Heat Capacity and Thermal Equilibrium

[MaryCate22]

Homework Statement

A 3.50-kg block of iron initially at 8.00 × 10^2 K is placed on top of a 6.25-kg block of copper initially at 4.00 × 10^2 K. Assume the blocks are thermally insulated from their surroundings but not from each other and that they constitute a closed system.

How much energy is transferred thermally from the iron to the copper as the two blocks come to thermal equilibrium?

Homework Equations

heat capacity=amount of energy transferred thermally (J)/resulting change in temperature
specific heat capacity (c) =amount of energy required to raise 1 kg of a certain material by 1 degree Kelvin (J/K*kg)

c of copper is 385, c of iron is 449

The Attempt at a Solution

Thermal equilibrium would be (400+800)/2=600. So during this process the copper would be raised 200 K. Using the specific heat capacity of copper 385 J/K*kg, I found that it would be 2406.25 J to raise 6.25 kgs of it by 1 K. To Raise 200 K it would be (2406.25)(200)=481250 J.

This is incorrect, however. What am I missing?

 

[haruspex]

Thermal equilibrium would be (400+800)/2=600.

No. Any other thoughts?

 

[MaryCate22]

No. Any other thoughts?

They do not contribute to the equilibrium temperature equally do they? I looked up a formula for finding equilibrium temp and otherwise worked it out the same. I got the right answer, thanks!

Formula for Equilibrium Temperature: c1m1(Tf-Ti)=c2m2(Tf-Ti)

-(449)(3.5)(Tf-800)=(385)(6.25)(Tf-400) Final Temperature= 558 K, so the copper is only raised 158 degrees.

 

[haruspex]

I got the right answer

Good job.

 

[HalfLostAndSearching]

There are a few things to consider in this problem. Firstly, the specific heat capacity values given are in J/K*kg, so they need to be converted to J/K in order to use them in the equation for heat capacity. This means that the specific heat capacity of copper is actually 2406.25 J/K and the specific heat capacity of iron is 4041 J/K.

Secondly, when the two blocks come to thermal equilibrium, the final temperature will not be exactly 600 K. The final temperature will be somewhere between 400 K and 800 K, and we cannot assume that it will be exactly in the middle. Thus, the change in temperature for both blocks needs to be calculated separately.

Using the formula for heat capacity, we can set up the following equation:
heat capacity of iron * change in temperature of iron = heat capacity of copper * change in temperature of copper

Substituting in the values we have, we get:
4041 J/K * (Tf - 800 K) = 2406.25 J/K * (Tf - 400 K)

Solving for Tf, we get Tf = 600 K, which is the final temperature that we assumed in our initial attempt at a solution.

Therefore, the change in temperature for the iron block is 600 K - 800 K = -200 K, and the change in temperature for the copper block is 600 K - 400 K = 200 K.

Now, using the specific heat capacity values in J/K (as mentioned above), we can calculate the amount of energy transferred from the iron to the copper:
Energy transferred = heat capacity of copper * change in temperature of copper
= 2406.25 J/K * 200 K
= 481,250 J

Thus, the amount of energy transferred thermally from the iron to the copper as they come to thermal equilibrium is 481,250 J.