To determine the order in which the samples reach [tex]\( 90.0^{\circ} \text{C} \)[/tex], we need to calculate the amount of heat required for each sample to go from room temperature (assumed to be [tex]\( 25.0^{\circ} \text{C} \)[/tex]) to [tex]\( 90.0^{\circ} \text{C} \)[/tex]. The heat required ([tex]\( q \)[/tex]) can be found using the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the sample.
- [tex]\( C_p \)[/tex] is the specific heat capacity.
- [tex]\( \Delta T \)[/tex] is the change in temperature.
Let's calculate [tex]\( \Delta T \)[/tex] first:
[tex]\[ \Delta T = 90.0^{\circ} \text{C} - 25.0^{\circ} \text{C} = 65.0^{\circ} \text{C} \][/tex]
Given data:
- Mass of each sample, [tex]\( m = 10 \, \text{g} \)[/tex]
- Specific heat capacities ([tex]\( C_p \)[/tex]):
- Aluminum (Al): [tex]\( 0.897 \, \text{J/(g°C)} \)[/tex]
- Silver (Ag): [tex]\( 0.234 \, \text{J/(g°C)} \)[/tex]
- Iron (Fe): [tex]\( 0.450 \, \text{J/(g°C)} \)[/tex]
- Zinc (Zn): [tex]\( 0.387 \, \text{J/(g°C)} \)[/tex]
Now we will calculate the heat required for each sample:
1. For Aluminum (Al):
[tex]\[ q_{\text{Al}} = 10 \, \text{g} \times 0.897 \, \text{J/(g°C)} \times 65.0^{\circ} \text{C} = 583.05 \, \text{J} \][/tex]
2. For Silver (Ag):
[tex]\[ q_{\text{Ag}} = 10 \, \text{g} \times 0.234 \, \text{J/(g°C)} \times 65.0^{\circ} \text{C} = 152.10 \, \text{J} \][/tex]
3. For Iron (Fe):
[tex]\[ q_{\text{Fe}} = 10 \, \text{g} \times 0.450 \, \text{J/(g°C)} \times 65.0^{\circ} \text{C} = 292.50 \, \text{J} \][/tex]
4. For Zinc (Zn):
[tex]\[ q_{\text{Zn}} = 10 \, \text{g} \times 0.387 \, \text{J/(g°C)} \times 65.0^{\circ} \text{C} = 251.55 \, \text{J} \][/tex]
Now we have the following heat requirements:
- [tex]\( q_{\text{Al}} = 583.05 \, \text{J} \)[/tex]
- [tex]\( q_{\text{Ag}} = 152.10 \, \text{J} \)[/tex]
- [tex]\( q_{\text{Fe}} = 292.50 \, \text{J} \)[/tex]
- [tex]\( q_{\text{Zn}} = 251.55 \, \text{J} \)[/tex]
To determine the order in which the samples will reach [tex]\( 90.0^{\circ} \text{C} \)[/tex], we order the samples by increasing heat required:
1. Silver (Ag): [tex]\( 152.10 \, \text{J} \)[/tex]
2. Zinc (Zn): [tex]\( 251.55 \, \text{J} \)[/tex]
3. Iron (Fe): [tex]\( 292.50 \, \text{J} \)[/tex]
4. Aluminum (Al): [tex]\( 583.05 \, \text{J} \)[/tex]
Thus, the order in which these samples will reach [tex]\( 90.0^{\circ} \text{C} \)[/tex] from first to last is:
[tex]\[ \boxed{\text{Ag , Zn , Fe , Al}} \][/tex]
