Anaesthetic Concentration

Date: 07/21/2003 at 17:02:43
From: Joanne
Subject: Concentration of drugs for anaesthetics

What is the % concentration when you constitute 100ml water with 2.5g 
of thiopentone?

I'm not sure that it is as simple as 2.5%

Date: 07/22/2003 at 05:56:57
From: Doctor Achilles
Subject: Re: Concentration of drugs for anaesthetics

Hi Joanne,

Thanks for writing to Dr. Math.

First of all, I should note that mixing anaesthetics is an extremely 
sensitive thing and it should only be done by a certified 
professional.

Having said that, you are correct, the answer is 2.5%. Because 1ml of 
water has a mass of 1g, 100ml will weigh 100g. 2.5g of solute in 100ml 
total will be approximately 2.5g of solute in 100g total.  
2.5/100 equals 0.025 or 2.5%.

Actually, to be more precise, the answer is:

  2.5% (w/v)

The "(w/v)" is short for "weight per volume." The reason is that the 
solute (anaesthetic) is measured by weight (or mass), and the solvent 
(water) is measured by volume. So the numerator of your fraction is 
weight and the denominator is volume.

I hope this helps. If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/23/2003 at 14:21:32
From: Joanne
Subject: Re: Concentration of drugs for anaesthetics

But when you add 2.5g to 100g of water, don't you end up with
102.5g of solution?  So shouldn't it be 2.5g divided by 102.5g? 

Date: 07/23/2003 at 15:42:26
From: Doctor Achilles
Subject: Re: Concentration of drugs for anaesthetics

Hi Joanne,

Notice how I said "2.5g of solute in 100ml *total*", rather
than "2.5g of solute in 100ml *of water*".

The appropriate way to mix a solution of a given
weight/volume is to first make a more concentrated solution.

In the case of making 100ml at 2.5% (w/v), what I would do
is get about 70ml of water and disolve 2.5g of solute.  I
would then adjust the volume with additional water until a
final volume of 100ml is reached.

If we assume that the final density is 1g/ml, then we will
have created a 2.5% solution (by weight).  However, because
our solution is (by definition) not 100% pure water what we
have instead is 2.5g in 100ml.  The notation "2.5% (w/v)" is
identical to saying "2.5g in 100ml of aqueous (i.e. mostly
water) solution".

We could make a solution of precisely 2.5g (by weight) if we
modified the protocol slightly:  Again, we could start with
70ml of water, add 2.5g of solute.  But now, instead of
bringing the final volume up to 100ml, we keep adding water
until the mass is 100g.

This would give a precise 2.5% solution.

In practice, the difference between making a 2.5% (by
weight) solution and a 2.5% (w/v) is negligible.  Even
though most solids have a density which can differ from
water by a factor of 10 or more, when they are disolved in
water, the solute will rarely differ from water by more than
a factor of 2.  Sucrose (table sugar) is a notable
exception. I don't know off-hand what its density is when
it is dissolved, but it is significantly greater than 1g/ml.

So let's say we have a solute which has a density of 2g/ml
when it is in aqueous form and for which 10g takes up a
volume of 3ml in aqueous form.  Let's say we want a 10%
solution.

If we make our solution 10% (by weight), then we will end up
with 100g solution that contains 10g of solute.

If we make our solution 10% (w/v), then we will have 10g of
solute in 100ml of solution.  But the mass of our solution
will be 105g.  So, by weight, we have created a (10g/105g)
9.5% solution.

If the density of the aqueous form of the solute were closer
to 1g/ml, this error would be less.

Moreover, the most important thing in chemistry and
biochemistry is reproducibility, and (w/v) solutions are
quite reproducible.  Additionally, if you mix a 5% (w/v)
solution of solute A with an equal volume of a 10% (w/v)
solution of solute B, then you will end up with a solution
which is 2.5% (w/v) for solute A and 5% (w/v) of solute B. 
(Solutions which are made strictly by weight also share this
property.)

So there is some error in (w/v) calcuations, but it is quite
consistent error.

The most precise way to talk about concentration is actually
molarity, which is briefly discussed here:

  pH Value
  http://mathforum.org/library/drmath/view/56359.html

But (w/v) is unambiguous as long as you specify that you are
giving percentages as (w/v).

Does that clear things up?

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/