# How does uncertainty apply to chemists?

Jan 15, 2016

Uncertainty applies in the sense that we as human beings are naturally imperfect. We will always have some form of uncertainty in any experiment we do, no matter how well we take precautions to not, so we should state how uncertain we are.

I will first assume that we are talking about random error and systematic error, and then proceed to the Heisenberg Uncertainty Principle.

RANDOM ERROR

Random error will always happen to some extent. If you do multiple trials, there is always the chance you will make human mistakes, such as with stopwatching or with measuring a specific mass onto a scale.

For example, timing is difficult to get spot-on.

The smaller the time-scale, the more difficult it is to be very precise.

Maybe you time $\text{15.00 s}$ one time and $\text{14.98 s}$ another time, and then $\text{15.02 s}$ for your third trial. Then, your standard deviation is $\pm \text{0.02 s}$. You may be, say, 95% confident that the expected result falls within the range $\text{14.98 - 15.02 s}$. (My personal fastest time in starting and stopping a typical stopwatch is $\text{0.12 s}$, for reference.)

Or, maybe you try to measure a small mass, which can be difficult to get spot-on as well.

The smaller the mass, the greater the uncertainty you have.

For example, I have had to measure a few $\mu g$ before, and it is just tough to get a low % uncertainty. The $\mu$-scale (both physically and conceptually) is small, so if you try to measure, say, $500$ $\mu g$, it's quite normal to get $520$ $\mu g$, or even $460$ $\mu g$ instead.

SYSTEMATIC ERROR

Systematic error is when something is wrong with your measurement tool, and it is consistently off.

Maybe you consistently measured distance from a specific reference point, and are precisely $x$ $\text{m}$ off from the correct distance by the same amount each time because of your offset. Then, if you try to measure, for instance, $1.00$, $2.00$ and $3.00$ $\text{m}$, you might get $1.02$, $2.02$, and $3.02$ $\text{m}$ instead. That is consistently off by $\text{0.02 m}$.

Or, maybe something more subtle is that you are wearing nitrile gloves and using a chemistry lab-grade scale. You'll introduce static onto the scale, which will cause the scale to display a continuously changing mass that is not accurate to your actual mass. This is called mass hysteresis.

To avoid this, you may have a static gun in the lab that you can use to remove the static. Then, you can take off your nitrile gloves and measure without gloves, and the scale shouldn't drift so much this time.

Consistency in measuring masses is especially an issue on small scales (such as a $\mu$-scale), because, wouldn't you know it, dust that you might not know is there can significantly alter your % accuracy.

HEISENBERG UNCERTAINTY PRINCIPLE

The Heisenberg Uncertainty Principle basically states that the more certainty $\Delta x$ you have about the position $x$, the less uncertainty $\Delta {p}_{x}$ you have about the momentum ${p}_{x}$, and vice versa. The relationship is:

\mathbf(DeltaxDeltap_x <= ℏ)

where ℏ = h//2pi is the reduced Planck's constant.

In a sense you have already seen sprinkles of this in this answer up above when I mentioned that smaller masses and smaller time scales lead to greater uncertainty.

In fact, a similar concept applies here.

In general, we usually are more certain (have lower uncertainty) about the momentum of an electron, so we tend to have high uncertainty about its position (Inorganic Chemistry, Miessler et al.).

We kind of get around that by making radial probability density plots to probabilistically define locations where an electron is statistically expected to be. We have ${a}_{0} {r}^{2} {R}_{n l}^{2} \left(r\right)$ on the y-axis and $r / {a}_{0}$ on the x-axis. Here, ${a}_{0}$ is the Bohr radius ($5.29177 \times {10}^{- 11}$ $\text{m}$).

To read this plot, it just says that the greatest chance you will have of finding electrons is at ~$1 {a}_{0}$ in the $1 s$ orbital, ~$5 {a}_{0}$ for the $2 s$ orbital, and ~$13 {a}_{0}$ for the $3 s$ orbital. Those are the "most populated" regions of electron density.

You have no chance of finding electrons at ~$2 {a}_{0}$ in the $2 s$ orbital, and at ~$2 {a}_{0}$ and ~$7 {a}_{0}$ in the $3 s$ orbital. Those are radial nodes.

(Obviously there is no chance either if you are completely outside the orbitals, e.g. $r \to \infty$)