A blacksmith at a historical reenactment is making an iron hinge. after placing the iron (0.750 kg) in the fire and shaping it with his hammer it is still at 200 C. he puts the iron piece in an 8 L bucket of water at 25 C. what is the final temp. of the iron (assume it is in an isolated system)?
show steps please!
To find the final temperature of the iron, we can use the principle of energy conservation.
The equation we'll use is:
m1 * c1 * (T1 - Tf) = m2 * c2 * (Tf - T2)
where:
m1 = mass of the iron (0.750 kg)
c1 = specific heat capacity of iron (450 J/kg·C)
T1 = initial temperature of the iron (200 °C)
Tf = final temperature of the iron (unknown)
m2 = mass of water (8 kg since 1 L of water is approximately 1 kg)
c2 = specific heat capacity of water (4186 J/kg·C)
T2 = temperature of the water (25 °C)
Plugging in the values, the equation becomes:
0.750 kg * 450 J/kg·C * (200 °C - Tf) = 8 kg * 4186 J/kg·C * (Tf - 25 °C)
Let's solve it step-by-step:
Step 1: Expand the equation:
337.5 J/°C * (200 °C - Tf) = 33488 J/°C * (Tf - 25 °C)
Step 2: Distribute the terms:
67500 J - 337.5 J*Tf = 33488 J*Tf - 837200 J
Step 3: Rearrange the equation:
337.5 J*Tf + 33488 J*Tf = 837200 J + 67500 J
Step 4: Combine like terms:
33825.5 J*Tf = 904700 J
Step 5: Divide both sides by 33825.5 J:
Tf = 904700 J / 33825.5 J
Step 6: Calculate Tf:
Tf ≈ 26.76 °C
Therefore, the final temperature of the iron is approximately 26.76 °C.
To find the final temperature of the iron, we can use the principle of heat transfer. The heat gained by the water is equal to the heat lost by the iron. This can be calculated using the equation:
Q = mcΔT
where:
Q is the amount of heat transferred,
m is the mass of the substance,
c is the specific heat capacity of the substance, and
ΔT is the change in temperature.
First, let's calculate the heat gained by the water. The specific heat capacity of water is approximately 4.18 J/g°C. Since we have 8 liters (8000 grams) of water, we can substitute these values into the equation:
Qwater = mwater * cwater * ΔTwater
Qwater = (8000 g) * (4.18 J/g°C) * (Tfinal - 25°C)
Next, we need to calculate the heat lost by the iron. The specific heat capacity of iron is approximately 0.45 J/g°C. Given that the mass of the iron is 0.750 kg (750 grams) and the initial temperature is 200°C, we can substitute these values into the equation:
Qiron = miron * ciron * ΔTiron
Qiron = (750 g) * (0.45 J/g°C) * (200°C - Tfinal)
Since it is an isolated system, the heat gained by the water is equal to the heat lost by the iron:
Qwater = Qiron
(8000 g) * (4.18 J/g°C) * (Tfinal - 25°C) = (750 g) * (0.45 J/g°C) * (200°C - Tfinal)
Now, let's solve this equation to find the final temperature of the iron. Here are the steps:
1. Expand the equation:
(33440 g*°C) * (Tfinal - 25°C) = (337.5 g*°C) * (200°C - Tfinal)
2. Distribute the terms on both sides:
33440 g*°C * Tfinal - 836000 g*°C = 67500 g*°C * (200°C - Tfinal)
3. Simplify further:
33440 g*°C * Tfinal - 836000 g*°C = 13500000 g*°C - 67500 g*°C * Tfinal
4. Move all terms containing Tfinal to one side and the others to the other side:
33440 g*°C * Tfinal + 67500 g*°C * Tfinal = 13500000 g*°C + 836000 g*°C
100940 g*°C * Tfinal = 14336000 g*°C
5. Divide both sides of the equation by the coefficient of Tfinal:
Tfinal = 14336000 g*°C / 100940 g*°C
Tfinal ≈ 142.05°C
Therefore, the final temperature of the iron is approximately 142.05°C.