Mixing ice, steam, and water and find equilibrium

In summary, the problem at hand involves mixing different quantities of steam, water, and ice and determining the final temperature and phase(s) of the resulting state. It is assumed that no energy is lost to the surroundings. Using the conservation of energy principle, the equations for each substance's change in energy are set up. After solving for the unknowns, it is found that none of the water boils and some of the steam condenses, resulting in a final temperature of 100 degrees Celsius with 0.485 kg of water and 0.065 kg of steam. The problem is then compared to a similar, simpler problem for a better understanding.
  • #1
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Homework Statement


.15 kg of steam at 100 degree Celcius is mixed with .2kg of water at 50 degree and .2 kg of ice at 0 degree. Determine the final temperature and phase(s) of the final state. Assume that no energy is lost to the surroundings.


Homework Equations





The Attempt at a Solution


I am thinking that since energy is conserved, the steam will lose energy while the ice and water will gain.

Therefore

-Q_steam + Q_water + Q_ice = 0

I know Q = mc*deltaT

The answer key I have says that the end result will be 100 degree with .485 kg water, and .065 kg steam. I would try to test if the all of the steam and ice condense and melt if I didn't know this.

So ice will all melt

Q_ice = .2*2100*(T-0) + .2*L_f

Q_water = .2*4190*(T-50)

Q_steam = .15*c*(T-100) + x*L_v

There are 3 possibilities, 1) none of the water boils 2) some of the water boils 3) all of the water boils. In this case, it is option 2. T must be 100

Q_ice = .2*2100*100 + .2*3.34*10^5 = 108800
Q_water = 41900
Q_steam = x*2.256*10^6

x = .0667 kg of steam turns into water

If this is the case, I have 1.43 kg of steam left and .4667 kg of water after the ice melt. This answer is however incorrect. Where did I go wrong?
 
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  • #2
i don't remember the specific values but for the heat capacities - did you use the values which had units J/g C or J/g K??

If you used the latter then you should have converted all your celcius figures to kelvins.

p.s. none of the water did boil after all, did it?
 
  • #3
No, if the final answer has .485 kg of water and .065 kg of steam, then none of the water boiled. So it's option 1. In fact, some of the steam condenses.

Qsteam = mLcond
Qwater = (0.2)cwater(50)
Qice = (0.2)Lfus + (0.2)cwater(100)

Wher Lcond is the latent heat of condensation for steam to water, Lfus is the latent heat of fusion (melting) for ice to water, and cwater is the specific heat capacity of water. Note Lcond is a negative quantity. You want to solve for m in the following equation:

mLcond + (0.2)Lfus + 30cwater = 0

It's easy, just plug in numbers and isolate, or better yet isolate then plug in numbers. You then argue that since the value for m is between 0 and 0.15 (you should get this if the book is right), your solution is consistent thus the only possibility. If you want a less ad hoc approach, you can compare

|0.15Lcond|

to

|0.2Lfus|

You should find that the first quantity is much bigger, hence in this set up, all of the ice will melt. So either all the steam condenses, none of it condenses, or some of it does. If all of it condenses, then it takes at least |0.15Lcond| away from the total energy, and the ice and water contribute at most 0.2Lfus + 30cwater to it, and you should find that there's no way in this situation for the total energy to be zero (it will have to be negative). If none of it condenses, well then it loses no energy and the ice and water strictly gain energy, so again there's no way for the total energy to be zero (it will have to be negative). Therefore, some of the steam must condense. Now you're justified in setting up the equations the way I did.
 
  • #4
Thank you very much for the help
 

1. What is the meaning of equilibrium in the context of mixing ice, steam, and water?

Equilibrium in this context refers to a state where the three phases of matter (ice, steam, and water) exist in a stable balance, with no net change or movement between them. This means that the rate at which ice melts and water evaporates is equal to the rate at which steam condenses and water freezes, resulting in a constant temperature and composition.

2. How can we achieve equilibrium when mixing ice, steam, and water?

To achieve equilibrium, the three phases of matter must be in close contact with each other and have a constant exchange of energy (heat) and matter (water molecules). This can be done by placing them in a closed container and allowing them to interact until no further changes occur.

3. What factors can affect the equilibrium between ice, steam, and water?

The factors that can affect the equilibrium include temperature, pressure, and the amount of each phase present. Changes in these factors can shift the equilibrium, causing a change in the relative amounts of ice, steam, and water present.

4. Is it possible to have all three phases (ice, steam, and water) present at the same time in equilibrium?

Yes, it is possible to have all three phases present at the same time in equilibrium. This can occur when there is a constant exchange of energy and matter between the phases, resulting in a stable balance between them.

5. How does the equilibrium between ice, steam, and water relate to the concept of phase change?

The equilibrium between ice, steam, and water is related to the concept of phase change because it involves the conversion of matter from one phase to another. In this case, the three phases exist in equilibrium, meaning that they are constantly undergoing phase changes but at equal rates, resulting in a constant overall composition.

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