Certainly! Let's solve the problem step by step.
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Given Data:
- Specific heat capacity of tin, [tex]\( Cp = 0.227 \, \text{J/g} \cdot ^\circ\text{C} \)[/tex]
- Initial temperature, [tex]\( T_{\text{initial}} = 15.0 \, ^\circ\text{C} \)[/tex]
- Final temperature, [tex]\( T_{\text{final}} = -10.0 \, ^\circ\text{C} \)[/tex]
- Energy released, [tex]\( q = 543 \, \text{J} \)[/tex]
Formula to Use:
[tex]\[ q = m \cdot Cp \cdot \Delta T \][/tex]
where:
- [tex]\( q \)[/tex] is the energy released or absorbed (in Joules)
- [tex]\( m \)[/tex] is the mass (in grams)
- [tex]\( Cp \)[/tex] is the specific heat capacity (in J/g·°C)
- [tex]\( \Delta T \)[/tex] is the change in temperature (in °C)
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Step-by-Step Solution:
1. Calculate the change in temperature [tex]\( \Delta T \)[/tex]:
[tex]\[
\Delta T = T_{\text{final}} - T_{\text{initial}} = -10.0 \, ^\circ\text{C} - 15.0 \, ^\circ\text{C} = -25.0 \, ^\circ\text{C}
\][/tex]
The change in temperature is [tex]\( -25.0 \, ^\circ\text{C} \)[/tex].
2. Rearrange the formula to solve for mass [tex]\( m \)[/tex]:
[tex]\[
m = \frac{q}{Cp \cdot \Delta T}
\][/tex]
3. Plug in the known values:
- [tex]\( q = 543 \, \text{J} \)[/tex]
- [tex]\( Cp = 0.227 \, \text{J/g} \cdot ^\circ\text{C} \)[/tex]
- [tex]\( \Delta T = -25.0 \, ^\circ\text{C} \)[/tex]
Note that since [tex]\(\Delta T\)[/tex] is negative, and we are interested in an absolute mass value, we'll take the absolute value of [tex]\(\Delta T\)[/tex] in the calculation.
[tex]\[
m = \frac{543 \, \text{J}}{0.227 \, \text{J/g} \cdot ^\circ\text{C} \times |-25.0 \, ^\circ\text{C}|} = \frac{543}{0.227 \times 25.0}
\][/tex]
4. Calculate the mass:
[tex]\[
m = \frac{543}{5.675} \approx 95.683 \, \text{g}
\][/tex]
5. Round the mass to three significant figures:
[tex]\[
m \approx 95.683 \, \text{g} \rightarrow 95.7 \, \text{g}
\][/tex]
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Final Answer:
The mass of the tin sample is [tex]\( 95.7 \)[/tex] grams (rounded to three significant figures).
