Let's solve the problem step-by-step:
1. Identify the given variables:
- Mass of each sample: [tex]\( m = 10 \)[/tex] grams
- Initial temperature: [tex]\( T_i = 25.0^{\circ} \text{C} \)[/tex] (room temperature)
- Final temperature: [tex]\( T_f = 90.0^{\circ} \text{C} \)[/tex]
- Specific heat capacities ([tex]\( C_p \)[/tex]):
- Aluminum (Al): [tex]\( C_{p, \text{Al}} = 0.897 \, \text{J/(g⋅°C)} \)[/tex]
- Silver (Ag): [tex]\( C_{p, \text{Ag}} = 0.234 \, \text{J/(g⋅°C)} \)[/tex]
- Iron (Fe): [tex]\( C_{p, \text{Fe}} = 0.450 \, \text{J/(g⋅°C)} \)[/tex]
- Zinc (Zn): [tex]\( C_{p, \text{Zn}} = 0.387 \, \text{J/(g⋅°C)} \)[/tex]
2. Calculate the temperature change ([tex]\( \Delta T \)[/tex]):
[tex]\[
\Delta T = T_f - T_i = 90.0^{\circ} \text{C} - 25.0^{\circ} \text{C} = 65.0^{\circ} \text{C}
\][/tex]
3. Calculate the heat required (q) for each metal:
[tex]\[
q = m \cdot C_p \cdot \Delta T
\][/tex]
- For Aluminum ([tex]\( q_{\text{Al}} \)[/tex]):
[tex]\[
q_{\text{Al}} = 10 \, \text{g} \cdot 0.897 \, \text{J/(g⋅°C)} \cdot 65.0^{\circ} \text{C} = 583.05 \, \text{J}
\][/tex]
- For Silver ([tex]\( q_{\text{Ag}} \)[/tex]):
[tex]\[
q_{\text{Ag}} = 10 \, \text{g} \cdot 0.234 \, \text{J/(g⋅°C)} \cdot 65.0^{\circ} \text{C} = 152.10 \, \text{J}
\][/tex]
- For Iron ([tex]\( q_{\text{Fe}} \)[/tex]):
[tex]\[
q_{\text{Fe}} = 10 \, \text{g} \cdot 0.450 \, \text{J/(g⋅°C)} \cdot 65.0^{\circ} \text{C} = 292.50 \, \text{J}
\][/tex]
- For Zinc ([tex]\( q_{\text{Zn}} \)[/tex]):
[tex]\[
q_{\text{Zn}} = 10 \, \text{g} \cdot 0.387 \, \text{J/(g⋅°C)} \cdot 65.0^{\circ} \text{C} = 251.55 \, \text{J}
\][/tex]
4. Determine the order in which the samples will reach [tex]\( 90.0^{\circ} \text{C} \)[/tex]:
The sample that requires the least amount of heat to reach the desired temperature will heat up the fastest.
Order of heat required:
- Silver (Ag) needs [tex]\( 152.10 \, \text{J} \)[/tex]
- Zinc (Zn) needs [tex]\( 251.55 \, \text{J} \)[/tex]
- Iron (Fe) needs [tex]\( 292.50 \, \text{J} \)[/tex]
- Aluminum (Al) needs [tex]\( 583.05 \, \text{J} \)[/tex]
Therefore, the order in which these samples will reach [tex]\( 90.0^{\circ} \text{C} \)[/tex] from first to last is:
[tex]\[
\text{Ag} , \text{Zn} , \text{Fe} , \text{Al}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\text{Ag} , \text{Zn} , \text{Fe} , \text{Al}}
\][/tex]
