You have collected exactly 2800 aluminum cans for recycling, each with a mass of 13.6 g. How much energy is needed to melt them if their initial temperature is 26.2◦C? Assume the specific heat, the latent heat and the melting point of aluminum are 899 J/kg ·◦ C,
3.97 × 105 J/kg and 660.4 ◦C, respectively. Answer in units of J
c=899 J/kg •◦ C, r= 3.97 × 105 J/kg
m=2800•0.0136= 38.08 kg
E = Q=cm(660.4-26.2) + r m = …
To calculate the energy needed to melt the aluminum cans, we need to break the problem down into two steps:
Step 1: Calculate the heat energy required to raise the temperature of the aluminum cans from 26.2°C to its melting point of 660.4°C.
The heat energy required to raise the temperature can be calculated using the specific heat formula:
Q1 = m * c * ΔT
Where:
Q1 = heat energy required
m = mass of aluminum cans (in kg)
c = specific heat of aluminum (in J/kg·°C)
ΔT = change in temperature (in °C)
Given:
Mass of one aluminum can = 13.6 g = 0.0136 kg (since 1 kg = 1000 g)
Specific heat of aluminum = 899 J/kg·°C
Initial temperature = 26.2°C
Melting point temperature = 660.4°C
ΔT = melting point temperature - initial temperature
ΔT = 660.4°C - 26.2°C
ΔT = 634.2°C
m = mass of one aluminum can * number of cans
m = 0.0136 kg * 2800 cans
m = 38.08 kg
Q1 = 38.08 kg * 899 J/kg·°C * 634.2°C
Q1 = 171,752.5616 J
Step 2: Calculate the heat energy required for melting the aluminum cans.
The heat energy required for the phase change (melting) can be calculated using the latent heat formula:
Q2 = m * L
Where:
Q2 = heat energy required
m = mass of aluminum cans (in kg)
L = latent heat of aluminum (in J/kg)
Given:
Latent heat of aluminum = 3.97 × 10^5 J/kg
Mass of aluminum cans = 38.08 kg
Q2 = 38.08 kg * 3.97 × 10^5 J/kg
Q2 = 1,510,576 J
Finally, the total energy required to melt the aluminum cans is the sum of Q1 and Q2:
Total energy = Q1 + Q2
Total energy = 171,752.5616 J + 1,510,576 J
Total energy = 1,682,328.5616 J
Therefore, the energy needed to melt the aluminum cans is approximately 1,682,328.5616 J.
To calculate the energy required to melt the aluminum cans, we need to consider two steps: raising the temperature of the cans to the melting point, and then melting them.
Step 1: Calculate the energy required to raise the temperature of the cans:
The specific heat (c) of aluminum is given as 899 J/kg · ◦C. This means that it takes 899 Joules of energy to raise the temperature of 1 kilogram of aluminum by 1 degree Celsius.
First, we need to calculate the total mass of aluminum cans. Since each can has a mass of 13.6 grams, the mass can be calculated as:
Mass = Number of cans * Mass per can
Mass = 2800 * 13.6 g
Mass = 38,080 g
Convert the mass to kilograms:
Mass = 38,080 g / 1000
Mass = 38.08 kg
Next, we need to calculate the temperature difference:
ΔT = Final temperature - Initial temperature
ΔT = 660.4 ◦C - 26.2 ◦C
ΔT = 634.2 ◦C
Now we can calculate the energy required to raise the temperature of the cans:
Energy = Mass * Specific heat * ΔT
Energy = 38.08 kg * 899 J/kg · ◦C * 634.2 ◦C
Step 2: Calculate the energy required to melt the aluminum cans:
The latent heat (L) of aluminum is given as 3.97 × 10^5 J/kg. This is the amount of energy required to change the state of 1 kilogram of aluminum from solid to liquid at its melting point.
The total energy required to melt the aluminum cans can be calculated as:
Energy = Mass * Latent heat
Energy = 38.08 kg * 3.97 × 10^5 J/kg
Adding the energy required for both steps together will give the total energy required to melt the aluminum cans.
Total energy = Energy for raising temperature + Energy for melting
Total energy = (38.08 kg * 899 J/kg · ◦C * 634.2 ◦C) + (38.08 kg * 3.97 × 10^5 J/kg)
Now, you can calculate the total energy required.