Calculation of density of a mixture

[TCBB]

Hi,

I am a bit confuse right now on how to calculate de density of a liquid mixture. It is because i already saw the same thing calculated in 3 different ways. For example for a 2 components mixture. Component a and b
1 - Dmix = xa * Da + xb * Db
2 - Dmix = 1/(xa * Da + xb *Db)
3 - Dmix = 1/((xa/Da)+ (xb / Db)

Dmix = density of mixture
xa and xb = masss fraction of component a and b
Da and Db = density of component a and b

Can any one clarify which way is the correct one?

Thanks in advance,

 

[25362]

The density of the mixture is, of course, equal to the total mass divided by the volume of the mixture.

As far as I recall (thermodynamics) the volume of the mixture is based on partial molar volumes and it may not be equal to the sum of the volumes of the components.

A classical case is ethanol + water.

Example, at ambient conditions:

70 cc of water + 30 cc of ethanol => 96.8 cc of the mixture.​

See thread798-107637.

 

[chemebabak]

The third one is correct:
D mix = 1/((xa/Da)+ (xb / Db)
D mix = total mass / (mass a / density a + mass b / density b)

Mathematically, it is easier to see:
1 / D mix = 1 / [total mass / (mass a / density a + mass b / density b)]
or
1 / D mix = (mass a / density a + mass b / density b) / total mass

Prove it: mass b = 0; mass b / density b = 0
Then, mass a / density a = volume a
Therefore, total mass / volume a = density of mixture

# 1 is false, but it is a little difficult to see:
Dmix = xa * Da + xb * Db = (ma * Da + mb * Db ) / mtotal
mb = 0, xb=0, xa=1, ma=mtotal, ma*Da = ma^2/Va;
Then Dmix = (ma^2/Va) / ma, not ma / Va.

#2
Dmix = 1/(xa * Da + xb *Db)
xb = 0, xa=1; Dmix = 1/Da - FALSE

---

Vapor density = ntotal * MWmix *RT*Z/P

(na*MWa + nb*MWb)/ntotal is mixture molar weight
n is moles

 

[25362]

IMHO, Chemebabak's selection would be correct only if the sum of A and B volumes equals the final volume of the mixture.:O)

 

[jmw]

As 25362 says, you may have to allow for shrinkage.
If these are hydrocarbons, good luck finding an equation.
Another example is Tomato juice and beer.
.



JMW

http://www.ViscoAnalyser.com

 

[sheiko]

It is indeed difficul to calculate Dmix for non ideal mixtures.

But just a thought.

Suppose that "i" is a component of the mixture.
Then, vi* = Mi/Di, with:
vi* being the molar volume of i.
Mi being the molecular weight of i.

If the mixture is ideal, then v = sumprod(xi, vi*)
If not, then there is an excess volume to add and:
v = sumprod(xi, vi*) + vE
=> vE = v - sumprod(xi, vi*)

I believe you can find experimental data of v (that is Dmix) and vi* (that is Di) at different composition and temperatures, so that you can correlate vE to xi and T.

Good luck.

"We don't believe things because they are true, things are true because we believe them."

 

[jmw]

Incidentally, my calculation (from an excel spreadsheet) is:

E8 = 100/((C9/C8)+(D9/D8))

C9 is the mass % of component A
C8 is the density of component A
D9 is the % mass component B
D8 is the density of component B

Note mine uses % and you quote mass fraction so your number 3 would be right if it had a complete set of parentheses.
Your number 2 is more likely for volume fractions not mass fractions:
E8 = ((C9*C8)+(D9*D8))/100
where C9 is the Volume % component A and D9 is the volume % component B.

So if mass %:
E8 = 100/((10*850)+(90*1010)) = 991.34kg/m3
and if volume %
E8 = ((11.663*850)+(83.337*1010))/100 = 991.34kg/m3

The spreadsheet is here and is protected but no password.

JMW

http://www.ViscoAnalyser.com