To determine the heat of the reaction when [tex]\( 7.5 \, \text{g} \)[/tex] of [tex]\( \text{KNO}_3 \)[/tex] dissociates in [tex]\( 49.0 \, \text{g} \)[/tex] of water in a coffee cup calorimeter, where the temperature changes from [tex]\( 20.4\,^\circ \text{C} \)[/tex] to [tex]\( 9.7\,^\circ \text{C} \)[/tex], follow the steps below:
### Step-by-Step Solution
1. Identify the temperature change: Calculate the change in temperature ([tex]\( \Delta T \)[/tex]):
[tex]\[
\Delta T = \text{final temperature} - \text{initial temperature}
\][/tex]
[tex]\[
\Delta T = 9.7\, ^\circ \text{C} - 20.4\, ^\circ \text{C} = -10.7\, ^\circ \text{C}
\][/tex]
2. Calculate the heat absorbed by the solution: Use the given specific heat capacity of the solution ([tex]\( C_{\text{soln}} = 4.18 \, \text{J/g} ^\circ \text{C} \)[/tex]) and the mass of the water ([tex]\( 49.0 \, \text{g} \)[/tex]):
[tex]\[
q_{\text{solution}} = \text{mass of water} \times C_{\text{soln}} \times \Delta T
\][/tex]
Substitute the values:
[tex]\[
q_{\text{solution}} = 49.0 \, \text{g} \times 4.18 \, \text{J/g} \, ^\circ \text{C} \times (-10.7 \, ^\circ \text{C})
\][/tex]
[tex]\[
q_{\text{solution}} = -2191.574 \, \text{J}
\][/tex]
3. Calculate the heat absorbed by the calorimeter: Use the calorimeter heat capacity ([tex]\( C_{\text{cal}} = 6.5 \, \text{J/} ^\circ \text{C} \)[/tex]):
[tex]\[
q_{\text{calorimeter}} = C_{\text{cal}} \times \Delta T
\][/tex]
Substitute the values:
[tex]\[
q_{\text{calorimeter}} = 6.5 \, \text{J/} ^\circ \text{C} \times (-10.7 \, ^\circ \text{C})
\][/tex]
[tex]\[
q_{\text{calorimeter}} = -69.55 \, \text{J}
\][/tex]
4. Calculate the total heat of the reaction: Sum the heat absorbed by the solution and the calorimeter:
[tex]\[
q_{\text{reaction}} = q_{\text{solution}} + q_{\text{calorimeter}}
\][/tex]
[tex]\[
q_{\text{reaction}} = -2191.574 \, \text{J} + (-69.55 \, \text{J})
\][/tex]
[tex]\[
q_{\text{reaction}} = -2261.124 \, \text{J}
\][/tex]
### Result
The heat of the reaction is [tex]\(-2261.124 \, \text{J}\)[/tex], indicating an endothermic process since the heat is absorbed (indicated by the negative sign).
