Answer :
To solve the problem of finding the specific heat capacity of quartz, we need to follow these steps:
1. Understand the Setup: We have a calorimeter with a quartz sample and water. The quartz starts at a higher temperature than the water. The final temperature of both, after heat exchange, is the same.
2. Given Values:
- Mass of quartz, [tex]\( m_{\text{quartz}} = 51.7 \, \text{g} \)[/tex]
- Initial temperature of quartz, [tex]\( T_{\text{initial, quartz}} = 91.7^\circ \text{C} \)[/tex]
- Mass of water, [tex]\( m_{\text{water}} = 100.0 \, \text{g} \)[/tex]
- Initial temperature of water, [tex]\( T_{\text{initial, water}} = 22.0^\circ \text{C} \)[/tex]
- Final temperature of both water and quartz, [tex]\( T_{\text{final}} = 27.6^\circ \text{C} \)[/tex]
- Specific heat capacity of water, [tex]\( c_{\text{water}} = 4.18 \, \text{J/(g}\cdot{^\circ}\text{C)} \)[/tex]
3. Calculate Heat Gained by Water:
[tex]\[
Q_{\text{water}} = m_{\text{water}} \times c_{\text{water}} \times (T_{\text{final}} - T_{\text{initial, water}})
\][/tex]
Plug in the values:
[tex]\[
Q_{\text{water}} = 100.0 \times 4.18 \times (27.6 - 22.0) = 2340.8 \, \text{J}
\][/tex]
4. Change in Temperature for Quartz:
[tex]\[
\Delta T_{\text{quartz}} = T_{\text{initial, quartz}} - T_{\text{final}} = 91.7 - 27.6 = 64.1^\circ \text{C}
\][/tex]
5. Calculate the Specific Heat Capacity of Quartz:
Since the heat lost by the quartz equals the heat gained by the water:
[tex]\[
Q_{\text{quartz}} = Q_{\text{water}}
\][/tex]
Using the formula:
[tex]\[
Q_{\text{quartz}} = m_{\text{quartz}} \times c_{\text{quartz}} \times \Delta T_{\text{quartz}}
\][/tex]
Solving for [tex]\( c_{\text{quartz}} \)[/tex]:
[tex]\[
c_{\text{quartz}} = \frac{Q_{\text{water}}}{m_{\text{quartz}} \times \Delta T_{\text{quartz}}}
\][/tex]
Substitute the known values:
[tex]\[
c_{\text{quartz}} = \frac{2340.8}{51.7 \times 64.1} = 0.706 \, \text{J/(g}\cdot{^\circ}\text{C)}
\][/tex]
6. Conclusion:
The specific heat capacity of quartz, based on this experiment, is approximately [tex]\( 0.706 \, \text{J/(g}\cdot{^\circ}\text{C)} \)[/tex], rounded to three significant figures.
1. Understand the Setup: We have a calorimeter with a quartz sample and water. The quartz starts at a higher temperature than the water. The final temperature of both, after heat exchange, is the same.
2. Given Values:
- Mass of quartz, [tex]\( m_{\text{quartz}} = 51.7 \, \text{g} \)[/tex]
- Initial temperature of quartz, [tex]\( T_{\text{initial, quartz}} = 91.7^\circ \text{C} \)[/tex]
- Mass of water, [tex]\( m_{\text{water}} = 100.0 \, \text{g} \)[/tex]
- Initial temperature of water, [tex]\( T_{\text{initial, water}} = 22.0^\circ \text{C} \)[/tex]
- Final temperature of both water and quartz, [tex]\( T_{\text{final}} = 27.6^\circ \text{C} \)[/tex]
- Specific heat capacity of water, [tex]\( c_{\text{water}} = 4.18 \, \text{J/(g}\cdot{^\circ}\text{C)} \)[/tex]
3. Calculate Heat Gained by Water:
[tex]\[
Q_{\text{water}} = m_{\text{water}} \times c_{\text{water}} \times (T_{\text{final}} - T_{\text{initial, water}})
\][/tex]
Plug in the values:
[tex]\[
Q_{\text{water}} = 100.0 \times 4.18 \times (27.6 - 22.0) = 2340.8 \, \text{J}
\][/tex]
4. Change in Temperature for Quartz:
[tex]\[
\Delta T_{\text{quartz}} = T_{\text{initial, quartz}} - T_{\text{final}} = 91.7 - 27.6 = 64.1^\circ \text{C}
\][/tex]
5. Calculate the Specific Heat Capacity of Quartz:
Since the heat lost by the quartz equals the heat gained by the water:
[tex]\[
Q_{\text{quartz}} = Q_{\text{water}}
\][/tex]
Using the formula:
[tex]\[
Q_{\text{quartz}} = m_{\text{quartz}} \times c_{\text{quartz}} \times \Delta T_{\text{quartz}}
\][/tex]
Solving for [tex]\( c_{\text{quartz}} \)[/tex]:
[tex]\[
c_{\text{quartz}} = \frac{Q_{\text{water}}}{m_{\text{quartz}} \times \Delta T_{\text{quartz}}}
\][/tex]
Substitute the known values:
[tex]\[
c_{\text{quartz}} = \frac{2340.8}{51.7 \times 64.1} = 0.706 \, \text{J/(g}\cdot{^\circ}\text{C)}
\][/tex]
6. Conclusion:
The specific heat capacity of quartz, based on this experiment, is approximately [tex]\( 0.706 \, \text{J/(g}\cdot{^\circ}\text{C)} \)[/tex], rounded to three significant figures.
