Gravity to Molarity question

[xico]

Is there a way to calculate between gravity or deg. Plato and moles?

I'm taking a chemistry lab and I would love to find ways to apply what I'm doing for a grade to what I love. Are there any experiments that you recommend for a learning chemist and brewer?

Thanks!

 

[bu_gee]

Sucrose adds 384 points per kilogram per liter. From there it should be a bit of simple math and light chem.

 

[ajdelange]

Degrees Plato is the strength of a sucrose solution of the same specific gravity as the wort/beer by weight in percent. Thus a 10 °P wort has the same specific gravity (1.0403) as a sucrose solution of 10 grams sucrose and 90 grams water (100 grams total). A mole of sucrose weighs 342.3 grams so 100 grams of a 10 °P sucrose solution contains 10/342.3 = 0.0293 moles (and is thus 0.0292/.09 = 0.325 molal - moles per kg of solvent) and 1 kg contains 100). As noted above the SG of a 10 °P solution is 1.0403 so its density is 0.998203*1.04003 = 1.03816 and a kg has volume 1/1.03816 = 0.962992 L. The density of water at 20 °C is 0.998203. Thus the molarity is 0.998203*1.04003/1.03843 = 100*.998203*1.04003/342.3 = 0.30328 M and the molality 10/90 = 0.11111

If that solution inverts then we have 100/180.16 moles each of glucose and fructose in 1 kg. With anything other than sucrose things are approximate as other sugars have specific gravities which are very close to but not exactly the same as that of sucrose and we have to take into account that in hydrolyzing each mole of sucrose (or any other dissacharide) kicks off a mole of water. Thus if you put 342.3 grams of sucrose (1 mole) into 1 kg of water and it inverts you have 1 mole glucose, 1 mole fructose, 1 mole of water (18 grams) from the inversion and the original 1 kg of water. The solution molality is not 1/1 = 1 but 1/1.018 = 0.982 with respect to each of glucose and fructose.

The only tricky part is figuring out how a 10 °P solution has specific gravity 1.04003. This is done 'officially' through the ASBC polynomnial which is a fit to the ASBC Tables derived from the Plato Commission tables. It is

°P = ((135.997*SG -630.272)*SG +1111.14)*SG -616.868

This gives °P in terms of the apparent specific gravity (20 °C/20°C) but we want it the other way round. There is no official inverse polynomial but one can use approximate formulae (the Lincoln equation). Best to just plug the ASBC formula into an Excel spreadsheet and play with SG until it calculates the °P you want or better still let the Solver do that for you automatically.

 

[rosenbrewery]

As stated above me, degrees Plato are an expression of the percent mass of sugar in the wort.

1 ºPlato = 1% sugar by mass = (1 gram of sugar)/(100 grams of wort)

Likewise, if you have wort at 10 ºPlato (approximates to 1.040 OG...nice for a session ale) then you'll have 10 grams of sugar per 100 grams of wort.

So to calculate the molarity of this hypothetical 10 ºPlato beer, all we have to do is some dimensional analysis.

Beer is a complex solution of many organic and inorganic substances, including multiple types of sugars, so to simplify things, I'm going to focus on maltose.

The molar mass of maltose = 342.3 g/mol

10 grams of maltose * (1 mol of maltose / 342.3g) ≈ 0.02921 mols ≈ 0.029 mols of maltose per 100 grams of wort

Now, using our specific gravity, which is just the density of the wort compared to water, we can calculate the volume in mL per 100 grams of wort


(Specific Gravity) = (Density of solution)/(Density of water)


1.040 (SG) = (Density of solution)/(Density of water) ---> 1.040 * 1mg/mL = Density of solution = 1.040 grams /1mL


100 g of wort * (1 mL/1.040g) = 96.1538461538 mL ≈ 96.15 mL (ignoring significant figures here)


Now putting it all together:

(0.029 mols of maltose)/(100 grams of wort) * (100 grams of wort / 96.15 mL) * (1000 mL / L) = 0.3016 mols of maltose per L of wort, or as us science nerds would say, we have a 0.3016 molar solution of maltose

Using all of this information, you could convert to gallons and scale up to a 5 gallon batch... figure out the total number of mols per 5 gallon batch... convert to grams of sugar per batch...

