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To solve this problem, we need to calculate the amount of heat (denoted as [tex]\(q\)[/tex]) given off in the reaction. We'll use the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where
- [tex]\( m \)[/tex] is the mass of the solution,
- [tex]\( C_p \)[/tex] is the specific heat capacity of the solution,
- [tex]\(\Delta T \)[/tex] is the change in temperature.
Given data:
- Volume of [tex]\( H_2SO_4 = 40.0 \, \text{mL} \)[/tex]
- Concentration of [tex]\( H_2SO_4 = 1.00 \, \text{M} \)[/tex]
- Volume of [tex]\( NaOH = 80.0 \, \text{mL} \)[/tex]
- Concentration of [tex]\( NaOH = 1.00 \, \text{M} \)[/tex]
- Initial temperature [tex]\( T_{\text{initial}} = 20.00 \, ^\circ \text{C} \)[/tex]
- Final temperature [tex]\( T_{\text{final}} = 29.20 \, ^\circ \text{C} \)[/tex]
- Mass of the solution [tex]\( m = 120.0 \, \text{g} \)[/tex]
- Specific heat capacity [tex]\( C_p = 4.184 \, \text{J/g} \cdot ^\circ \text{C} \)[/tex]
Step-by-step solution:
1. Calculate the temperature change ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 29.20\, ^\circ \text{C} - 20.00\, ^\circ \text{C} = 9.20\, ^\circ \text{C} \][/tex]
2. Calculate the heat (q) using the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
[tex]\[ q = 120.0\, \text{g} \cdot 4.184\, \text{J/g} \cdot ^\circ \text{C} \cdot 9.20\, ^\circ \text{C} \][/tex]
[tex]\[ q = 120.0 \cdot 4.184 \cdot 9.20 \][/tex]
[tex]\[ q = 4619.136\, \text{J} \][/tex]
3. Convert the heat from Joules to kilojoules:
[tex]\[ q_{\text{kJ}} = \frac{q}{1000} \][/tex]
[tex]\[ q_{\text{kJ}} = \frac{4619.136}{1000} \][/tex]
[tex]\[ q_{\text{kJ}} = 4.619136\, \text{kJ} \][/tex]
Therefore, the amount of heat given off in the reaction is approximately [tex]\( 4.62 \, \text{kJ} \)[/tex].
Among the given options, the correct answer is:
[tex]\[ 4.62 \, \text{kJ} \][/tex]
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where
- [tex]\( m \)[/tex] is the mass of the solution,
- [tex]\( C_p \)[/tex] is the specific heat capacity of the solution,
- [tex]\(\Delta T \)[/tex] is the change in temperature.
Given data:
- Volume of [tex]\( H_2SO_4 = 40.0 \, \text{mL} \)[/tex]
- Concentration of [tex]\( H_2SO_4 = 1.00 \, \text{M} \)[/tex]
- Volume of [tex]\( NaOH = 80.0 \, \text{mL} \)[/tex]
- Concentration of [tex]\( NaOH = 1.00 \, \text{M} \)[/tex]
- Initial temperature [tex]\( T_{\text{initial}} = 20.00 \, ^\circ \text{C} \)[/tex]
- Final temperature [tex]\( T_{\text{final}} = 29.20 \, ^\circ \text{C} \)[/tex]
- Mass of the solution [tex]\( m = 120.0 \, \text{g} \)[/tex]
- Specific heat capacity [tex]\( C_p = 4.184 \, \text{J/g} \cdot ^\circ \text{C} \)[/tex]
Step-by-step solution:
1. Calculate the temperature change ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 29.20\, ^\circ \text{C} - 20.00\, ^\circ \text{C} = 9.20\, ^\circ \text{C} \][/tex]
2. Calculate the heat (q) using the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
[tex]\[ q = 120.0\, \text{g} \cdot 4.184\, \text{J/g} \cdot ^\circ \text{C} \cdot 9.20\, ^\circ \text{C} \][/tex]
[tex]\[ q = 120.0 \cdot 4.184 \cdot 9.20 \][/tex]
[tex]\[ q = 4619.136\, \text{J} \][/tex]
3. Convert the heat from Joules to kilojoules:
[tex]\[ q_{\text{kJ}} = \frac{q}{1000} \][/tex]
[tex]\[ q_{\text{kJ}} = \frac{4619.136}{1000} \][/tex]
[tex]\[ q_{\text{kJ}} = 4.619136\, \text{kJ} \][/tex]
Therefore, the amount of heat given off in the reaction is approximately [tex]\( 4.62 \, \text{kJ} \)[/tex].
Among the given options, the correct answer is:
[tex]\[ 4.62 \, \text{kJ} \][/tex]
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