Measuring partial molar volumes

[Brutulf]

Greetings.

 

Yesterday I performed a lab expIeriment in physical chemistry to determine the partial molar volumes for mixtures of water and acetone. I think I'm beginning to wrap my head around the theory, but there is one thing which I'm having a bit of trouble with. In the lab notes, a formula for the molar quantity of each component in a given mixture was given, along with the task to prove it. I'm having a hard time doing that, also I suspect I might be suffering from some basic misunderstanding about what exactly is meant by some of the variables (the lab notes can be somewhat unclear on defining them, IMHO).

 

The formula given for a two-component system is:

 

[math]n_{1} = \frac{kW}{V_{1}^{*}(k\rho_{1}+\rho_{2})}[/math]

 

[math]n_{2} = \frac{W}{V_{2}^{*}(k\rho_{1}+\rho_{2}}[/math]

 

 

where

 

-n is the amount in moles of each component

-W is the total mass of the mixture (which we measured using a pyknometre)

-k is the relation between the measured volumes of the components (a fraction)

-rho 1 and 2 is the densities of the components

-V₁^* and V₂^* is the partial molar volumes of the pure components 1 and 2. (that is, just the molar volume of pure water and pure acetone respectively, so -V₁^* would be the molar mass of water divided by its density...-I think?)

 

If anyone could take a shot at explaining this, it would be greatly appreciated.

 

Edit: Just a note on the procedure we used. We measured out various volumes of water and acetone which we mixed, then we weighted these solutions in a pyknometre, so we got mass readings for the same volume of the various mixtures. So, we can detirmine "V unmix" if we know the amount in moles of each component in the mixture in the pyknometre, which the equations above should help with...

 

Edit 2: Ok, I checked out that the equation gives the right amount of moles. A justification would still be useful though.

 

Note: I posted this same thread on the xkcd.com science forum earlier. I guess this site might have some more traffic though.

[vedmecum]

can you please write in short what actually you want ?

[CaptainPanic]

What is the actual question? We ask not because we're lazy to answer, but because asking the right question leads to the right answer.

 

A molar volume has the units m3/mol. It is the volume of 6.022*10^23 molecules of a particular substance...

 

[math]n_{1} = \frac{kW}{V_{1}^{*}(k\rho_{1}+\rho_{2})}[/math]

-n is in [mol]

-W is in [kg]

-k is dimensionless

-rho 1 and 2 are in [kg/m3]

-V1* and V2* are in [?]

 

Checking the formula so far, we can actually derive the units of the V1 and V2, using a dimension analysis.

 

[math][mol] = \frac{[-][kg]}{V_{1}^{*}([-][kg/m3])}[/math]

 

Formulas always have the same units before and after the = sign.

The right side of the formula can only be in [mol] if you give the units [m3/mol] to the V1.

 

Quite possibly, you use imperial/UK/US units. You can convert it all yourself.

[Brutulf]

What is the actual question? We ask not because we're lazy to answer, but because asking the right question leads to the right answer.

 

 

Ok. The question - or rather, task - is to derive/prove the equation(s). I guess this boils down to algebraically manipulating known amounts: Molar mass, density, sample mass, etc. Still, I was never any good at algebra, and at a loss as to where to start/attack the problem from. :/

 

Formulas always have the same units before and after the = sign.

The right side of the formula can only be in [mol] if you give the units [m3/mol] to the V1.

 

Quite so. Dimensional analysis posed no problem, and the equation also evidently gives the correct answer, but it is not so evident to me, why/how?

 

Quite possibly, you use imperial/UK/US units. You can convert it all yourself.

 

Thank the heavens, no! SI all the way, baby.

 

And to summarize: All the following are known, experimentally:

 

-Total sample volume after mixing

-Total sample mass ('W')

-The ratio between the volumes of component 1 and 2 that were mixed (for example, for the first mixture we measured out 1 mL water and 29 mL acetone, so k = 1/29, when water is component 1)

-The densities of the components

-The molar volume of the components

-The molar mass of the components

 

Now, how can I go about using these to construct the expression above?

 

Thanks for the reply, by the way.

[Brutulf]

Phew, I figured it out (with some help). So just for reference:

 

Starting with a mass balance for the measurement-vessel, we have:

 

[math]

W_{tot} = \rho _1 V_1 + \rho _2 V_2[/math]

 

where V1 and V2 are the measured volumes before mixing, and multiplying on both sides by

[math]

\frac{{V_1}}{{V_2 }} = k

[/math]

 

we get

[math]

W_{tot} k = \rho _1 V_1\frac{{V_1}}{{V_2}} + \rho _2 V_2 \frac{{V_1 }}{{V_2}}

[/math]

 

 

[math]

W_{tot} k = \rho_{1}V_{1}k + \rho_{2}V_{1}

[/math]

 

 

[math]

W_{tot} k = V_1 \left( {\rho _1 k + \rho _2 } \right)

[/math]

 

We thus obtain the volume before mixing of component 1:

 

 

[math]

V_1 = \frac{{W_{tot} k}}{{\left( {\rho _1 k + \rho _2 } \right)}}

[/math]

 

 

and finally, dividing the volume before mixing with the molar volume, we get the molar amount:

 

[math]

n_1 = \frac{{V_1 }}{{V_m }} = \frac{{W_{tot} k}}{{V_m \left( {\rho _1 k + \rho _2 } \right)}}

[/math]

 

 

 

 

Hooray!