To solve this problem, we need to calculate the amount of heat required for each element to change its temperature from the initial temperature [tex]\(25.0^{\circ} C\)[/tex] to the final temperature [tex]\(90.0^{\circ} C\)[/tex]. The heat required ([tex]$q$[/tex]) can be calculated using the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the sample (in grams),
- [tex]\( C_p \)[/tex] is the specific heat capacity (in J/(g·°C)),
- [tex]\( \Delta T \)[/tex] is the change in temperature (in °C).
Given:
- Mass ([tex]\( m \)[/tex]) of each sample = 10 g,
- Initial temperature = [tex]\(25.0^{\circ} C\)[/tex],
- Final temperature = [tex]\(90.0^{\circ} C\)[/tex],
- Change in temperature ([tex]\( \Delta T \)[/tex]) = [tex]\( 90.0 - 25.0 = 65.0^{\circ} C\)[/tex].
For each element, we have:
1. Aluminum (Al):
- Specific heat capacity [tex]\( C_p \)[/tex] = 0.897 J/(g·°C)
- Heat required [tex]\( q_{Al} = 10 \cdot 0.897 \cdot 65.0 = 582.05 \)[/tex] J
2. Silver (Ag):
- Specific heat capacity [tex]\( C_p \)[/tex] = 0.234 J/(g·°C)
- Heat required [tex]\( q_{Ag} = 10 \cdot 0.234 \cdot 65.0 = 152.1 \)[/tex] J
3. Iron (Fe):
- Specific heat capacity [tex]\( C_p \)[/tex] = 0.450 J/(g·°C)
- Heat required [tex]\( q_{Fe} = 10 \cdot 0.450 \cdot 65.0 = 292.5 \)[/tex] J
4. Zinc (Zn):
- Specific heat capacity [tex]\( C_p \)[/tex] = 0.387 J/(g·°C)
- Heat required [tex]\( q_{Zn} = 10 \cdot 0.387 \cdot 65.0 = 251.55 \)[/tex] J
Next, we will sort the elements by the amount of heat required ([tex]\( q \)[/tex]) to reach [tex]\( 90.0^{\circ} C \)[/tex] in ascending order:
[tex]\[ q_{Ag} < q_{Zn} < q_{Fe} < q_{Al} \][/tex]
Thus, the order in which the elements will reach [tex]\( 90.0^{\circ} C \)[/tex] first to last is:
[tex]\[ \text{Ag, Zn, Fe, Al} \][/tex]
Comparing this result with the given options, we select:
[tex]\( \boxed{2} \)[/tex] Ag , Zn , Fe , Al
