Help me with my math?

[Thoroughbred_Med]

So I'm working on a free-standing question in TBR GChem :

"If a 20.0g piece of metal at 75 C with a heat capacity of 0.50 cal/g*K were added to 40g H20 at 25 C, then what would the equilibrium temperature be, assuming no loss of heat to the environment?"

I know how to set up the problem using MCdeltaT water = MCdeltaT metal

so, (40)(1.00)(Tf - 25)K = (20)(0.50)(Tf-75)K

= 40(Tf - 25) = 10(Tf - 75)

= 40Tf - 1000 = 10Tf - 750

= 30Tf = 250

Tf = 8.3 degrees Celcius ... but that's not right


In TBR's answer explanation, they swap the temperature terms in the metal so the math looks like,

(40)(1.00)(Tf - 25)K = (20)(0.50)(75 - Tf)

I always thought deltaT equaled (Tfinal - Tinitial) ??? You get the right answer ( 35 degrees Celsius) if you follow their math but I just don't understand how they can put the Tinitial term first in such an expression?

Can y'all help me out?
Thanks!

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[aldol16]

MCdeltaT water = MCdeltaT metal

It's because your formula, quoted above, is wrong. Remember that these expressions represent the heat changes of their respective substances. So think of it in terms of words: the heat gained by the water must be equal to the heat lost from the metal, if no energy is dissipated from the system. In other words, the directionality of heat flow from one substance to the other is opposite from that going the other way. Therefore, the equality above cannot be correct because it suggests that the direction of heat change is the same for both substances when in fact heat is lost from the metal and gained by the water. This means that you need a negative sign on one side above.

Another way to look at this is conservation of energy. In a closed system, the sum of all energy changes must equal zero. That is m*C*delta T(water) + m*C*delta T(metal) = 0. Then the expression works out. I find it easier to start from this expression since it's the purest expression of energy conservation and you don't have to worry about thinking about signs.

 

[Thoroughbred_Med]

It's because your formula, quoted above, is wrong. Remember that these expressions represent the heat changes of their respective substances. So think of it in terms of words: the heat gained by the water must be equal to the heat lost from the metal, if no energy is dissipated from the system. In other words, the directionality of heat flow from one substance to the other is opposite from that going the other way. Therefore, the equality above cannot be correct because it suggests that the direction of heat change is the same for both substances when in fact heat is lost from the metal and gained by the water. This means that you need a negative sign on one side above.

Another way to look at this is conservation of energy. In a closed system, the sum of all energy changes must equal zero. That is m*C*delta T(water) + m*C*delta T(metal) = 0. Then the expression works out. I find it easier to start from this expression since it's the purest expression of energy conservation and you don't have to worry about thinking about signs.

perfect explanation! Thank you! I agree, the expression has the most clarity when considering conservation of energy.