Charge Is Quantized — Question 7

Question 7

A fractional mean ionic charge does not require fractionally charged ions

A sample contains 500 positive ions, each carrying either $+e$ or $+2e$. Their total charge is $+800e$. No negative ions or electrons are included in this counted sample.

1. Find the number of ions in each charge state.

2. Find the mean charge per ion. If 100 randomly selected ions are measured in many repetitions, explain whether every group must have total charge $160e$ and derive the mean group charge.

Question 7 - Solution

A fractional mean ionic charge does not require fractionally charged ions

(a) Method and calculation

$$N_1+N_2=500,\quad N_1+2N_2=800.$$$$N_2=300,\quad N_1=200,\quad\langle q\rangle=800e/500=1.60e.$$

(b) Interpretation and further calculation

$$Q_{100}=(100+N_{2,100})e,\qquad N_{2,100}\in\{0,1,\ldots,100\}.$$$$\langle Q_{100}\rangle=100\langle q\rangle=160e.$$Every group has an integer charge, but different groups can contain different numbers of doubly charged ions. The mean need not equal each observation.

Physical check

The individual ion charge and an average over a mixed population are different quantities.