What is the optimal amount of water to use for rinsing after brushing teeth?

[RayPhysics]

PHYSICS of Brushing your teeth:
I spent a few weeks a few years ago, five minutes here and there, trying to figure out what is the optimal amount of water that you should put in your mouth to get the maximum rinsing effect after you brush your teeth. Do you fill your mouth half full, 1/3 full or what? If you want to rinse out a bottle with water then how full do you fill it before you shake it? With less water the water has more distance to travel. With more water there is more mass involved. Are you looking for energy transfer or momentum, one involves the square of the velocity and one only varies linearly with velocity. This kind of question actually comes up when you analyze tankers full of fuel and motorcycle design where they look at making high speed turns. The forces are not negligeable.

 

[Bobbywhy]

RayPhysics, Welcome to Physics Forums!

Your question about "What is the optimal amount of water that you should put in your mouth to get the maximum rinsing effect after you brush your teeth?" seems a good candidate for experiment. Have you considered ways you may learn the answer through experiments that you design and then measure the results? Using the scientific method the optimal amount of water to rinse your mouth with may be determined.

Here on Physics Forums there are many Scientists and Engineers ready and willing to assist any true searcher. Some effort on your part to show what you've already tried to discover the answer would be helpful for others here to provide useful suggestions.

Cheers, Bobbywhy

 

[256bits]

You did not say what you are trying to rinse out of the bottle. To clean out a substance such as alcohol from the bottle would necessitate a greater amount of water so as to achieve a greater dilution. Something such as wet sand would require less water so that agitation would keep the particles suspended while you are draining the bottle.

 

[sophiecentaur]

yes. The application needs to be specified exactly because there are so many variables. Cleaning out paint brushes is a classic situation when the white spirit is in short supply.

 

[Borek]

Define "optimal", otherwise it won't be before 12th page that some will start to realize everyone is talking about different "optimal" which is why they can't agree on anything

 

[CWatters]

I think a similar question was asked in New Scientist some years ago. I don't remember the answer sorry.

 

[sophiecentaur]

There's a bit of Maths needed here, I think. For the simple case of dilution of volume 1 of paint in the brush when you have volume V of thinners.
You have so much solvent and divide it into n equal portions of V/n
The first mixing / cleaning will dilute the paint by
1/(1+V/n)
You then pour away all but the volume 1, that remains in the brush

After n mixings you have a dilution of

(1/(1+V/n))ⁿ

I plotted some results from Excel for V = 4,8,16 and for a range of n
I hadn't realized but, even when you have a small amount of solvent, it seems that you can do far better with multiple dilutions. The the smaller the amount of mixture left in the the brush, of course, the better use you make of the solvent.

 

[Borek]

I hadn't realized but, even when you have a small amount of solvent, it seems that you can do far better with multiple dilutions.

You just pried open door that is frequently used by every chemist in the world. This approach is a basis of the liquid-liquid extraction technique.

 

[sophiecentaur]

You live and learn - Excel strikes again. I'd never have been bothered to do it by hand and I'm too rusty to be doing it 'by algebra'.

 

[jbriggs444]

I plotted some results from Excel for V = 4,8,16 and for a range of n
I hadn't realized but, even when you have a small amount of solvent, it seems that you can do far better with multiple dilutions. The the smaller the amount of mixture left in the the brush, of course, the better use you make of the solvent.

In the limit with V large, the optimal dilution should be by a factor of 1/e at each cycle.

http://en.wikipedia.org/wiki/Radix_economy

 

[sophiecentaur]

I suppose e has to rear its ugly head in a situation like this.

What I found more interesting is that, even if you only have an equal amount of solvent, you can get significantly better dilution by using, say ten dilutions, just adding 1/10 of the solvent you've got each time.- i.e 38% concentration instead of 50%. If you just have 4 times the volume of solvent, you can get a concentration of only 3% with 12 dilutions. Actually, this does agree with my experience of cleaning brushes and just splashing small amounts of thinner on them but doing it several times.
PF does make you think!

 

[jbriggs444]

In the limit with V large, the optimal dilution should be by a factor of 1/e at each cycle.

http://en.wikipedia.org/wiki/Radix_economy

My conclusion here was incorrect. If you take the cost of a factor of n dilution as n then the assumptions of radix economy would apply and the optimum would be a dilution by a factor of e each time.

But if you, more reasonably, take the cost of a factor of n dilution as n-1 then the optimum is to add as little solvent as you can manage at each cycle. This corresponds to "continous compounding". This fits with Sophiecentaur's observations.