 

[ajdelange]

I left a whole lot of important stuff out last night and have gone back and edited it in. Perhaps the most important thing, important enough to justify a separate post, is that you can't compute the molarity of a wort from it's SG or °P because a wort isn't a sucrose solution but rather a spectrum of sugars, maltose, rafinose, glucose, fructose, sucrose.... We found in #3 that a 10 °P sucrose solution is 0.30328 M/L. The best we can say with accuracy about wort which measures 10 °P is that it has the same specific gravity as a 0.30328 M sucrose solution. We could reason to ourselves that a 10 °P sucrose solution contains 2*0.30328 M 6-carbon sugar units if uninverted and about 2% less than that if inverted. As other dissacharides, trisaccharides and even soluble starches approximately follow the Plato table we could then say that approximately the molarity of a 10 °P wort is 0.60656 moles equivalent 6-carbon sugar per liter.

 

[bu_gee]

<Blink>

1000g/342g/mol = 2.92 mol.

1 kg of sugar in 1L is 2.92 molar with an SG of 1.384.

384/2.92 = 132. So a 1 molar sucrose mix has an SG of 1.132.

While I'm not certain of the pkl (ppg in si) of anhydrous maltose, you're almost certainly in the same ballpark and you might even be within the precision of the available significant figures.

You just can't get anhydrous maltose so you can't do the same easy calculations.

Your question leads to a tough slog in general because you're talking in mass per volume whereas the hobby is generally interested in volume per volume.

 

[ajdelange]

1000g/342g/mol = 2.92 mol.

I think you have missed the point. It is not a question as to what any particular sugar concentration calculates out to but rather as to how to interpret that. I am not referring to water of hydration here but rather the water that is released when the 1-6 bond is broken. Sucrose has a molecular weight of 342.3, glucose and fructose 180.16. Note that 2*180.16 - 342.3 = 18.02, the molecular weight of a water molecule. As we cannot be certain of the extent of inversion of sucrose (of which there isn't much in wort anyway) we are uncertain about the amount of solvent added to the mix by inversion. Similarly there is a question as to how much of maltose has been hydrolyzed at any give time.

1 kg of sugar in 1L is 2.92 molar with an SG of 1.348.

The apparent SG (20/20) of a 1 kg sucrose in 1 L of solution is 1.37026 (20 °C) while for glucose it is 1.36032, for fructose it is 1.36758 and for invert sugar 1.36429. All close to one another certainly but not exactly the same. Also note that the molarities of the glucose and fructose solutions are close to, but not exactly, twice the molarity of the sucrose solution.

348/2.92 = 119. So a 1 molar sucrose mix has an SG of 1.119.

A 1 M sucrose solution would contain 342.3 grams of sucrose per liter and would have apparent SG (20/20) of 1.13079 but I'm not sure what point you are trying to make here. Brewers don't work with solutions this strong (all SG numbers I've given here come from the 'official' ICUMSA polynomial). The point I was trying to make is that bond water and differences in SG vs. w/w strength among sugars make it impossible to precisely determine the number of moles of hexose in a solution with a give °P value. Not to mention that not everything in 'extract' is sugar.

While I'm not certain of the pkl (ppg in si) of anhydrous maltose, you're almost certainly in the same ballpark and you might even be within the precision of the available significant figures.

You just can't get anhydrous maltose so you can't do the same easy calculations.

I don't know what this means either. We aren't talking about adding maltose to anything but rather taking a wort, measuring its specific gravity and trying to draw conclusions about the moles of something it might contain.

