To determine the initial temperature of the gold sample, we can use the formula relating heat transfer to mass, specific heat, and temperature change:
[tex]\[ Q = m \cdot C \cdot \Delta T \][/tex]
Where:
- [tex]\( Q \)[/tex] is the heat released ([tex]\( 53.25 \)[/tex] J),
- [tex]\( m \)[/tex] is the mass of the sample ([tex]\( 1.55 \)[/tex] g),
- [tex]\( C \)[/tex] is the specific heat capacity ([tex]\( 0.129 \)[/tex] J/(g°C)),
- [tex]\( \Delta T \)[/tex] is the change in temperature (°C).
First, we need to determine the change in temperature [tex]\( \Delta T \)[/tex].
[tex]\[ \Delta T = \frac{Q}{m \cdot C} \][/tex]
Plugging in the given values:
[tex]\[ \Delta T = \frac{53.25}{1.55 \cdot 0.129} \][/tex]
[tex]\[ \Delta T \approx 266.32 \, \text{°C} \][/tex]
With the change in temperature calculated, we then find the initial temperature [tex]\( T_{\text{initial}} \)[/tex] using the final temperature [tex]\( T_{\text{final}} \)[/tex] and the relationship:
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
Rearranging to solve for [tex]\( T_{\text{initial}} \)[/tex]:
[tex]\[ T_{\text{initial}} = T_{\text{final}} - \Delta T \][/tex]
Given:
[tex]\[ T_{\text{final}} = 73.3 \, \text{°C} \][/tex]
Thus:
[tex]\[ T_{\text{initial}} = 73.3 - 266.32 \][/tex]
[tex]\[ T_{\text{initial}} \approx -193.02 \, \text{°C} \][/tex]
Therefore, the initial temperature of the gold sample was approximately -193.02 °C.
