To find the electrostatic potential energy (U) of two electrons separated by a distance of 3.0 Å in vacuum, we can use the formula for electrostatic potential energy:
\[
U = \frac{k \cdot q_1 \cdot q_2}{r}
\]
where:
- \( U \) is the electrostatic potential energy,
- \( k \) is Coulomb's constant, approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \),
- \( q_1 \) and \( q_2 \) are the charges of the electrons (both are equal to the elementary charge, \( e = 1.6 \times 10^{-19} \, \text{C} \)),
- \( r \) is the distance between the charges.
### Step 1: Identify the values
- Charge of an electron, \( q_1 = q_2 = -1.6 \times 10^{-19} \, \text{C} \)
- Distance, \( r = 3.0 \, \text{Å} = 3.0 \times 10^{-10} \, \text{m} \)
- Coulomb's constant, \( k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)
### Step 2: Substitute the values into the formula
Substituting the values into the formula:
\[
U = \frac{(9 \times 10^9) \cdot (-1.6 \times 10^{-19}) \cdot (-1.6 \times 10^{-19})}{3.0 \times 10^{-10}}
\]
### Step 3: Calculate the numerator
Calculating the numerator:
\[
(9 \times 10^9) \cdot (1.6 \times 10^{-19}) \cdot (1.6 \times 10^{-19}) = 9 \times 10^9 \cdot 2.56 \times 10^{-38} = 23.04 \times 10^{-29}
\]
### Step 4: Calculate the potential energy
Now substituting back into the equation:
\[
U = \frac{23.04 \times 10^{-29}}{3.0 \times 10^{-10}} = 7.68 \times 10^{-19} \, \text{J}
\]
### Step 5: Convert to electron volts
To convert joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[
\text{Energy in eV} = \frac{7.68 \times 10^{-19}}{1.6 \times 10^{-19}} = 4.8 \, \text{eV}
\]
### Final Answer
Thus, the electrostatic potential energy of two electrons separated by 3.0 Å in vacuum is:
- In joules: \( 7.68 \times 10^{-19} \, \text{J} \)
- In electron volts: \( 4.8 \, \text{eV} \)
