Answer :
To determine how much energy is needed to raise the temperature of [tex]\(10 \, \text{g}\)[/tex] of iron and aluminum each by [tex]\(1^{\circ} \text{C}\)[/tex], we can start by using the formula for heat energy:
[tex]\[ Q = m \cdot C \cdot \Delta T \][/tex]
where:
- [tex]\( Q \)[/tex] is the heat energy (in joules, [tex]\(J\)[/tex]),
- [tex]\( m \)[/tex] is the mass (in grams, [tex]\( g \)[/tex]),
- [tex]\( C \)[/tex] is the specific heat capacity (in [tex]\( \frac{J}{g \cdot {}^{\circ}C} \)[/tex]),
- [tex]\( \Delta T \)[/tex] is the change in temperature (in [tex]\( {}^{\circ}C \)[/tex]).
Step-by-step calculation:
1. Determine the energy needed for iron:
For iron (Fe):
- Mass ([tex]\( m \)[/tex]) = [tex]\( 10 \, g \)[/tex]
- Specific heat capacity ([tex]\( C_{Fe} \)[/tex]) = [tex]\( 0.450 \, \frac{J}{g \cdot {}^{\circ}C} \)[/tex]
- Temperature change ([tex]\( \Delta T \)[/tex]) = [tex]\( 1^{\circ}C \)[/tex]
Plug these values into the formula:
[tex]\[ Q_{Fe} = 10 \, g \cdot 0.450 \, \frac{J}{g \cdot {}^{\circ}C} \cdot 1^{\circ}C = 4.5 \, J \][/tex]
2. Determine the energy needed for aluminum:
For aluminum (Al):
- Mass ([tex]\( m \)[/tex]) = [tex]\( 10 \, g \)[/tex]
- Specific heat capacity ([tex]\( C_{Al} \)[/tex]) = [tex]\( 0.900 \, \frac{J}{g \cdot {}^{\circ}C} \)[/tex]
- Temperature change ([tex]\( \Delta T \)[/tex]) = [tex]\( 1^{\circ}C \)[/tex]
Plug these values into the formula:
[tex]\[ Q_{Al} = 10 \, g \cdot 0.900 \, \frac{J}{g \cdot {}^{\circ}C} \cdot 1^{\circ}C = 9.0 \, J \][/tex]
3. Compare the energy needed for iron and aluminum:
The energy required for aluminum is [tex]\( 9.0 \, J \)[/tex], while the energy required for iron is [tex]\( 4.5 \, J \)[/tex]. To compare these amounts, calculate the ratio of the energy required for aluminum to the energy required for iron:
[tex]\[ \text{Ratio} = \frac{Q_{Al}}{Q_{Fe}} = \frac{9.0 \, J}{4.5 \, J} = 2.0 \][/tex]
This means aluminum requires twice as much energy as iron to raise the temperature of the same mass by the same amount. Therefore, we conclude:
Iron needs half as much energy as aluminum.
Answer: Fe needs half as much energy as Al.
[tex]\[ Q = m \cdot C \cdot \Delta T \][/tex]
where:
- [tex]\( Q \)[/tex] is the heat energy (in joules, [tex]\(J\)[/tex]),
- [tex]\( m \)[/tex] is the mass (in grams, [tex]\( g \)[/tex]),
- [tex]\( C \)[/tex] is the specific heat capacity (in [tex]\( \frac{J}{g \cdot {}^{\circ}C} \)[/tex]),
- [tex]\( \Delta T \)[/tex] is the change in temperature (in [tex]\( {}^{\circ}C \)[/tex]).
Step-by-step calculation:
1. Determine the energy needed for iron:
For iron (Fe):
- Mass ([tex]\( m \)[/tex]) = [tex]\( 10 \, g \)[/tex]
- Specific heat capacity ([tex]\( C_{Fe} \)[/tex]) = [tex]\( 0.450 \, \frac{J}{g \cdot {}^{\circ}C} \)[/tex]
- Temperature change ([tex]\( \Delta T \)[/tex]) = [tex]\( 1^{\circ}C \)[/tex]
Plug these values into the formula:
[tex]\[ Q_{Fe} = 10 \, g \cdot 0.450 \, \frac{J}{g \cdot {}^{\circ}C} \cdot 1^{\circ}C = 4.5 \, J \][/tex]
2. Determine the energy needed for aluminum:
For aluminum (Al):
- Mass ([tex]\( m \)[/tex]) = [tex]\( 10 \, g \)[/tex]
- Specific heat capacity ([tex]\( C_{Al} \)[/tex]) = [tex]\( 0.900 \, \frac{J}{g \cdot {}^{\circ}C} \)[/tex]
- Temperature change ([tex]\( \Delta T \)[/tex]) = [tex]\( 1^{\circ}C \)[/tex]
Plug these values into the formula:
[tex]\[ Q_{Al} = 10 \, g \cdot 0.900 \, \frac{J}{g \cdot {}^{\circ}C} \cdot 1^{\circ}C = 9.0 \, J \][/tex]
3. Compare the energy needed for iron and aluminum:
The energy required for aluminum is [tex]\( 9.0 \, J \)[/tex], while the energy required for iron is [tex]\( 4.5 \, J \)[/tex]. To compare these amounts, calculate the ratio of the energy required for aluminum to the energy required for iron:
[tex]\[ \text{Ratio} = \frac{Q_{Al}}{Q_{Fe}} = \frac{9.0 \, J}{4.5 \, J} = 2.0 \][/tex]
This means aluminum requires twice as much energy as iron to raise the temperature of the same mass by the same amount. Therefore, we conclude:
Iron needs half as much energy as aluminum.
Answer: Fe needs half as much energy as Al.
