A 7 gallon bucket of water of mass 26.502 kg and temperature 100◦C is added to a 62 gallon gallon tub of water of mass 234.732 kg having a temperature of 10◦C.
What is the final equilibrium temperature of the mixture? Assume there is no loss of heat to the surroundings.
Answer in units of ◦C.
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To find the final equilibrium temperature of the mixture, we can use the principle of energy conservation. The total amount of energy in the system before and after mixing should remain the same.
The energy transferred by the hot water (7-gallon bucket of water) to the cold water (62-gallon tub of water) can be calculated using the formula:
Q = mcΔT
where Q is the heat transferred, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature.
For the hot water, with a mass of 26.502 kg and a temperature change from 100◦C to the final equilibrium temperature, let's call it Tf:
Q1 = m1cΔT1
Q1 = (26.502 kg)(4.18 kJ/kg◦C)(100◦C - Tf)
For the cold water, with a mass of 234.732 kg and a temperature change from 10◦C to Tf:
Q2 = m2cΔT2
Q2 = (234.732 kg)(4.18 kJ/kg◦C)(Tf - 10◦C)
Since there is no loss of heat to the surroundings, the heat transferred from the hot water to the cold water is equal to the heat gained by the cold water. Therefore, we can set up an equation:
Q1 = Q2
(26.502 kg)(4.18 kJ/kg◦C)(100◦C - Tf) = (234.732 kg)(4.18 kJ/kg◦C)(Tf - 10◦C)
Now, let's solve this equation to find the final equilibrium temperature, Tf.