Sure, let's break down the solution step-by-step.
1. Identify the given values:
- Mass of the metal, [tex]\( m \)[/tex]: 0.782 kg
- Heat loss, [tex]\( Q \)[/tex]: 4.04 kJ
- Temperature change, [tex]\( \Delta T \)[/tex]: 43.2°C
2. Convert the units:
- Convert the heat loss from kilojoules to joules:
[tex]\( Q = 4.04 \text{ kJ} \times 1000 \text{ J/kJ} = 4040 \text{ J} \)[/tex]
- Convert the mass from kilograms to grams:
[tex]\( m = 0.782 \text{ kg} \times 1000 \text{ g/kg} = 782 \text{ g} \)[/tex]
3. Use the formula for specific heat capacity:
The formula to calculate specific heat capacity ([tex]\( c \)[/tex]) is:
[tex]\[
Q = mc\Delta T
\][/tex]
Rearrange the formula to solve for [tex]\( c \)[/tex]:
[tex]\[
c = \frac{Q}{m \Delta T}
\][/tex]
4. Substitute the values into the equation:
[tex]\[
c = \frac{4040 \text{ J}}{782 \text{ g} \times 43.2^\circ \text{C}}
\][/tex]
5. Compute the specific heat capacity:
[tex]\[
c = \frac{4040}{33782.4} \approx 0.1196 \, \frac{\text{J}}{\text{g}^\circ \text{C}}
\][/tex]
Hence, the specific heat capacity of the metal is approximately [tex]\( 0.1196 \, \frac{\text{J}}{\text{g}^\circ \text{C}} \)[/tex].
