Entropy Of Mixing and Specific Heat

[Point Conception]

Homework Statement

100g of lead, specific heat .0345 cal/g/C at 100 C is mixed with 200 g of water at 20 C.
Find the difference in entropy of system at end from value before mixing.
Note: I think the book has wrong answer of 1.42 cal/K

Homework Equations

Before mixing entropy for water = 4000 c/293 K = 13.65 c/K
Entropy for lead = 100g * .0345 c/g/C * 100C = 345 c/373K = .925 c/K
Total before mixing = 14.57c/K

The Attempt at a Solution

The lead loses 276 calories going from 100 C to 20 C
So now the water temp is 4276 c/ 200g * 1c/g/C = 294.38K entropy =
4276 c/ 294.38K=14.52c/K
For the mixed in lead = 69 c/.0345 c/g /C= 2000gC /100g= 293K S = 69c/293k = .23c/K
So total after mixing = 14.75c/K
ΔS = 14.75c/K - 14.57c/K = .18c/K
I think the book got the wrong answer with : 4345 c/293 K = 14.82 c/K
And 345 c/293K = 1.177 c/K for total 15.99 c/K - 14.57c/K = 1.42 c/K

 

[haruspex]

The lead loses 276 calories going from 100 C to 20 C

Will the lead go to 20C?

 

[Point Conception]

Will the lead go to 20C?

No it will go to the water temp. 294.38K, 21.38 C losing 271.2 cal, Did the lead first lose 276c then gain back 4.8 cal ? - Some confusion on this mixing of specific heats since I am reading that the lead is in powder forum.
78.62 C * .0345 c/g/C * 100 g = 271.2 cal. For entropy value of 73.8 c/294.38K =.25c/k This still does not get close to the book answer of ΔS = 1.42 c/K for after mix - before mix. By the way the book only gave answer. I just plugged in those values in trying to determine how they got that by error. But it gave the after mix value 15.99 c/K - before mix 14.57c/K = 1.42c/K (book)
But it looks wrong since the 345 calories are applied twice

 

[haruspex]

I get a different answer again. I don't believe this is valid:

Before mixing entropy for water = 4000 c/293 K = 13.65 c/K

You cannot say there is some absolute quantity of entropy in a body. You can only discuss changes in entropy. If a body with heat capacity H cal/C drops in temperature from T to T-δT (absolute) then it loses HδT of heat and HδT/T of entropy. Integrating, ΔS = H ln(T1/T0). If I apply that formula to this problem I get a net increase in entropy of 0.107 cal/K.

 

[Point Conception]

I get a different answer again. I don't believe this is valid:

You cannot say there is some absolute quantity of entropy in a body. You can only discuss changes in entropy. If a body with heat capacity H cal/C drops in temperature from T to T-δT (absolute) then it loses HδT of heat and HδT/T of entropy. Integrating, ΔS = H ln(T1/T0). If I apply that formula to this problem I get a net increase in entropy of 0.107 cal/K.

It was my understanding that in a case like this for water and lead that S = Joules/K
So I don't understand why 4000c/293K & 345c/373 K are not valid before mix.
After the mix there is a total of 4345 cal in 300 gr. mixture.
With 200 g having C_p =1 c/g/C and 100 gr with C_p = .0345 c/g/C
This problem and answer of S₂ - S₁ = 1.42 c/K are from my Halliday Resnick textbook.
Maybe the problem is on what the temperature is in S₂ from above.
Could you work that ? thanks

 

[rude man]

Luckily I happen to have the Resnick & Halliday text. I looked up the problem and its answer. The OP correctly described the problem but the answer is +0.11 cal/K.

I computed +0.13 cal/K but that is undoubtedly due to my approximate temperature calculations. So all is well.

 

[haruspex]

Luckily I happen to have the Resnick & Halliday text. I looked up the problem and its answer. The OP correctly described the problem but the answer is +0.11 cal/K.

I computed +0.13 cal/K but that is undoubtedly due to my approximate temperature calculations. So all is well.

So the OP quotes the wrong book answer? Different edition perhaps?
0.11 fits with my 0.107, so are we agreed that the ln() equation I derived is the right one to use?

 

[rude man]

So the OP quotes the wrong book answer? Different edition perhaps?
0.11 fits with my 0.107, so are we agreed that the ln() equation I derived is the right one to use?

Yes.

I checked the answer carefully, and it's 0.11. I assume the OP just picked the wrong problem answer although none of the answers in the relevant chapter (Chapt. 25 in my edition) were anything near what the OP quoted (1.42).

 

[Point Conception]

Yes this is chapter 25 problem #15 in a very early combined 1st & 2nd edition 1960.
I got in a used bookstore in Thailand. Since I am out of the U.S. The correct answer in book .1cal/K. was in error in the #17 answer place.
With T₂ = 294.36 K I got .11c/K for S₂-S₁ = mC_p ln T₂/T₁

Problem # 17 : A brass rod is in contact thermally with a heat reservoir at 127°C at one end
and 27°C at the other end. What is ΔS for the conduction of 1200 calories through the rod,
end to end ? Is this correct ? ΔS = (1200 cal)( ln 400K/300K) = 345c/K
Not close to 1.42 c/K ?

 

[haruspex]

ΔS = (1200 cal)( ln 400K/300K) = 345c/K

No, this is different. The entropy change is in general ∫dQ/T. In the OP, T varied with Q, so you got the ln() formula. This time T is constant.

 

[rude man]

What is delta S for the cold reservoir?
What is delta S for the hot reservoir?
What is delta S for the brass bar?
Add them up!

You need to think about why this is not a dQ/T integration! More importantly, you need to think about why the other problem was an integration. What were you physically doing in the other problem? Remember, the integral has to be along a reversible path.

PS I did not get 1.42 cal/K for this problem either. @Haruspex?

 

[haruspex]

PS I did not get 1.42 cal/K for this problem either. @Haruspex?

No, but at least it's the right order of magnitude this time.

 

[rude man]

No, but at least it's the right order of magnitude this time.

Yay team - sort of.

 

[Point Conception]

1200 calories = Q
T₁ = 400K
T₂ = 300K

Q/T₂- Q/T₁ = 1c/K ΔS

If this is correct then book has #15 & #17 in error

 

[haruspex]

Q/T₂- Q/T₁ = 1c/K ΔS

That's what I get.

 

[rude man]

1200 calories = Q
T₁ = 400K
T₂ = 300K

Q/T₂- Q/T₁ = 1c/K ΔS

If this is correct then book has #15 & #17 in error

Bravo!

 

[Point Conception]

was[/I] an integration. What were you physically doing in the other problem? Remember, the integral has to be along a reversible path.

The first problem was integration since ΔS = ∫₁² = dQ/T with T varing is based on ∫₁² 1/x dx = ln x₂/x₁
The second problem with T constant = 1/T ∫₁² dQ = Q/T

Question : In the first mixing problem with a plot of Q on y-axis and T on x-axis
and ΔS the area under curve. Would this plot look like the inverse y=1/x curve ?
By the way I did not mean to imply that the 1960 text was my text in 1960

 

[haruspex]

with a plot of Q on y-axis and T on x-axis
and ΔS the area under curve. Would this plot look like the inverse y=1/x curve ?

Since S = ∫dQ/T, for S to be the area under the curve the natural arrangement would be y = 1/T, x = Q. Since, in the first question, Q = kT then yes, that would give a curve like y = 1/x.