Question

how much energy is needed to raise the temperature of 10 g of iron compared to 10 g of aluminum, each by 1 °c?
$c_{fe} = 0.450 \frac{j}{g \cdot ^\circ c}$ $c_{al} = 0.900 \frac{j}{g \cdot ^\circ c}$
al needs twice as much energy as fe.
fe needs 0.450 times less energy than al.
fe needs 0.450 times more energy than al.
fe needs twice as much energy as al.

Explanation:

Step1: Recall the formula for heat energy

The formula for the heat energy \( Q \) required to raise the temperature of a substance is \( Q = mc\Delta T \), where \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

Step2: Calculate \( Q \) for iron (\( Q_{Fe} \))

Given \( m_{Fe} = 10 \, \text{g} \), \( c_{Fe} = 0.450 \, \frac{\text{J}}{\text{g} \cdot ^\circ\text{C}} \), and \( \Delta T_{Fe} = 1 \, ^\circ\text{C} \).
Substitute into the formula: \( Q_{Fe} = 10 \times 0.450 \times 1 = 4.5 \, \text{J} \).

Step3: Calculate \( Q \) for aluminum (\( Q_{Al} \))

Given \( m_{Al} = 10 \, \text{g} \), \( c_{Al} = 0.900 \, \frac{\text{J}}{\text{g} \cdot ^\circ\text{C}} \), and \( \Delta T_{Al} = 1 \, ^\circ\text{C} \).
Substitute into the formula: \( Q_{Al} = 10 \times 0.900 \times 1 = 9.0 \, \text{J} \).

Step4: Compare the energies

Find the ratio \( \frac{Q_{Fe}}{Q_{Al}} = \frac{4.5}{9.0} = 0.5 \) (or \( \frac{c_{Fe}}{c_{Al}} = \frac{0.450}{0.900} = 0.5 \), since \( m \) and \( \Delta T \) are the same for both). This means \( Q_{Fe} = 0.5 Q_{Al} \), or \( Q_{Al} = 2 Q_{Fe} \). So aluminum needs twice as much energy as iron.

Answer:

A. Al needs twice as much energy as Fe.