Your question leads to a tough slog in general because you're talking in mass per volume whereas the hobby is generally interested in volume per volume.

Again confused because I didn't actually pose a question. The brewing community is concerned about extract. Plato and SG are sufficient for management of that.

 

[ACbrewer]

OP: you might find this useful
http://en.wikipedia.org/wiki/Brix

There are at least 3 scales used (Gravity/density, Brix and Plato). I personally like Brix as the best because it does a weight%/weight - as in [sugar mass]/[total mass] and should be temp independent. Plato is also weight%/weight and should be identical to Brix, but for some reason about the 3rd significant figure they differ. My personal preference could be that I learned SG for Beers and began working some with Brix for wines (although I usually use SG there also)

 

[ajdelange]

The reason the Kaiser established the Plato Kommission is because the Brix scale was known to be innacurate in the 4th decimal place and beyond. Plato did exactly what Brix did, i.e. measure the specific gravity of solutions of known strength w/w but he did it at difference ref temperature and did it more carefully. The Plato tables go out to 6 decimal places. I think I put the details in the Wikipedia article. Unless you are using an oscillating tube densitomer or pycnometer either scale should serve you well but if you are an accuracy freak use the Plato scale. The brewing industry does.

 

[bu_gee]

I wasn't talking about hydrolyzed disaccharides at all. I'm making a rough guess as to the gravity of a 1M disaccharide solution (mainly maltose in beer) based on the only anhydrous disaccharide I have available to me (sucrose).

I apparently decided to use 348 in my calculations instead of 384 because I'm apparently dyslexic.

It is good you're confused because I didn't say you asked a question. I was referring to the OPs question.

 

[ACbrewer]

The reason the Kaiser established the Plato Kommission is because the Brix scale was known to be innacurate in the 4th decimal place and beyond. Plato did exactly what Brix did, i.e. measure the specific gravity of solutions of known strength w/w but he did it at difference ref temperature and did it more carefully. The Plato tables go out to 6 decimal places. I think I put the details in the Wikipedia article. Unless you are using an oscillating tube densitomer or pycnometer either scale should serve you well but if you are an accuracy freak use the Plato scale. The brewing industry does.

Ok from what I read on Wikipedia and what you have here, the definition of 1 deg Plato, Brix and the other systems is w%/w of sucrose in solution (usually 1 deg = 1gram in 100 total).
What the difference is comparing these numbers back to SG and they all have a little variance there.

 

[ajdelange]

What the difference is comparing these numbers back to SG and they all have a little variance there.

If you can put that in English I'll try to respond.

 

[ACbrewer]

Ok from what I read on Wikipedia and what you have here, the definition of 1 deg Plato, Brix and the other systems is w%/w of sucrose in solution (usually 1 deg = 1gram in 100 total).
What the difference is comparing these numbers back to SG and they all have a little variance there.

If you can put that in English I'll try to respond.

What I meant was the definition of each is the same (1deg is 1 gr sucrose in 100 gram solution -99gr water).
The difference in values as they relate to SG is that P to SG or Br to SG were experimentally driven and limited by the measuring devices (both temp and hydrometer) of the time.


NOTE:
This is why it would make more sense to do recipes in weights/masses than in volumes. I mean most flour recipes actually call for sifted flour which has a different volume than unsifted. And most brewers by now should know that a cup of sugar doesn't have consistent weight. Not much of a problem at 1cup, but with skeeter pee calling for 16 cups or 7#, it makes more sense to put in 7#

 

[ajdelange]

Yes, that makes much more sense not. All three (Balling, Brix, Plato) made solutions of known strength (x grams of sucrose, y grams of water ==> 100*x/(x + y) °). This is not a trivial task. The sucrose must be dry and one has to work very quickly so that water doesn't have a chance to evaporate. When I do it I tare a volumetric flask, add the sucrose and weigh then add water to a bit below the mark, slap in the plug, atemperate, make up to the mark and weigh again. The first measurement gives me x, the second x + y. I have never been able to make a 10.000 °P solution but I have gotten pretty close.

