How much heat in joules must be added to 0.413 kg of aluminum to change it from a solid at 122 °C to a liquid at 660 °C (its melting point)? The latent heat of fusion for aluminum is 4.0 x 105 J/kg.
To calculate the amount of heat required to change the phase of a substance, we need to consider two things: the heat required to raise the temperature from the solid to the melting point, and the heat required to change the phase from a solid to a liquid.
Here's the step-by-step process to calculate it:
Step 1: Calculate the heat required to raise the temperature from 122 °C to the melting point:
The specific heat capacity of aluminum is 0.897 J/g·°C. To convert the mass to grams, we multiply by 1000:
0.413 kg × 1000 g/kg = 413 g
The temperature change is:
660 °C - 122 °C = 538 °C
The heat required is calculated using the formula:
Q = m × c × ΔT
where
Q = heat transferred (in joules),
m = mass (in grams),
c = specific heat capacity (in joules per gram per degree Celsius), and
ΔT = temperature change (in degrees Celsius).
Plugging the values into the formula:
Q1 = 413 g × 0.897 J/g·°C × 538 °C.
Step 2: Calculate the heat required to change the phase from solid to liquid:
The latent heat of fusion for aluminum is 4.0 × 10^5 J/kg. We use the mass of aluminum in kilograms for this calculation, so no conversion is needed.
The heat required is calculated using the formula:
Q2 = m × L
where
Q2 = heat transferred (in joules),
m = mass (in kilograms), and
L = latent heat of fusion (in joules per kilogram).
Plugging the values into the formula:
Q2 = 0.413 kg × 4.0 × 10^5 J/kg.
Step 3: Calculate the total heat required by adding both Q1 and Q2:
Total heat required = Q1 + Q2.
Now, let's calculate the values:
Q1 = 413 g × 0.897 J/g·°C × 538 °C = 202,651.194 J.
Q2 = 0.413 kg × 4.0 × 10^5 J/kg = 1.652 × 10^5 J.
Total heat required = 202,651.194 J + 1.652 × 10^5 J = 3.852 × 10^5 J.
Therefore, the amount of heat required to change 0.413 kg of aluminum from a solid at 122 °C to a liquid at its melting point of 660 °C is approximately 3.852 × 10^5 joules.