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Answer:
[tex]\( 30 \, \text{J/C} \)[/tex]
Explanation:
The heat lost by the hot water is equal to the heat gained by the cold water plus the heat absorbed by the calorimeter. The calorimeter constant represents the heat capacity of the calorimeter.
Solving:
Formula for hot water:
[tex]q_{\text{hot}} = m_{\text{hot}} \cdot c_{\text{water}} \cdot (T_{\text{initial, hot}} - T_{\text{final}})[/tex]
[tex]\hrulefill[/tex]
Given:
[tex]\item \( m_{\text{hot}} = 50 \, \text{ml} = 50 \, \text{g} \) (The density of water is \( 1 \, \text{g/ml}) \ \item \( c_{\text{water}} = 4.18 \, \text{J/g*C} \) (specific heat capacity of water) \item \( T_{\text{initial, hot}} = 85 \, \text{C} \) \item \( T_{\text{final}} = 53 \, \text{C} \)[/tex]
[tex]\hrulefill[/tex]
[tex]q_{\text{hot}} = 50 \, \text{g} \cdot 4.18 \, \text{J/gC} \cdot (85 \, \text{C} - 53 \, \text{C})\\ \\ q_{\text{hot}} = 50 \cdot 4.18 \cdot 32\\\\ \boxed{q_{\text{hot}} = 6692 \, \text{J}\\}[/tex]
[tex]\hrulefill[/tex]
Formula for cold water:
[tex]\[ q_{\text{cold}} = m_{\text{cold}} \cdot c_{\text{water}} \cdot (T_{\text{final}} - T_{\text{initial, cold}}) \][/tex]
[tex]\hrulefill[/tex]
Given:
[tex]\item \( m_{\text{cold}} = 50 \, \text{ml} = 50 \, \text{g} \) \item \( T_{\text{initial, cold}} = 25 \, \text{C} \)[/tex]
[tex]\hrulefill[/tex]
[tex]q_{\text{cold}} = 50 \, \text{g} \cdot 4.18 \, \text{J/gC} \cdot (53 \, \text{C} - 25 \, \text{C})\\ q_{\text{cold}} = 50 \cdot 4.18 \cdot 28\\ \boxed{q_{\text{cold}} = 5852 \, \text{J}}[/tex]
[tex]\hrulefill[/tex]
Calorimeter constant:
[tex]\[ C_{\text{cal}} = \frac{q_{\text{cal}}}{\Delta T} \] \\\\ Where \( \Delta T \) is the change in temperature of the water, which is the same for both hot and cold water: \[[/tex]
[tex]\hrulefill[/tex]
[tex]\Delta T = T_{\text{final}} - T_{\text{initial, cold}} = 53 \, \text{C} - 25 \, \text{C} = 28 \, \text{C} \\ C_{\text{cal}} = \frac{840 \, \text{J}}{28 \, \text{C}}\\ \boxed{ C_{\text{cal}} = 30 \, \text{J/C}}[/tex]
[tex]\hrulefill[/tex]
[tex]\text{Therefore, the calorimeter constant is \( 30 \, \text{J/C} \).}[/tex]