So hyperspacepirate(youtube) is doing some cool stuff, and im trying to work out the math.
I've been able to rework the calculations for pressure and tempature easy enough, but for 2 days I tried to rework the algebra to solving for 'n'
until I googled it and read it cant be done algebraicly and requires iterative approximation.
Let me figure out how to write the math all pretty like
$$(P+\frac{an^2}{V^2})(V-nb)= nRT$$
So I punched in my values and equation into wolframalpha and got this nice little graph with 3 solutions.
Can anyone elaborate on the subject?
At first glance, i would guess that these are values for three different phases(gas,liquid,solid).
For my project im trying to calculate how many grams of a gas is required to reach a desired partial pressure within a closed volume in a mixed gas
system.
To be more specific, a mixed refrigerant system.
As the first (higher boiling point) refrigerant is cooled enough to stay in liquid phase in the evaporator, it will reduce pressure in the condencer.
So I want to calculate how much of each refrigerant is required to maintain enough pressure to condence.
[Edited on 27-6-2023 by Rainwater]DraconicAcid - 26-6-2023 at 18:02
If you multiply it out, you get a third order polynomial for n. That will always have three solutions. it has nothing to do with three phases- it's
just math.
Just like a quadratic equation will have two solutions and a linear equation will have one solution.Keras - 27-6-2023 at 00:40
If you multiply it out, you get a third order polynomial for n. That will always have three solutions. it has nothing to do with three phases- it's
just math.
Nah, cubic polynomials can have either 1 or 3 real roots. A real polynomial’s roots are always complex conjugates (easy to verify that if z is a
root, then z̅ is a root too), so they go in pairs, except when they are real (i.e. their own conjugates).
For example (z² + 1)(z - 1) = z³ - z² + z - 1 has only one real root, 1, and two more complex roots, ± i.
[Edited on 27-6-2023 by Keras]Rainwater - 27-6-2023 at 01:56
Do these 3 values give give the same pressure, temperature and volume?
Because that breaks my brain. What im seeing, mathematically. Ya, but no way.
Ok, dumb moment again. I see what i was doing wrong. So simple I missed it.
I wasnt solving for n, i was solving for nRT. About 4 days wasted
If I simplifie the equation to link
$$\frac{(P+\frac{an^2}{V^2})(V-nb)}{RT}= n$$
Then I still get 3 different answers, no way
Edit:using nitrogen gas at in this example, 111.55 kelvin, 15.799 atm
[Edited on 27-6-2023 by Rainwater]
To clarify, can anyone point me to an example that shows how the number of moles is calculated?
[Edited on 27-6-2023 by Rainwater]Keras - 27-6-2023 at 05:20
Get a good book on physical chemistry. Atkin's is one of the best.Rainwater - 27-6-2023 at 07:00
Get a good book on physical chemistry. Atkin's is one of the best.
Thank you for the reference.
So after some googleing, i discovered at those temps and pressures,
what van der waals is predicting is the phase transition to a liquid.
As more mass is added, it is changed into a liquid maintaining the pressure and tempature,
but now there is ratio of liquid/gas in the appratus
Essentially altering the volume and moles in the gas phase to maintain pressure and tempature until a critical point is reached when pressure must
increase
By increasing the temperature above the condensation point of the gas, only 1 solution is valid. As the temp approaches this point a range of
solutions become valid do to the liquid/vapor ratio