Part 2 involves measuring the specific gravity. Again this is not trivial and involves weighing a tared pycnometer filled with very pure water, getting absolutely every molecule of that water out and then weighing the test solution. You don't measure SG to 6 decimal places with a hydrometer. There are choices as to whether you want apparent or true SG and what reference temperature you use. It is a relatively easy matter to interconvert among these. The ASBC table is mathematically converted from the Plato table, for example.

Some us do actually work in weight rather than volume.

 

[ACbrewer]

Some us do actually work in weight rather than volume.

Most of us work in both.... X# of malt/honey/sugar/fermentable and then we have that in Y gallons/liters of wort/must

and 10.000 P would be 5 sig figs, not a simple task by any measure. Generally in most applications, that would be considered 'rounding error'... although I'm aware there are other applications that require a much tighter tolerance.

 

[ajdelange]

Most of us work in both.... X# of malt/honey/sugar/fermentable and then we have that in Y gallons/liters of wort/must

Clearly but actually it was you who suggested that weight might be more convenient. I have load cells under my kettle so at the conclusion of a boil (in addition to knowing how many pounds of water were evaporated) I have the mass of the wort which, multipled by °P and divided by 100 gives me pounds extract with no further effort. In any event if you want to know extract you must know the density of the solution as both °P, from which w/w concentration is determined depends on that as does mass which you can get by a volume calculation or a simple weighing. Recall that in the early days of British brewing mashing efficiency was measured in 'brewers pounds per quarter' which, IMO, has it all over 'furlongs per fortnight' as an example the English people's ability to come up with obscure ways of measuring things.

and 10.000 P would be 5 sig figs, not a simple task by any measure. Generally in most applications, that would be considered 'rounding error'... although I'm aware there are other applications that require a much tighter tolerance.

In this application it is the specific gravity that was measured by Plato's people to six decimal places though there is little point in doing that if the strength of the solution is not known to at least that level of significance. It is certainly easy enough to weigh out 10.0000 grams of sucrose on a decent laboratory balance but it must be kept in mind that this is the apparent weight and the same must be done in the pycnometer weighings but the calculations are straightforward enough. What is difficult is being sure that there is no moisture pickup nor evaporation from samples as they are being prepared and placed in the pycnometer (or today, densitometer). I would not have wanted to work for Leo Plato!

 

[ACbrewer]

Not convenient, more sense for purposes of accuracy... however, after about 3 sig figs more accuracy is not needed for most measures. Yeah landing on the Moon, Mars orbit insertion requires more, but we are boiling 5 gallons of liquid... ok ~19Kg (1L), and at 3 sig figs we are accurate to the gram, at 5 we are accurate to 10mg.... I can see BMC, heck even Boston Brewing wanting to keep track of things to 4 or maybe 5 figures, they are talking $$$ at that point.

Plus I'm not sure we have an easy way of doing %w/w... but we do have a way of getting density fairly easily to about 3 decimal places.

It is pretty cool you have the ability to measure everything in your system by weight er mass.

 

[ajdelange]

Plus I'm not sure we have an easy way of doing %w/w... but we do have a way of getting density fairly easily to about 3 decimal places.

° Plato, easily calculated from density (ASBC polynomial, Lincoln equation), is w/w%

 

[bu_gee]

It all goes back to Ray's Rule of Precision: Measure with a micrometer, mark with chalk, cut with an axe.

I measured out 200.g of sucrose and 1800.g of water to presumably make a 10.0% w/w mix. The hydrometer reading came out at 1.039 SG / 9.5 brix. I made sure my mix was right at 60°F as was the calibration of my hydrometer.

Which piece of my equipment sucked? All of it. Despite all the weighing allegedly being precise to 3 sig figs, it wasn't. Plus, where do you count sig figs on the SG scale? Is 1.039 really 2 figures or is it 4? 9.5 is definitely 2 on the brix scale.

