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Sagot :
To find the masses of both substances (Pyrex glass and sand), we'll follow a systematic approach making use of the heat absorbed, the specific heat capacities, and the temperature changes.
### Part A: Mass of Pyrex Glass
1. Given Data:
- Heat absorbed ([tex]\(q\)[/tex]) = [tex]\(1.98 \times 10^3 \text{ J}\)[/tex]
- Initial temperature ([tex]\(T_i\)[/tex]) = [tex]\(25.0^{\circ} \text{C}\)[/tex]
- Final temperature for glass ([tex]\(T_f\)[/tex]) = [tex]\(55.3^{\circ} \text{C}\)[/tex]
- Specific heat capacity of Pyrex glass ([tex]\(c\)[/tex]) = [tex]\(0.75 \text{ J/g} \cdot ^{\circ} \text{C}\)[/tex]
2. Temperature Change ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T_{\text{glass}} = T_f - T_i = 55.3^{\circ} \text{C} - 25.0^{\circ} \text{C} = 30.3^{\circ} \text{C} \][/tex]
3. Using the Heat Formula:
The formula to relate heat absorbed to mass, specific heat capacity, and temperature change is:
[tex]\[ q = m \cdot c \cdot \Delta T \][/tex]
Rearranging the formula to solve for mass ([tex]\(m\)[/tex]):
[tex]\[ m = \frac{q}{c \cdot \Delta T} \][/tex]
4. Calculating the Mass:
[tex]\[ m_{\text{glass}} = \frac{1.98 \times 10^3 \text{ J}}{0.75 \text{ J/g} \cdot ^{\circ} \text{C} \times 30.3^{\circ} \text{C}} = 87.13 \text{ g} \][/tex]
5. Final Answer:
The mass of the Pyrex glass is [tex]\(87.13 \text{ g}\)[/tex].
### Part B: Mass of Sand
1. Given Data:
- Heat absorbed ([tex]\(q\)[/tex]) = [tex]\(1.98 \times 10^3 \text{ J}\)[/tex]
- Initial temperature ([tex]\(T_i\)[/tex]) = [tex]\(25.0^{\circ} \text{C}\)[/tex]
- Final temperature for sand ([tex]\(T_f\)[/tex]) = [tex]\(62.0^{\circ} \text{C}\)[/tex]
- Specific heat capacity of sand ([tex]\(c\)[/tex]) = [tex]\(0.84 \text{ J/g} \cdot ^{\circ} \text{C}\)[/tex]
2. Temperature Change ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T_{\text{sand}} = T_f - T_i = 62.0^{\circ} \text{C} - 25.0^{\circ} \text{C} = 37.0^{\circ} \text{C} \][/tex]
3. Using the Heat Formula:
Rearranging the heat formula to solve for mass ([tex]\(m\)[/tex]):
[tex]\[ m = \frac{q}{c \cdot \Delta T} \][/tex]
4. Calculating the Mass:
[tex]\[ m_{\text{sand}} = \frac{1.98 \times 10^3 \text{ J}}{0.84 \text{ J/g} \cdot ^{\circ} \text{C} \times 37.0^{\circ} \text{C}} = 63.71 \text{ g} \][/tex]
5. Final Answer:
The mass of the sand is [tex]\(63.71 \text{ g}\)[/tex].
This process allows us to determine the mass of both Pyrex glass and sand given the heat absorbed and the specific heat capacities.
### Part A: Mass of Pyrex Glass
1. Given Data:
- Heat absorbed ([tex]\(q\)[/tex]) = [tex]\(1.98 \times 10^3 \text{ J}\)[/tex]
- Initial temperature ([tex]\(T_i\)[/tex]) = [tex]\(25.0^{\circ} \text{C}\)[/tex]
- Final temperature for glass ([tex]\(T_f\)[/tex]) = [tex]\(55.3^{\circ} \text{C}\)[/tex]
- Specific heat capacity of Pyrex glass ([tex]\(c\)[/tex]) = [tex]\(0.75 \text{ J/g} \cdot ^{\circ} \text{C}\)[/tex]
2. Temperature Change ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T_{\text{glass}} = T_f - T_i = 55.3^{\circ} \text{C} - 25.0^{\circ} \text{C} = 30.3^{\circ} \text{C} \][/tex]
3. Using the Heat Formula:
The formula to relate heat absorbed to mass, specific heat capacity, and temperature change is:
[tex]\[ q = m \cdot c \cdot \Delta T \][/tex]
Rearranging the formula to solve for mass ([tex]\(m\)[/tex]):
[tex]\[ m = \frac{q}{c \cdot \Delta T} \][/tex]
4. Calculating the Mass:
[tex]\[ m_{\text{glass}} = \frac{1.98 \times 10^3 \text{ J}}{0.75 \text{ J/g} \cdot ^{\circ} \text{C} \times 30.3^{\circ} \text{C}} = 87.13 \text{ g} \][/tex]
5. Final Answer:
The mass of the Pyrex glass is [tex]\(87.13 \text{ g}\)[/tex].
### Part B: Mass of Sand
1. Given Data:
- Heat absorbed ([tex]\(q\)[/tex]) = [tex]\(1.98 \times 10^3 \text{ J}\)[/tex]
- Initial temperature ([tex]\(T_i\)[/tex]) = [tex]\(25.0^{\circ} \text{C}\)[/tex]
- Final temperature for sand ([tex]\(T_f\)[/tex]) = [tex]\(62.0^{\circ} \text{C}\)[/tex]
- Specific heat capacity of sand ([tex]\(c\)[/tex]) = [tex]\(0.84 \text{ J/g} \cdot ^{\circ} \text{C}\)[/tex]
2. Temperature Change ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T_{\text{sand}} = T_f - T_i = 62.0^{\circ} \text{C} - 25.0^{\circ} \text{C} = 37.0^{\circ} \text{C} \][/tex]
3. Using the Heat Formula:
Rearranging the heat formula to solve for mass ([tex]\(m\)[/tex]):
[tex]\[ m = \frac{q}{c \cdot \Delta T} \][/tex]
4. Calculating the Mass:
[tex]\[ m_{\text{sand}} = \frac{1.98 \times 10^3 \text{ J}}{0.84 \text{ J/g} \cdot ^{\circ} \text{C} \times 37.0^{\circ} \text{C}} = 63.71 \text{ g} \][/tex]
5. Final Answer:
The mass of the sand is [tex]\(63.71 \text{ g}\)[/tex].
This process allows us to determine the mass of both Pyrex glass and sand given the heat absorbed and the specific heat capacities.
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