To determine the amount of energy required to raise the temperature of 10 grams of iron and 10 grams of aluminum by [tex]\(1^{\circ} C\)[/tex], we can use the heat energy formula:
[tex]\[ Q = m \cdot c \cdot \Delta T \][/tex]
where:
- [tex]\( Q \)[/tex] is the heat energy.
- [tex]\( m \)[/tex] is the mass.
- [tex]\( c \)[/tex] is the specific heat capacity.
- [tex]\( \Delta T \)[/tex] is the temperature change.
Given:
- Mass of iron ([tex]\( m_{Fe} \)[/tex]) = [tex]\( 10 \)[/tex] grams
- Mass of aluminum ([tex]\( m_{Al} \)[/tex]) = [tex]\( 10 \)[/tex] grams
- Temperature change ([tex]\( \Delta T \)[/tex]) = [tex]\( 1^{\circ} C \)[/tex]
- Specific heat capacity of iron ([tex]\( c_{Fe} \)[/tex]) = [tex]\( 0.450 \frac{ J }{ g \cdot{ }^{\circ} C } \)[/tex]
- Specific heat capacity of aluminum ([tex]\( c_{Al} \)[/tex]) = [tex]\( 0.900 \frac{ J }{ g \cdot{ }^{\circ} C } \)[/tex]
First, calculate the energy needed to raise the temperature of iron:
[tex]\[ Q_{Fe} = m_{Fe} \cdot c_{Fe} \cdot \Delta T \][/tex]
[tex]\[ Q_{Fe} = 10 \, \text{g} \cdot 0.450 \, \frac{ \text{J} }{ \text{g} \cdot ^{\circ} \text{C} } \cdot 1\, ^{\circ} \text{C} \][/tex]
[tex]\[ Q_{Fe} = 4.5 \, \text{J} \][/tex]
Next, calculate the energy needed to raise the temperature of aluminum:
[tex]\[ Q_{Al} = m_{Al} \cdot c_{Al} \cdot \Delta T \][/tex]
[tex]\[ Q_{Al} = 10 \, \text{g} \cdot 0.900 \, \frac{ \text{J} }{ \text{g} \cdot ^{\circ} \text{C} } \cdot 1\, ^{\circ} \text{C} \][/tex]
[tex]\[ Q_{Al} = 9.0 \, \text{J} \][/tex]
Now, we compare the energy required for iron and aluminum. The ratio of the energy needed for iron to the energy needed for aluminum is:
[tex]\[ \text{Energy ratio} = \frac{Q_{Fe}}{Q_{Al}} \][/tex]
[tex]\[ \text{Energy ratio} = \frac{4.5 \, \text{J}}{9.0 \, \text{J}} \][/tex]
[tex]\[ \text{Energy ratio} = 0.5 \][/tex]
This means that iron needs half as much energy as aluminum to raise the temperature of 10 grams by [tex]\(1^{\circ} C\)[/tex]. Therefore, the correct answer is:
D. Fe needs half as much energy as Al
