One mole of a substance contains `6.02 xx 10^(23)` protons and an equal number of electrons. If the protons could somehow be separated from the electrons and placed in very small, individual containers separated by `1.00 xx 10^(3)m`, what would be the magnitude of the electrostatic force exerted by one box on the other ?

Options

A. `8.7 xx 10^(8) N`
B. `9.5 xx 10^(9)N`
C. `2.2 xx 10^(10)N`
D. `8.3 xx 10^(13) N`

Answer

To find the magnitude of the electrostatic force exerted by one box on the other, we can follow these steps:

### Step 1: Determine the charge of one mole of protons and electrons
One mole of protons or electrons contains Avogadro's number of particles, which is \(6.02 \times 10^{23}\).

The charge of one proton is \(+1.6 \times 10^{-19}\) coulombs, and the charge of one electron is \(-1.6 \times 10^{-19}\) coulombs.

Thus, the total charge for one mole of protons (Q1) is:
\[ Q_1 = (6.02 \times 10^{23}) \times (1.6 \times 10^{-19}) = 9.632 \times 10^{4} \text{ C} \]

And the total charge for one mole of electrons (Q2) is:
\[ Q_2 = (6.02 \times 10^{23}) \times (-1.6 \times 10^{-19}) = -9.632 \times 10^{4} \text{ C} \]

### Step 2: Use Coulomb's Law to calculate the force
Coulomb's Law states that the electrostatic force \(F\) between two point charges is given by:
\[ F = k \frac{|Q_1 \cdot Q_2|}{r^2} \]
where:
- \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\),
- \(r\) is the distance between the charges, which is given as \(1.00 \times 10^{3} \, \text{m}\).

Substituting the values:
\[ F = 8.99 \times 10^9 \frac{|(9.632 \times 10^{4}) \cdot (-9.632 \times 10^{4})|}{(1.00 \times 10^{3})^2} \]

### Step 3: Calculate the force
Calculating the numerator:
\[ |Q_1 \cdot Q_2| = (9.632 \times 10^{4})^2 = 9.28 \times 10^{9} \text{ C}^2 \]

Now substituting back into the force equation:
\[ F = 8.99 \times 10^9 \frac{9.28 \times 10^{9}}{(1.00 \times 10^{3})^2} \]
\[ = 8.99 \times 10^9 \frac{9.28 \times 10^{9}}{1.00 \times 10^{6}} \]
\[ = 8.99 \times 10^9 \times 9.28 \times 10^{3} \]
\[ = 8.34 \times 10^{13} \text{ N} \]

### Final Answer
The magnitude of the electrostatic force exerted by one box on the other is approximately:
\[ F \approx 8.34 \times 10^{13} \text{ N} \]

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