SOLUTION: If I have four gallons of 50 degree farenheit water and I need to add one more gallon of water what does the temperature of the last gallon need to be if the whole mixture ends up

Question 378673: If I have four gallons of 50 degree farenheit water and I need to add one more gallon of water what does the temperature of the last gallon need to be if the whole mixture ends up at 70 degrees.
This is used to mix up home brewed beer where the ideal temperature is 70 to add the yeast. The last gallon actually has a sugar solution.
The entire specific gravity of the mix ends up at 1035 just in case that needs to be figured in.
My name is Gerard and I am not smarter than a fifth grader.
Found 2 solutions by Alan3354, richard1234:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
If I have four gallons of 50 degree farenheit water and I need to add one more gallon of water what does the temperature of the last gallon need to be if the whole mixture ends up at 70 degrees.
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If you assume that mixing equal volumes of water gives the mix the average of the 2 temperatures, which seems likely, tho I haven't thought about this before:
Then you have to deal with units of "degree-gallons", something new to me.
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4 gals @ 50� = 200 deg-gallons
The mix desired is 5 gallons @ 70� = 350 deg-gallons
The gallon to be added has to have 150 deg-gallons of heat, so it has to be 150�

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Keep in mind that since we're measuring heat and heat transfer, temperature needs to be measured in Kelvins as we can accurately measure the amount of heat and kinetic energy within the two mixtures.
50 F = 283 Kelvins (rounded to the nearest Kelvin)
70 F = 294 Kelvins
(283 Kelvins)*(4 gallons) + (x Kelvins)*(1 gallon) = (294 Kelvins)*(5 gallons)
As the units are consistent (Kelvins*gallons), we can solve this as a normal algebra problem and obtain x = 338 Kelvins, i.e. the temperature of the fifth gallon is 338 Kelvins, or 149 degrees Fahrenheit.
However, I solved the problem using the fifth gallon being water. If you're using a sugar mixture, it might have a different heat capacity but I anticipate no significant difference in heat capacity as the sugar is dissolved.
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