At a fabrication plant, a hot metal forging has a mass of 66.8 kg and a specific heat capacity of 437 J/(kg C°). To harden it, the forging is quenched by immersion in 785 kg of oil that has a temperature of 25.0 °C and a specific heat capacity of 2770 J/(kg C°). The final temperature of the oil and forging at thermal equilibrium is 61.7 °C. Assuming that heat flows only between the forging and the oil, determine the initial temperature in degrees Celsius of the forging.
At final thermal equilibrium, the heat gained by the oil equals the heat lost by the metal. Write that as an equation and solve for the initial temperature of the metal, To.
(To - 61.7)*66.8*437 = (61.7-25)*785*2770
To solve this problem, we can use the principle of conservation of energy. The amount of heat gained by the oil is equal to the amount of heat lost by the forging, assuming no heat is lost to the surroundings.
The equation to calculate heat transfer is:
Q = mcΔT
Where:
Q is the heat transfer (in joules)
m is the mass (in kg)
c is the specific heat capacity (in J/(kg C°))
ΔT is the change in temperature (in C°)
Let's calculate the heat lost by the forging and the heat gained by the oil.
Heat lost by the forging:
Q_forging = m_forging * c_forging * (T_forging_initial - T_equilibrium)
Heat gained by the oil:
Q_oil = m_oil * c_oil * (T_equilibrium - T_oil_initial)
Since the system reaches thermal equilibrium, the amount of heat lost by the forging (Q_forging) is equal to the amount of heat gained by the oil (Q_oil). Therefore, we can set these two equations equal to each other:
m_forging * c_forging * (T_forging_initial - T_equilibrium) = m_oil * c_oil * (T_equilibrium - T_oil_initial)
Let's plug in the given values and solve for T_forging_initial.
m_forging = 66.8 kg
c_forging = 437 J/(kg C°)
T_equilibrium = 61.7 °C
m_oil = 785 kg
c_oil = 2770 J/(kg C°)
T_oil_initial = 25.0 °C
66.8 kg * 437 J/(kg C°) * (T_forging_initial - 61.7 °C) = 785 kg * 2770 J/(kg C°) * (61.7 °C - 25.0 °C)
Now we can solve for T_forging_initial.