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 calculate the final temperature of the iron, we can use the principle of the conservation of energy. The energy lost by the iron as it cools down is equal to the energy gained by the water as it heats up.
First, let's determine the initial and final heat energies of the iron and water:
1. Calculate the initial heat energy of the iron:
Q_iron = m_iron * c_iron * ΔT_iron
where:
m_iron = mass of iron = 0.750 kg
c_iron = specific heat capacity of iron = 450 J/kg°C (approximately)
ΔT_iron = change in temperature of the iron = final temperature - initial temperature = T_final - 200°C
2. Calculate the heat energy gained by the water:
Q_water = m_water * c_water * ΔT_water
where:
m_water = mass of water = 8 L = 8 kg (since 1 L of water has a mass of 1 kg)
c_water = specific heat capacity of water = 4186 J/kg°C (approximately)
ΔT_water = change in temperature of the water = final temperature - initial temperature = T_final - 25°C
According to the principle of conservation of energy:
Q_iron = -Q_water (as energy lost by iron is equal to energy gained by water)
Now, substitute the values into the equation:
m_iron * c_iron * ΔT_iron = -m_water * c_water * ΔT_water
0.750 kg * 450 J/kg°C * (T_final - 200°C) = -8 kg * 4186 J/kg°C * (T_final - 25°C)
Simplify the equation:
337.5 (T_final - 200) = -33488 (T_final - 25)
Now, expand the equation:
337.5 T_final - 67500 = -33488 T_final + 837200
Combine the like terms:
337.5 T_final + 33488 T_final = 837200 + 67500
33825.5 T_final = 904700
Finally, solve for T_final:
T_final = 904700 / 33825.5
T_final ≈ 26.78°C
Therefore, the final temperature of the iron is approximately 26.78°C.
To solve this problem, we can use the principle of heat transfer, which states that heat gained (or lost) by an object is equal to the heat lost (or gained) by the surrounding objects. The formula for heat transfer is:
Q = mcΔT
Where:
Q = heat transferred
m = mass of the object (in kg)
c = specific heat capacity of the object (in J/kg°C)
ΔT = change in temperature (in °C)
Let's break down the problem step by step:
Step 1: Calculate the heat gained by the water.
The heat gained by the water can be calculated using the formula mentioned above. The specific heat capacity of water is approximately 4186 J/kg°C.
Q_water = m_water × c_water × ΔT_water
Since the water is initially at 25°C and the final temperature is unknown, we can substitute the values into the equation:
Q_water = 8 kg × 4186 J/kg°C × (T_final - 25°C) --(1)
Step 2: Calculate the heat lost by the iron piece.
Similarly, the heat lost by the iron piece can be calculated using the same formula. However, the specific heat capacity of iron is different from water. The specific heat capacity of iron is approximately 450 J/kg°C.
Q_iron = m_iron × c_iron × ΔT_iron
Since the iron piece is initially at 200°C and the final temperature is unknown, we can substitute the values into the equation:
Q_iron = 0.75 kg × 450 J/kg°C × (200°C - T_final) --(2)
Step 3: Equate the heat transfer equations.
According to the principle of heat transfer, the heat gained by the water is equal to the heat lost by the iron piece in an isolated system.
Q_water = Q_iron
Substituting equations (1) and (2), we get:
8 kg × 4186 J/kg°C × (T_final - 25°C) = 0.75 kg × 450 J/kg°C × (200°C - T_final)
Step 4: Solve for T_final.
Simplifying the equation:
33488 J/°C × (T_final - 25°C) = 33750 J/°C × (200°C - T_final)
33488 T_final - 33750 × 25 = 33750 × 200 - 33750 T_final
33488 T_final + 843750 = 6750000 - 33750 T_final
67000 T_final = 6750000 - 843750
T_final = (6750000 - 843750) / 67000
T_final ≈ 72.002°C
Therefore, the final temperature of the iron piece is approximately 72.002°C.
Note: The approximation is due to rounding off the decimal places in the final calculation.