How much energy is needed to raise the temperature of 10 g of iron compared to 10 g of aluminum, each by [tex]\(1^{\circ}C\)[/tex]?

Given:
[tex]\(C_{Fe} = 0.450 \frac{J}{g \cdot ^{\circ}C}\)[/tex]
[tex]\(C_{Al} = 0.900 \frac{J}{g \cdot ^{\circ}C}\)[/tex]

A. Fe needs 0.450 times less energy than Al
B. Fe needs 0.450 times more energy than Al
C. Fe needs twice as much energy as Al
D. Fe needs half as much energy as Al



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.