How Do You Relate Entropies to Temperatures in Water Mixing?

[gfd43tg]

I don't know how to relate the entropies, S1 and S2, to the temperatures in my entropy balance

 

[rude man]

Can't say I understand your attempt.

In order to determine entropy differences you have to come up with one or more reversible processes. Then you can compute the difference in entropies before & after.

Hint: take the hotter water solution and reversibly reduce its temperature from its original temperature to the final temperature of the two solutions. Compute ΔS_hotter.

Do the same for the cooler solution.

You wind up with two solutions of equal temperature, then you can mentally just pour them together without any further changes in any thermodynamic coordinate.

Answer will be ΔS_cooler - ΔS_hotter.

 

[gfd43tg]

I am just doing an entropy balance on my control volume, being the tank where the two liquids are mixed. I just can't find an absolute value of entropy for S1 and S2

 

[rude man]

You never deal with absolute entropies in introductory physics. You deal in differences in entropy between two (or more) states.

So the hotter water's entropy changes by ΔS_h and the colder by ΔS_c. The total change in entropy is ΔS_c + ΔS_h.

 

[gfd43tg]

I was able to solve it. The tricky thing at the end is to get it in the form they want, you have a squared term on top, so you have to take out a square from the top and bottom to make the bottom have the square root in it.

For the part with two masses being different, if I add them together, how can I get one expression for the total entropy? I can get the mass specific total entropy change, but not sure about the total entropy change.

 

[rude man]

I was able to solve it. The tricky thing at the end is to get it in the form they want, you have a squared term on top, so you have to take out a square from the top and bottom to make the bottom have the square root in it.

yes, I noticed that too.

For the part with two masses being different, if I add them together, how can I get one expression for the total entropy? I can get the mass specific total entropy change, but not sure about the total entropy change.

I don't know how exactly you did the math. If you did it the way I suggested it's a straightforward extension of the method. You compute the two entropy changes independently and then add them. You don't need specific entropies. Of course, the final temperature will not be the mean of the two starting temperatures.

Post your math if you want to.

EDIT Sorry, I hadn't noticed mass-specific heat of water already used in part (a). Of course, use it again for part (b). This is usually written with a lower-case c which is what threw me off in the originally provided answer.