Heat Capacity Question

[Perkins]

I was working on a heat capacity question, and the book said that:

.Since the heat capacity of aluminum at 1300˚C (0.232 cal/g·˚C) is about twice that of the heat capacity of copper at 1300˚C (0.118 cal/g·˚C), when equal parts by mass of aluminum and copper are mixed, the final temperature should be closer to the initial temperature of aluminum (1310˚C) than to the initial temperature of copper (1300˚C)... .

I was wondering why this is true. I thought high heat capacity means it takes more heat to increase the temperature, so it seems to me that Al with higher heat capacity and higher initial temperature kind of cancels out.

your help is much appreciated!

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[Rabolisk]

I thought high heat capacity means it takes more heat to increase the temperature

This is true.

so it seems to me that Al with higher heat capacity and higher initial temperature kind of cancels out.

I'm not sure I understand what you are saying...

 

[Perkins]

hmm I'm not too sure what i'm saying either. To be honest I'm not what to make of this sentence:

.Since the heat capacity of aluminum at 1300˚C (0.232 cal/g·˚C) is about twice that of the heat capacity of copper at 1300˚C (0.118 cal/g·˚C), when equal parts by mass of aluminum and copper are mixed, the final temperature should be closer to the initial temperature of aluminum (1310˚C) than to the initial temperature of copper (1300˚C)..

What really gets me is when it says "the final temperature should be closer to the initial temperature of aluminum." I'm not sure how heat capacity would tell us that.

It seems like if aluminum takes more heat to change it's temperature, but I don't see how that results in the final temperature being close to the temperature of aluminum. All that I can see from that is: It takes more heat to raise the temperature of aluminum by one unit of temperature (i.e. a celsius or kelvin)

 

[tttgo]

This is how I interpret it:
Aluminum has a higher heat capacity, so more heat is needed to heat it to a temp of 1310. Let's say it takes 2a (made up unit) to heat it to 1310 degrees. That means it would take a little less than a (let's call it .9a) to heat up copper to 1300 (since 1300<1310). Now if you put the two together, the temperatures will try to balance. You are averaging 2a and .9a amount of heat. Isn't 1.45 closer than 2a (for aluminum) than .9a?

 

[Rabolisk]

For another example, consider this. You have two samples of water of equal mass. One is at 10 degrees, and the other is at 20 degrees. If you mix the two, then the final temperature should be 15 degrees, assuming no heat loss to the surroundings. Agree?

Now replace the 10 degree sample of water with a sample of copper of equal mass. Mix the copper and the water. With no heat loss to the surroundings, heat will be transferred from water to copper, lowering water's temperature and increasing copper's temperature. But the heat capacity of water is nearly about 10 times that of copper. That means that if enough heat is transferred to decrease water's temperature by 0.5 degrees, copper's temperature will be raised by 5 degrees. Even though the heat lost by water and the heat gained by copper are equal, the change in temperature is very different (and it is this that is measured by heat capacity). Can you see why the final temperature will be closer to water's initial temperature (20) than copper's (10)?

 

[tttgo]

My above post doesn't make sense. Let's say 1310 degree Al and 1300 degree Cu are placed in solution. If both held the same amount of heat (amt of heat to heat it to starting temp), the average would be 1305 right? Since Al has higher heat capacity, it releases twice as much heat as Cu does when in the solution. Since the 2 beats 1, the solution (say it's water gets its temp raised by however many units of heat. Also, note that each unit of heat from Al is hotter than Cu's because it's 1310 vs 1300.

 

[Perkins]

Thanks for the explanation rabolisk and tttgo!

From both of your explanations it seems to boil down to this:

heat capacity measures the ease of heat transfer. A higher heat capacity means it takes more heat to increase temperature of the material by 1 degree Celsius (or Kelvin)--- HOWEVER higher heat capacity ALSO means that the material releases more heat per every 1 degree Ceslsius/Kelvin temperature drop it undergoes.

Thus final temperature of the solution shifts closer to the initial temperature of the higher heat capacity object because this object's temperature won't change too much despite the heat exchange. The lower heat capacity object's temperature changes more, and is "accommodating." to fit the temperature of the higher heat capacity object.

So then if Aluminum where 1300 deg C to start with while Copper was instead 1310 deg C to start with, we would expect final temperature to be lower than 1305 because the copper would lose a lot of heat to the Aluminum but the Aluminum wouldn't increase it's temperature by much while the Copper temperature has decreased quite a bit.

Is this summary / interpretation correct?

 

[indianjatt]

Here's another approach to the problem using conservation of energy (below). It can be time consuming, but hopefully the solution makes more sense.

Assume 1 g of each.

.232 cal/oC(Te-1310) = - .118 cal/ oC (Te - 1300)
.232 cal/oC(Te) - 303.92 cal = -.118 cal/oC(Te) + 153.4 cal
.350 cal/oC(Te) = 457.32 cal
Te = 1306.63 oC