The International System of Units — Question 6

Question 6

One mole of what?

The Avogadro constant is exactly $N_{\mathrm A}=6.022\,140\,76\times10^{23}\ \mathrm{mol^{-1}}$. The relation between the number $N$ of specified entities and their amount $n$ is $N=nN_{\mathrm A}$.

Two sealed samples, X and Y, each contain exactly $N_{\mathrm A}(1\ \mathrm{mol})$ molecules. Every X molecule consists of two atoms; every Y molecule consists of three atoms. Assume the molecules remain intact and no other particles are present. For a simple mass comparison, take the mass of each Y molecule to be twice that of each X molecule.

1. State the amount of molecules in each sample and the amount of constituent atoms in each. Explain how two different amounts can correctly describe the same sample.

2. Determine the mass ratio $m_Y/m_X$. Assess the claim “one mole always has one standard mass.”

3. A label reads only “amount: $1\ \mathrm{mol}$.” Explain why it is incomplete here. Write unambiguous labels for both molecular samples.

4. Does exact $N_{\mathrm A}$ imply that an ordinary experimentally prepared sample labelled $1.000\ \mathrm{mol}$ contains an exactly known number of molecules?

Question 6 - Solution

One mole of what?

(a) Specify the entities before counting.

$$n_{X,\mathrm{atoms}}=2n_{X,\mathrm{molecules}}=2\ \mathrm{mol},\qquad n_{Y,\mathrm{atoms}}=3\ \mathrm{mol}.$$

Each sample has molecular amount $N/N_{\mathrm A}=\boxed{1\ \mathrm{mol}}$. The atom counts are $2N$ in X and $3N$ in Y, so their amounts of constituent atoms are respectively $\boxed{2\ \mathrm{mol}}$ and $\boxed{3\ \mathrm{mol}}$.

There is no contradiction: “molecules” and “constituent atoms” identify different entities. One molecule counts as one molecular entity even when it contains several atomic entities. Amount of substance is not the same quantity as number of entities: $n$ has unit mole, whereas $N$ is a count with unit one.

(b) Equal numbers need not have equal masses.

Let the mass of one X molecule be $\mu$. Then $$m_X=N\mu,\qquad m_Y=N(2\mu),\qquad \boxed{m_Y/m_X=2}.$$ The mole fixes an entity count per amount, not a universal mass. Equal molecular amounts of different substances can have different masses. In this model the mass ratio follows from the supplied molecular-mass relation, not from the definition of the mole alone.

(c) Repair the labels.

Suitable labels are “$n(\text{X molecules})=1\ \mathrm{mol}$” and “$n(\text{Y molecules})=1\ \mathrm{mol}$,” with X and Y identified by chemical species on a real label. A bare amount does not tell the reader whether molecules, atoms, ions, or some other specified entities are intended.

(d) Exact definition, measured sample.

$$\delta N=N_{\!A}\delta n,\qquad\frac{\delta N}{N}=\frac{\delta n}{n}\quad(N_{\!A}\text{ exact}).$$

No. The exact counts in the prompt are ideal assumptions. A prepared sample’s amount may depend on measured mass, purity, and composition. An exact $N_{\mathrm A}$ makes $N=nN_{\mathrm A}$ exact as a relation, but an uncertain $n$ still gives an uncertain inferred $N$. Printed decimal places alone are not an uncertainty statement.