Avogadro's Number — Question 5

Question 5

Dissociation changes particle count while conserving nuclei

Use $N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$, exact. Initially a sample contains .800 mol of diatomic molecules $X_2$. A fraction $f=.250$ of the original molecules dissociates into two atoms each; no nuclei enter or leave. Undissociated molecules and free atoms are both counted as particles.

1. Find the remaining molecular amount, free-atom amount and total particle count.

2. Derive total particle amount in terms of $f$. If a later measurement gives 1.20 mol of particles, infer the dissociated fraction and check the conserved nucleus count.

Question 5 - Solution

Dissociation changes particle count while conserving nuclei

(a) Method and calculation

$$n_{X_2}=n_0(1-f)=.600\,\mathrm{mol},\quad n_X=2n_0f=.400\,\mathrm{mol}.$$$$n_{\mathrm{particles}}=1.00\,\mathrm{mol},\quad N_{\mathrm{particles}}=N_A.$$

(b) Interpretation and further calculation

$$n_{\mathrm{particles}}=n_0(1-f)+2n_0f=n_0(1+f).$$$$f=1.20/.800-1=.500,\quad 2n_{X_2}+n_X=2n_0=1.60\,\mathrm{mol\ of\ nuclei}.$$Particles increase from .800 to 1.20 mol while the number of nuclei stays fixed.

Physical check

The allowed range is $n_0\le n_{\mathrm{particles}}\le2n_0$ because $0\le f\le1$.