Sure, let's solve the question step-by-step.
1. Convert the mass of the metal from kilograms to grams:
The given mass of the metal is 0.491 kg. To convert this to grams, we use the following conversion factor:
[tex]\[
1 \, \text{kg} = 1000 \, \text{g}
\][/tex]
So,
[tex]\[
0.491 \, \text{kg} \times 1000 \, \text{g/kg} = 491 \, \text{g}
\][/tex]
2. Convert the heat lost from kilojoules to joules:
The given heat lost is 4.08 kJ. To convert this to joules, we use the following conversion factor:
[tex]\[
1 \, \text{kJ} = 1000 \, \text{J}
\][/tex]
So,
[tex]\[
4.08 \, \text{kJ} \times 1000 \, \text{J/kJ} = 4080 \, \text{J}
\][/tex]
3. Given temperature change:
The temperature change is provided as [tex]\( 22.4^{\circ} \text{C} \)[/tex].
4. Calculate the specific heat capacity:
The specific heat capacity [tex]\( c \)[/tex] can be found using the formula:
[tex]\[
q = m \cdot c \cdot \Delta T
\][/tex]
where:
\begin{itemize}
\item [tex]\( q \)[/tex] is the heat lost, which we calculated to be 4080 J,
\item [tex]\( m \)[/tex] is the mass of the sample, which is 491 g,
\item [tex]\( \Delta T \)[/tex] is the change in temperature, which is [tex]\( 22.4^{\circ} \text{C} \)[/tex].
\end{itemize}
Rearranging the formula to solve for [tex]\( c \)[/tex]:
[tex]\[
c = \frac{q}{m \cdot \Delta T}
\][/tex]
Substituting in the known values:
[tex]\[
c = \frac{4080 \, \text{J}}{491 \, \text{g} \times 22.4^{\circ} \text{C}}
\][/tex]
Doing the division:
[tex]\[
c \approx 0.3709630491707885 \, \frac{J}{g^{\circ}C}
\][/tex]
Thus, the specific heat capacity of the metal is approximately [tex]\( 0.371 \, \frac{J}{g^{\circ}C} \)[/tex] (rounded to three decimal places).
