Answer :
To determine the order in which the elements will reach [tex]\(90.0^\circ \text{C}\)[/tex], we need to calculate the amount of heat required for each sample to reach this temperature from room temperature ([tex]\(25.0^\circ \text{C}\)[/tex]).
Given:
- Mass of each sample: [tex]\(10 \text{g}\)[/tex]
- Initial temperature: [tex]\(25.0^\circ \text{C}\)[/tex]
- Final temperature: [tex]\(90.0^\circ \text{C}\)[/tex]
The specific heats ([tex]\(C_p\)[/tex]) for the elements are:
- Aluminum (Al): [tex]\(0.897 \ \frac{J}{(g \cdot {}^\circ C)}\)[/tex]
- Silver (Ag): [tex]\(0.234 \ \frac{J}{(g \cdot {}^\circ C)}\)[/tex]
- Iron (Fe): [tex]\(0.450 \ \frac{J}{(g \cdot {}^\circ C)}\)[/tex]
- Zinc (Zn): [tex]\(0.387 \ \frac{J}{(g \cdot {}^\circ C)}\)[/tex]
The formula to calculate the heat required ([tex]\(q\)[/tex]) for each sample is:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where
- [tex]\(m\)[/tex] is the mass,
- [tex]\(C_p\)[/tex] is the specific heat,
- [tex]\(\Delta T\)[/tex] is the change in temperature ([tex]\(90.0^\circ \text{C} - 25.0^\circ \text{C} = 65.0^\circ \text{C}\)[/tex]).
Let's compute the heat required for each element:
1. For Aluminum:
[tex]\[ q_{\text{Al}} = 10 \text{g} \times 0.897 \ \frac{J}{(g \cdot {}^\circ C)} \times 65.0^\circ \text{C} = 583.050 \ \text{J} \][/tex]
2. For Silver:
[tex]\[ q_{\text{Ag}} = 10 \text{g} \times 0.234 \ \frac{J}{(g \cdot {}^\circ C)} \times 65.0^\circ \text{C} = 152.100 \ \text{J} \][/tex]
3. For Iron:
[tex]\[ q_{\text{Fe}} = 10 \text{g} \times 0.450 \ \frac{J}{(g \cdot {}^\circ C)} \times 65.0^\circ \text{C} = 292.500 \ \text{J} \][/tex]
4. For Zinc:
[tex]\[ q_{\text{Zn}} = 10 \text{g} \times 0.387 \ \frac{J}{(g \cdot {}^\circ C)} \times 65.0^\circ \text{C} = 251.550 \ \text{J} \][/tex]
Now, sort the elements by the amount of heat required in ascending order:
1. [tex]\(152.100 \ \text{J}\)[/tex] (Ag)
2. [tex]\(251.550 \ \text{J}\)[/tex] (Zn)
3. [tex]\(292.500 \ \text{J}\)[/tex] (Fe)
4. [tex]\(583.050 \ \text{J}\)[/tex] (Al)
So, the order in which the elements will reach [tex]\(90.0^\circ \text{C}\)[/tex] from first to last is:
[tex]\[ \text{Ag}, \text{Zn}, \text{Fe}, \text{Al} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\text{Ag , Zn , Fe , Al}} \][/tex]
Given:
- Mass of each sample: [tex]\(10 \text{g}\)[/tex]
- Initial temperature: [tex]\(25.0^\circ \text{C}\)[/tex]
- Final temperature: [tex]\(90.0^\circ \text{C}\)[/tex]
The specific heats ([tex]\(C_p\)[/tex]) for the elements are:
- Aluminum (Al): [tex]\(0.897 \ \frac{J}{(g \cdot {}^\circ C)}\)[/tex]
- Silver (Ag): [tex]\(0.234 \ \frac{J}{(g \cdot {}^\circ C)}\)[/tex]
- Iron (Fe): [tex]\(0.450 \ \frac{J}{(g \cdot {}^\circ C)}\)[/tex]
- Zinc (Zn): [tex]\(0.387 \ \frac{J}{(g \cdot {}^\circ C)}\)[/tex]
The formula to calculate the heat required ([tex]\(q\)[/tex]) for each sample is:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where
- [tex]\(m\)[/tex] is the mass,
- [tex]\(C_p\)[/tex] is the specific heat,
- [tex]\(\Delta T\)[/tex] is the change in temperature ([tex]\(90.0^\circ \text{C} - 25.0^\circ \text{C} = 65.0^\circ \text{C}\)[/tex]).
Let's compute the heat required for each element:
1. For Aluminum:
[tex]\[ q_{\text{Al}} = 10 \text{g} \times 0.897 \ \frac{J}{(g \cdot {}^\circ C)} \times 65.0^\circ \text{C} = 583.050 \ \text{J} \][/tex]
2. For Silver:
[tex]\[ q_{\text{Ag}} = 10 \text{g} \times 0.234 \ \frac{J}{(g \cdot {}^\circ C)} \times 65.0^\circ \text{C} = 152.100 \ \text{J} \][/tex]
3. For Iron:
[tex]\[ q_{\text{Fe}} = 10 \text{g} \times 0.450 \ \frac{J}{(g \cdot {}^\circ C)} \times 65.0^\circ \text{C} = 292.500 \ \text{J} \][/tex]
4. For Zinc:
[tex]\[ q_{\text{Zn}} = 10 \text{g} \times 0.387 \ \frac{J}{(g \cdot {}^\circ C)} \times 65.0^\circ \text{C} = 251.550 \ \text{J} \][/tex]
Now, sort the elements by the amount of heat required in ascending order:
1. [tex]\(152.100 \ \text{J}\)[/tex] (Ag)
2. [tex]\(251.550 \ \text{J}\)[/tex] (Zn)
3. [tex]\(292.500 \ \text{J}\)[/tex] (Fe)
4. [tex]\(583.050 \ \text{J}\)[/tex] (Al)
So, the order in which the elements will reach [tex]\(90.0^\circ \text{C}\)[/tex] from first to last is:
[tex]\[ \text{Ag}, \text{Zn}, \text{Fe}, \text{Al} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\text{Ag , Zn , Fe , Al}} \][/tex]
