.0 \, \mathrm{g} \times 134 \, \mathrm{J/g} = 6700 \, \mathrm{J} \]

Calculate the Total Energy Required

Add the heat required to raise the temperature to the heat required to melt the copper: \[ 20,345.5 \, \mathrm{J} + 6,700 \, \mathrm{J} = 27,045.5 \, \mathrm{J} \]

Key concepts

specific heat capacity

Specific heat capacity is a crucial concept in heat energy calculations. It is the amount of heat energy required to raise the temperature of one gram of a substance by one degree Celsius. In the given exercise, we have used the specific heat capacity of copper, which is 0.385 J/g°C. This means that for every gram of copper, you need 0.385 joules of energy to raise its temperature by 1°C.
To calculate the energy needed to increase the temperature of a substance, you can use the formula:
\[ q = mc\Delta T \]
where:

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

In our case, we had 50 grams of copper, and the temperature changed from 25°C to 1083°C, a change of 1058°C. Substituting these values into the formula, we get:
\[ q = 50 \, \mathrm{g} \times 0.385 \, \mathrm{J/g^{\circ}C} \times 1058^{\circ} \mathrm{C} = 20,345.5 \, \mathrm{J} \] This shows how specific heat capacity helps in understanding the energy required to change the temperature of a substance.

heat of fusion

The heat of fusion is the amount of energy required to change a substance from a solid to a liquid at its melting point without changing its temperature. For copper, the heat of fusion is 134 J/g. This means that 134 joules of energy are needed to melt one gram of copper.
In our exercise, we needed to melt 50 grams of copper. To find the energy required, we use the formula:
\[ q = mL \]
where:

• \( q \) is the heat energy
• \( m \) is the mass in grams
• \( L \) is the heat of fusion

By substituting the known values, we get:
\[ q = 50.0 \, \mathrm{g} \times 134 \, \mathrm{J/g} = 6,700 \, \mathrm{J} \] This amount of energy is specifically for melting the copper, distinct from the energy required to heat it to the melting point.

temperature change

Understanding temperature change is fundamental when performing heat energy calculations. In any process where heat is added or removed, the temperature change of the substance is a key factor.
The temperature change \( \Delta T \) is calculated as:
\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \]
In our exercise, the initial temperature \( T_{\text{initial}} \) was 25°C, and the final temperature \( T_{\text{final}} \) was 1083°C. Thus, the temperature change \( \Delta T \) was:
\[ \Delta T = 1083^{\circ}\mathrm{C} - 25^{\circ}\mathrm{C} = 1058^{\circ}\mathrm{C} \]
This significant temperature change required a substantial amount of energy to heat the copper from room temperature to its melting point. By understanding the temperature change, along with specific heat capacity, we can determine how much energy is needed to achieve the desired state. This step is crucial in the overall process of calculating heat energy requirements.