At a fabrication plant, a hot metal forging has a mass of 78.5 kg and a specific heat capacity of 426 J/(kg C°). To harden it, the forging is quenched by immersion in 788 kg of oil that has a temperature of 30.3 °C and a specific heat capacity of 2770 J/(kg C°). The final temperature of the oil and forging at thermal equilibrium is 55.4 °C. Assuming that heat flows only between the forging and the oil, determine the initial temperature in degrees Celsius of the forging.

To solve this problem, we can use the principle of energy conservation. The heat lost by the hot forging is equal to the heat gained by the oil during the cooling process. The equation that represents this principle is:

Q_lost = Q_gained

Where:
Q_lost = heat lost by the forging
Q_gained = heat gained by the oil

The formula to calculate the heat lost/gained is:

Q = m * c * ΔT

Where:
Q = heat (in joules)
m = mass (in kg)
c = specific heat capacity (in J/(kg C°))
ΔT = change in temperature (in C°)

1. Calculate the heat lost by the forging (Q_lost):
Q_lost = m_forging * c_forging * ΔT_forging

2. Calculate the heat gained by the oil (Q_gained):
Q_gained = m_oil * c_oil * ΔT_oil

As the problem states that the final temperature at thermal equilibrium is 55.4 °C, we can assume that the heat lost by the forging is equal to the heat gained by the oil. So we have:

Q_lost = Q_gained

Now let's substitute the formulas for heat lost/gained:

m_forging * c_forging * ΔT_forging = m_oil * c_oil * ΔT_oil

Rearranging the formula, we have:

ΔT_forging = (m_oil * c_oil * ΔT_oil) / (m_forging * c_forging)

Now, let's substitute the given values into the equation and solve for ΔT_forging:

m_oil = 788 kg
c_oil = 2770 J/(kg C°)
ΔT_oil = 55.4 °C - 30.3 °C = 25.1 °C
m_forging = 78.5 kg
c_forging = 426 J/(kg C°)

ΔT_forging = (788 kg * 2770 J/(kg C°) * 25.1 °C) / (78.5 kg * 426 J/(kg C°))

ΔT_forging = (6886612 J * °C) / (33471 J)

ΔT_forging = 205.73 °C

Now, to determine the initial temperature of the forging, we need to subtract the change in temperature from the final temperature:

Initial temperature = Final temperature - ΔT_forging

Initial temperature = 55.4 °C - 205.73 °C

Initial temperature ≈ -150.33 °C

Therefore, the initial temperature of the forging is approximately -150.33 °C.

To solve this problem, we can use the principle of conservation of energy. We know that the heat gained by the oil is equal to the heat lost by the forging.

The formula for heat transfer is:

Q = mcΔT

where:
Q is the heat transferred
m is the mass of the substance
c is the specific heat capacity of the substance
ΔT is the change in temperature

Let's calculate the heat gained by the oil and the heat lost by the forging.

For the oil:
Q_oil = mcΔT
Q_oil = (788 kg) * (2770 J/(kg*C°)) * (55.4°C - 30.3°C)

For the forging:
Q_forging = mcΔT
Q_forging = (78.5 kg) * (426 J/(kg*C°)) * (55.4°C - T_initial)

Since the heat gained by the oil is equal to the heat lost by the forging, we can set Q_oil equal to Q_forging and solve for T_initial.

(788 kg)(2770 J/(kg*C°))(55.4°C - 30.3°C) = (78.5 kg)(426 J/(kg*C°))(55.4°C - T_initial)

Now we can solve for T_initial.

(788 kg)(2770 J/(kg*C°))(55.4°C - 30.3°C) = (78.5 kg)(426 J/(kg*C°))(55.4°C - T_initial)

Simplifying the equation:
T_initial ≈ 48.6°C

Therefore, the initial temperature of the forging is approximately 48.6°C.