To determine the mass of the tin sample, we start by clearly understanding and applying the problem's given details alongside the formula [tex]\( q = m C_p \Delta T \)[/tex].
Given values:
- Specific heat capacity ([tex]\( C_p \)[/tex]) of tin: [tex]\( 0.227 \, \text{J} / \text{g} \cdot {}^{\circ}C \)[/tex]
- Energy released ([tex]\( q \)[/tex]): [tex]\( 543 \, \text{J} \)[/tex]
- Initial temperature ([tex]\( T_{\text{initial}} \)[/tex]): [tex]\( 15.0^{\circ}C \)[/tex]
- Final temperature ([tex]\( T_{\text{final}} \)[/tex]): [tex]\( -10.0^{\circ}C \)[/tex]
Step-by-step solution:
1. Calculate the change in temperature ([tex]\( \Delta T \)[/tex]):
[tex]\[
\Delta T = T_{\text{final}} - T_{\text{initial}}
\][/tex]
[tex]\[
\Delta T = -10.0^{\circ}C - 15.0^{\circ}C
\][/tex]
[tex]\[
\Delta T = -25.0^{\circ}C
\][/tex]
The change in temperature is [tex]\(-25.0^{\circ}C\)[/tex].
2. Rearrange the formula [tex]\( q = m C_p \Delta T \)[/tex] to solve for mass ([tex]\( m \)[/tex]):
[tex]\[
m = \frac{q}{C_p \Delta T}
\][/tex]
3. Substitute the known values into the formula:
[tex]\[
m = \frac{543 \, \text{J}}{0.227 \, \text{J} / \text{g} \cdot {}^{\circ}C \times (-25.0^{\circ}C)}
\][/tex]
4. Perform the calculation:
[tex]\[
m = \frac{543}{0.227 \times (-25.0)}
\][/tex]
[tex]\[
m = \frac{543}{-5.675}
\][/tex]
[tex]\[
m \approx -95.68281938325991 \, \text{g}
\][/tex]
5. Round the mass to three significant figures:
[tex]\[
m \approx -95.683 \, \text{g}
\][/tex]
However, since mass cannot be negative, it indicates we consider the absolute value when dealing with magnitude:
[tex]\[
m = 95.683 \, \text{g}
\][/tex]
Therefore, the mass of the tin sample is [tex]\(\boxed{95.683} \, \text{g}\)[/tex].
