Change in Entropy When Mixing Water at Different Temperatures

In summary, the change in entropy when mixing water at different temperatures is a result of the redistribution of thermal energy between the warmer and cooler water. As the two bodies of water mix, the overall entropy increases due to the increased randomness and disorder in the system. This process aligns with the second law of thermodynamics, where entropy tends to increase in isolated systems. The final equilibrium temperature reached after mixing reflects the balance of energy and the resultant change in entropy can be calculated using thermodynamic principles.
  • #1
domephilis
54
6
Homework Statement
Exercise 11.30 Find the change in entropy if 500g of water at 80C is added to 300g of water at 20C. (Fundamentals of Physics I by R. Shankar)
Relevant Equations
After re-reading the book, I did figure out what I was supposed to do. Take both waters through a series of reservoirs to bring them down to their final temperature while allowing for a quasi-static process. Thus, . And I did get the correct answer, 3.2 cal/K.

But, before that, I tried a different method; and, my question is why I'm wrong. The method is as follows. I also thought that I needed a reversible (or quasi-static) process. So I imagined the hotter water slowly dripping into the colder water while allowing equilibrium at almost every moment. Then, . T(m) can be easily determined through some calorimetric work. We can then integrate both sides to get the change in entropy. (Units are in calories, grams, and Kelvin. I have omitted the specific heat because it is one for water.) Hence, . A similar logic applies to the change in entropy of the hotter water being cooled down by the cold water. Here, I assume that entropy is additive (which was given in the book). The final answer I got was 14.77 cal/K. Setting aside the gory details of the calculations, is there anything wrong with the logic?
 
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  • #2
Your alternative path is not reversible. Can you see why?
 
  • #3
That’s my confusion. I added the water in slowly allowing for equilibrium at almost every moment. If I wanted to reverse a step, I can just pour that drop back in. I am mimicking the quasi-static process described in the book for a gas (i.e. slowly take out grains of sand which compresses a piston while being at equilibrium after each step …).
 
  • #4
domephilis said:
That’s my confusion. I added the water in slowly allowing for equilibrium at almost every moment. If I wanted to reverse a step, I can just pour that drop back in. I am mimicking the quasi-static process described in the book for a gas (i.e. slowly take out grains of sand which compresses a piston while being at equilibrium after each step …).
I’m not sure this is right, but here’s my guess. When I pour the hot to the cold, I heat the cold water somewhat and gets warmer water. When I try to reverse it by pouring the water back, that bit back is not the 80C that it used to be and will cool the hotter water. Hence, we end up with a result different than our original. Does that sound right?
 
  • #5
domephilis said:
I’m not sure this is right, but here’s my guess. When I pour the hot to the cold, I heat the cold water somewhat and gets warmer water. When I try to reverse it by pouring the water back, that bit back is not the 80C that it used to be and will cool the hotter water. Hence, we end up with a result different than our original. Does that sound right?
Yes. Very good.
 
  • #6
Chestermiller said:
Yes. Very good.
I see. Thank you very much. Have a nice day.
 

FAQ: Change in Entropy When Mixing Water at Different Temperatures

What is entropy and how does it relate to mixing water at different temperatures?

Entropy is a measure of the disorder or randomness in a system. When mixing water at different temperatures, the overall entropy of the system usually increases. This is because the mixing process leads to a more uniform distribution of energy among the molecules, resulting in higher disorder and thus higher entropy.

How do you calculate the change in entropy when mixing two volumes of water at different temperatures?

The change in entropy (ΔS) when mixing two volumes of water can be calculated using the formula: ΔS = m * c * ln(T_final / T_initial), where m is the mass of the water, c is the specific heat capacity, T_final is the final equilibrium temperature, and T_initial is the initial temperature of the water. You would calculate the change in entropy for each volume of water and then sum these values to get the total change in entropy.

Does the final temperature of the mixed water affect the change in entropy?

Yes, the final temperature of the mixed water does affect the change in entropy. A larger difference between the initial temperatures of the two water samples generally leads to a greater increase in entropy upon mixing. This is because the energy distribution becomes more random and disordered as the two temperatures equilibrate.

Is the change in entropy when mixing water always positive?

In general, the change in entropy when mixing water at different temperatures is positive. This is because the process of mixing leads to an increase in disorder. However, if you were to consider a hypothetical scenario where mixing results in a more ordered state (which is not typical for water), the change in entropy could be zero or negative, but this is not observed in practical situations.

What role does the specific heat capacity of water play in the change in entropy during mixing?

The specific heat capacity of water plays a crucial role in determining the amount of heat absorbed or released during the mixing process. Since specific heat capacity indicates how much heat is required to change the temperature of a substance, it influences the final temperature achieved after mixing and subsequently affects the change in entropy. Higher specific heat capacity means that water can absorb more heat without a significant change in temperature, which can lead to more pronounced entropy changes during mixing.

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