The magnitude of the electrostatic force between two identical ions that are separated by a distance of 7.8 × 10-10 m is 97.10 × 10-9 N. (a) What is the charge (in Coulombs) of each ion? (b) How many electrons are “missing” from each ion (thus giving the ion its charge imbalance)?
F=k•q1•q2/r²= k•q²/r².
q=sqrt(F• r²/k) = ....
N=q/e=....
Answer
To find the charge of each ion and the number of missing electrons, we'll need to use the formula for the magnitude of the electrostatic force:
\(F = \frac{{k \cdot |q_1 \cdot q_2|}}{{r^2}}\)
Where:
- \(F\) is the magnitude of the electrostatic force (97.10 × 10^-9 N in this case)
- \(k\) is the electrostatic constant (which is approximately 8.99 × 10^9 N⋅m^2/C^2)
- \(q_1\) and \(q_2\) are the charges of each ion (in Coulombs)
- \(r\) is the distance between the ions (7.8 × 10^-10 m in this case)
(a) To find the charge of each ion, we can rearrange the equation:
\(q_1 \cdot q_2 = \frac{{F \cdot r^2}}{{k}}\)
Now, we substitute the given values and solve for \(q_1 \cdot q_2\):
\(q_1 \cdot q_2 = \frac{{97.10 \times 10^{-9} \cdot (7.8 \times 10^{-10})^2}}{{8.99 \times 10^9}}\)
\(q_1 \cdot q_2 = 0.0673\) Coulombs squared
Since the two ions are identical, \(q_1 = q_2 = q\), and we can rewrite the equation:
\(q^2 = 0.0673\) Coulombs squared
Taking the square root of both sides, we find the charge of each ion:
\(q = \sqrt{0.0673}\) Coulombs
(b) To determine the number of missing electrons, we can calculate the excess or deficit of charge on each ion. Electrons carry a fundamental charge of approximately -1.602 × 10^-19 Coulombs.
The charge imbalance of each ion can be expressed as:
\(q = n \cdot q_e\)
Where:
- \(q\) is the charge of each ion (in Coulombs)
- \(n\) is the number of missing or excess electrons
- \(q_e\) is the charge of a single electron (approximately -1.602 × 10^-19 Coulombs)
Rearranging the equation, we can solve for \(n\):
\(n = \frac{q}{q_e}\)
Substituting the value of \(q\) we calculated earlier:
\(n = \frac{\sqrt{0.0673}}{-1.602 \times 10^{-19}}\)
Calculating this expression gives us the number of missing or excess electrons on each ion.