# Weighing by difference

When preparing a known mass of something (e.g. when producing a standard solution) it is often that in my school lab that we first use a weighing boat to measure the mass and then once transferred into a beaker we reweigh the boat and calculate the mass transferred.

• Do we this even though we zero the scale anyway?

Now to my knowledge would this not increase the uncertainty as instead of just using the beaker on the scale and adding the mass directly which would lead to one uncertainty we instead weigh it twice which would thus have 2x the uncertainty in the scale.

• Could somebody explain where my reasoning is wrong and why this method is more accurate?
• Beakers are heavy. Lots of balances can't handle that much weight. Also what if your beaker had stuff in it already and you added to too much? – Zhe Jun 23 '17 at 19:35
• Also, the weight of the container may change, e.g. because two similar pieces of filter paper may weigh slightly different amounts, or the same beaker, though used again, could have a drop of water or precipitate clinging to it after washing, or a small chip removed. – DrMoishe Pippik Jun 23 '17 at 20:41
• Would the problem proposed by DrMoishe not be solved by zeroing the apparatus since we are only using it once? – Jake Rose Jun 23 '17 at 21:22

Although there are several sources of weighing error that would be minimized by using the weighing by difference method (as discussed in the comments), I want to illustrate what I see as the limiting problem, that of the relative uncertainty of the measurement, which roughly scales (no pun intended, really) with the mass being measured.

Lets use a bit of an exaggerated situation for illustration. In particular, we'll say that we have one balance that can measure down to $\pu{0.001g}$, and can handle up to $\pu{100g}$ total mass. In reality you would likely have a high-mass balance for weighing directly into the beaker, or a low-mass balance for using the weighing boat. We are just using one really good balance to simplify this illustration, but the concept is the same either way.

Lets say our beaker has a mass of $\pu{50g}$, our weighing boat is $\pu{1g}$, and we are to measure out $\pu{1g}$ of material.

The beaker plus the material to be measured will then have a measured mass of ~$\pu{51.00 +/- 0.05g}$. This means that the relative uncertainty in the measured mass of the material is no better than $\pu{5\%}$($\pu{0.05g}$ out of $\pu{1g}$).

The weighing boat plus the material to be measured out will have a measured mass of ~$\pu{2.000 +/- 0.002g}$. This means that the relative uncertainty in the measured mass of the material is $\pu{0.2\%}$ ($\pu{0.002g}$ out of $\pu{1g}$).

So, even with two measurements rather than one, using the weighing boat method results in much less total uncertainty in the measured mass of your material as compared to weighing a small quantity of material directly into a large beaker.

• This assumes that the balance error scales linearly as the total mass. Quite reasonable though you may want to justify. – Zhe Jun 23 '17 at 22:13
• Where do the uncertainty values come from? does the uncertainty in the mass increase the uncertainty in the measurement? – Jake Rose Jun 24 '17 at 12:57

Most chemists in the lab I have seen so far (caution: sample distribution may be skewed) usually do not reweigh whatever they weighed into after transferring into the reaction vessel or beaker. For most substances, this is justifyable since you can see if significant bits remain in the boat or weighing device you used. Also, typically the mass remaining in the boat will be in the dimension of $\pu{0.1 mg}$ or less and thus not already in the same dimension of the scale’s uncertainty. In rare cases, especially when I see lots of a substance remaining, I will reweigh the boat or weighing vessel, though.

As to the question of weighing into a boat or the beaker this is more complicated. For one, the uncertainty usually increases roughly linearly with the weight of the sample measured. Many scales in the lab I’m working in now have five sub-gram digits but once the total weight is larger than eighty grams or so, the final digit is hidden. The smaller and lighter the vessel you use, the less it affects the total weight on the scales and thus also the less inaccurate your weighing is.

Weighing directly into the beaker or flask can also have practical disadvantages. If — what is often the case in synthesis — you need to weigh more than one compound, you should keep them separate in the weighing step since you will probably have to add and remove to get to the mass you want to have. You naturally cannot remove a pure sample from a mixture (‘so which of the white crystals am I taking out now?’). Furthermore, these reaction vessels typically allow for a somewhat easy addition of solids into them, but make it very hard to remove a bit of solid from them, e.g. because you can’t access the solids with a spatula through the opening. These problems come on top of the scaling uncertainty, making weighing directly into a flask more and more unsuitable.