3 sig fig measurement on the front end should easily be able to yield 2 on the back.

So where does this leave the average home brewer? Not caring. At the end of the day, I only care about SG to calculate my ethanol content v%/v.

 

[ajdelange]

Given that you are able to measure weight to 1 ppt (part per thousand, 3 digit significance) the uncertainty in the strength of the mix is about 0.0127 °P (1.27 ppt) corresponding to an uncertainty of 0.000053 sg so clearly your hydrometer is the major source of error (assuming you balance is really as good as 1 ppt).

Now if you believe that the formulas used by home brewers to calculate ABV as a function of OG and FG (AE) are realistic, that is when you are really dreaming.

 

[bu_gee]

Now if you believe that the formulas used by home brewers to calculate ABV as a function of OG and FG (AE) are realistic, that is when you are really dreaming.

Considering the crappiness of the measuring equipment, they are probably of a similar precision.

 

[ajdelange]

They are as precise as the machine with which you calculate the numbers. It's just that they aren't very accurate. The other problem is that the OG you measure as you fill your fermenter isn't the effective OG of the beer as you have either measured before or after adding a starter and some water is swept out along with the CO2. Commercial operators don't compute ABV from OG and FG. They compute OG from ABV and TE.

Also note that no one says you must continue to use the $1.80 hydrometer that came with the home brewing kit your brother in law gave you for Christmas 20 yrs ago. High precision narrow range hydrometers are available at reasonable prices. They can be read to about 0.05 °P and, in my experience with them, are about that accurate.

 

[bu_gee]

They are as precise as the machine with which you calculate the numbers. It's just that they aren't very accurate. The other problem is that the OG you measure as you fill your fermenter isn't the effective OG of the beer as you have either measured before or after adding a starter and some water is swept out along with the CO2. Commercial operators don't compute ABV from OG and FG. They compute OG from ABV and TE.

Also note that no one says you must continue to use the $1.80 hydrometer that came with the home brewing kit your brother in law gave you for Christmas 20 yrs ago. High precision narrow range hydrometers are available at reasonable prices. They can be read to about 0.05 °P and, in my experience with them, are about that accurate.

That doesn't seem like a tool necessary for this hobby, especially considering the problems with the equations and the sugar mix uncertainty you noted before. At an industrial scale, that may be different.

Precision and accuracy are both nice things, but the use still needs to justify the cost and the headache.

 

[ajdelange]

That doesn't seem like a tool necessary for this hobby,

You don't actually need a hydrometer at all. Beer was brewed for thousands of years without them. But I'm guessing that thousands of home brewers use them because I've always got them from home brew suppliers.

...especially considering the problems with the equations and the sugar mix uncertainty you noted before.

There is no problem with sugar mixing as you don't do that in brewing. You mash grain and then measure the amount of sugar extracted from the mash. There is a problem with the ABV = k*(OE - AE) formulas that abound because they don't recognize that k is a function of OE. You don't have to use a lousy formula any more than you have to use a lousy hydrometer. It's just that most home brewers don't care and so choose both. Some of us are interested in the technical aspects of brewing rather than just getting a cheap buzz and so choose more accurate methods and equipment.

At an industrial scale, that may be different.

At the industrial scale hydrometers are generally not considered accurate enough.

Precision and accuracy are both nice things, but the use still needs to justify the cost...

What on god's green earth ever gave you that idea?

...and the headache.

What headache? It's as easy to use well made narrow range hydrometer as it is a cheap wide range one. It's as easy to plug the numbers into an accurate formula as an inaccurate one.

What is a headache is cleaning a U-tube densitometer but I don't think you will find too many of those in use by home brewers.

 

[xico]

Thanks all for the thoughtful responses. I've been on the road for work the last two weeks and haven't been able to give it the time it deserves. The end of this week I will work through it all.