To determine the heat of the reaction for the dissociation of [tex]\( KNO_3 \)[/tex] in water in a coffee cup calorimeter, we need to follow several steps involving the specific heat capacities of the solution and the calorimeter, as well as the temperature change.
### Step-by-Step Solution:
1. Given Data:
- Mass of [tex]\( KNO_3 \)[/tex]: [tex]\( 7.5 \, \text{g} \)[/tex]
- Mass of water: [tex]\( 49.0 \, \text{g} \)[/tex]
- Initial temperature: [tex]\( 20.4^\circ \mathrm{C} \)[/tex]
- Final temperature: [tex]\( 9.7^\circ \mathrm{C} \)[/tex]
- Specific heat capacity of solution ([tex]\( C_{\text{soln}} \)[/tex]): [tex]\( 4.18 \, \text{J/g}^\circ \mathrm{C} \)[/tex]
- Heat capacity of the calorimeter ([tex]\( C_{\text{cal}} \)[/tex]): [tex]\( 6.5 \, \text{J/}^\circ \mathrm{C} \)[/tex]
2. Calculate the change in temperature ([tex]\( \Delta T \)[/tex]):
[tex]\[
\Delta T = \text{Final Temperature} - \text{Initial Temperature} = 9.7^\circ \mathrm{C} - 20.4^\circ \mathrm{C} = -10.7^\circ \mathrm{C}
\][/tex]
3. Calculate the heat absorbed by the solution ([tex]\( q_{\text{soln}} \)[/tex]):
The total mass of the solution is the sum of the mass of [tex]\( KNO_3 \)[/tex] and the mass of water:
[tex]\[
\text{Total mass of solution} = 7.5 \, \text{g} + 49.0 \, \text{g} = 56.5 \, \text{g}
\][/tex]
Using the specific heat capacity of the solution and the change in temperature:
[tex]\[
q_{\text{soln}} = \text{Total mass of solution} \times C_{\text{soln}} \times \Delta T
\][/tex]
[tex]\[
q_{\text{soln}} = 56.5 \, \text{g} \times 4.18 \, \text{J/g}^\circ \mathrm{C} \times (-10.7^\circ \mathrm{C}) = -2527.019 \, \text{J}
\][/tex]
4. Calculate the heat absorbed by the calorimeter ([tex]\( q_{\text{cal}} \)[/tex]):
[tex]\[
q_{\text{cal}} = C_{\text{cal}} \times \Delta T
\][/tex]
[tex]\[
q_{\text{cal}} = 6.5 \, \text{J/}^\circ \mathrm{C} \times (-10.7^\circ \mathrm{C}) = -69.55 \, \text{J}
\][/tex]
5. Calculate the total heat of reaction ([tex]\( q_{\text{rxn}} \)[/tex]):
The heat of the reaction is the negative sum of the heat absorbed by the solution and the calorimeter. Since the temperature decreased, the system released heat, so [tex]\( q_{\text{rxn}} \)[/tex] should be positive:
[tex]\[
q_{\text{rxn}} = -(q_{\text{soln}} + q_{\text{cal}})
\][/tex]
[tex]\[
q_{\text{rxn}} = -(-2527.019 \, \text{J} - 69.55 \, \text{J}) = 2596.569 \, \text{J}
\][/tex]
### Conclusion:
The heat of the reaction, [tex]\( q_{\text{rxn}} \)[/tex], is [tex]\( +2596.569 \, \text{J} \)[/tex].
