What Is the Correct Equation for Equilibrium Temperature of Two Fluids?

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    Equilibrium Fluids
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The discussion focuses on calculating the final equilibrium temperature when mixing two bodies of water at different temperatures. The key equation used is based on the principle of heat transfer, where the heat lost by the hot water equals the heat gained by the cold water. A common mistake identified was the incorrect application of the equation, specifically not accounting for the negative sign when one body loses heat. After correcting this error, the final equilibrium temperature was successfully calculated. The correct approach emphasizes the importance of proper sign conventions in thermal equilibrium calculations.
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Homework Statement
A 5-gallon container of water (1 gal = 3.79 liter) at 212 degrees F is added to 50 gallons of water at 50 degrees F. What is the final equilibrium temperature in degrees C?
Relevant Equations
Qlost = Qgained
Not really sure how to start this one.
 
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trying_mybest said:
Homework Statement:: A 5-gallon container of water (1 gal = 3.79 liter) at 212 degrees F is added to 50 gallons of water at 50 degrees F. What is the final equilibrium temperature in degrees C?
Relevant Equations:: Qlost = Qgained

Not really sure how to start this one.
If the final temperature is T degrees F, what are the values of Qlost and Qgained?
 
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Final temp should be ˚C.

I used m1C∆T1 = m2C∆T2

I keep getting 0 for the final temp when it's supposed to be 18 C
 
trying_mybest said:
I keep getting 0 for the final temp when it's supposed to be 18 C
And you'd like someone to guess where you are going wrong?
trying_mybest said:
Final temp should be ˚C.
Yes, but it will be simpler to do one conversion at the end than two at the start.
 
trying_mybest said:
Final temp should be ˚C.

I used m1C∆T1 = m2C∆T2

I keep getting 0 for the final temp when it's supposed to be 18 C
That equation looks fine but what do you put for ##\Delta T_1## and ##\Delta T_2##?
 
m1 = 5 gal * 3.79 L/gal * 1000 g/L = 18,750 g
m2 = 50 gal * 3.79 L/gal * 1000 g/L = 187,500 g

m1*C*(Tf - T1) = m2*C*(Tf - T2)

m1CTf - m1CT1 = m2CTf - m2CT2

m1CTf - m2CTf = m1CT1 - m2CT2
Tf(m1 - m2) = m1T1 - m2T2
Tf = (m1T1 - m2T2) / (m1 - m2)
Tf = (18,750*212 - 187,500*50) / (18,750 - 187,500)
Tf = -5,400,000 / -168,750 = 32 ˚F = 0 ˚C
 
Well, i think the mistake lies right at the start.
$$m_1C(T_f-T_1)=m_2C(T_f-T_2)$$ is not correct, the correct is $$-m_1C(T_f-T_1)=m_2C(T_f-T_2)$$ because one body is losing heat (so the heat will be negative) and the other is gaining heat (so the heat will be positive).
 
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Delta2 said:
Well, i think the mistake lies right at the start.
$$m_1C(T_f-T_1)=m_2C(T_f-T_2)$$ is not correct, the correct is $$-m_1C(T_f-T_1)=m_2C(T_f-T_2)$$ because one body is losing heat (so the heat will be negative) and the other is gaining heat (so the heat will be positive).

Thank you! I caught that mistake earlier, fixed it, and got the correct answer.
 
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