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Loren has samples of the elements listed below at room temperature. He exposes the samples to the same heat source until each sample reaches a temperature of [tex][tex]$90.0^{\circ} C$[/tex][/tex].

[tex]\[
\begin{array}{l}
10 \text{ g of } Al (s) \left( C_{p} = 0.897 \text{ J} / \left( \text{g} \cdot {}^{\circ} \text{C} \right) \right) \\
10 \text{ g of } Ag (s) \left( C_{p} = 0.234 \text{ J} / \left( \text{g} \cdot {}^{\circ} \text{C} \right) \right) \\
10 \text{ g of } Fe (s) \left( C_{p} = 0.450 \text{ J} / \left( \text{g} \cdot {}^{\circ} \text{C} \right) \right) \\
10 \text{ g of } Zn (s) \left( C_{p} = 0.387 \text{ J} / \left( \text{g} \cdot {}^{\circ} \text{C} \right) \right)
\end{array}
\][/tex]

From first to last, which lists the order in which these samples will reach [tex]$90.0^{\circ} C$[/tex]?

A. Ag, Zn, Fe, Al
B. Al, Fe, Zn, Ag
C. Al, Fe, Ag, Zn
D. Ag, Al, Zn, Fe

Sagot :

GinnyAnswer
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]
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