Question

A hydrogen atom has a single proton at its center and a single electron at a distance of approximately \(0.0529 \mathrm{nm}\) from the proton. (a) What is the electric potential energy in joules? (b) What is the significance of the sign of the answer?

Short answer

Answer: The electric potential energy of the hydrogen atom is approximately -4.28 x 10^(-18) J. The negative sign indicates that the electron and proton are bound together by an attractive force, which keeps the electron in orbit around the proton and stabilizes the hydrogen atom.

Step-by-step solution

Identify the given values

Given values: - Distance between proton and electron, \(r = 0.0529 \, \mathrm{nm}\) - Charge of proton, \(q_{1} = +1.6 \times 10^{-19} \, \mathrm{C}\) - Charge of electron, \(q_{2} = -1.6 \times 10^{-19} \, \mathrm{C}\) We also know that: - Coulomb's constant, \(k = 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2}\) Note that we need to convert the given distance from nanometers to meters.

Convert distance to meters

To convert the distance from nanometers to meters, we use the following relation: 1 nm = \(10^{-9}\) m Hence, the distance in meters is: \(r = 0.0529 \cdot 10^{-9}\) m

Calculate the electric potential energy

Using the formula for electric potential energy, we can now calculate the energy: \(U = \frac{k \cdot q_{1} \cdot q_{2}}{r}\) Substitute the known values into the formula: \(U = \frac{(8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2}) \cdot (1.6 \times 10^{-19} \, \mathrm{C}) \cdot (-1.6 \times 10^{-19} \, \mathrm{C})}{(0.0529 \cdot 10^{-9}\, \mathrm{m})}\) Now, perform the calculation: \(U = -4.28 \times 10^{-18} \, \mathrm{J}\) The electric potential energy of the hydrogen atom is approximately \(-4.28 \times 10^{-18} \,\mathrm{J}\).

Discuss the significance of the sign

The negative sign of the answer indicates that the electric potential energy is negative. This implies that the electron and proton are bound together by an attractive force since the electron has a negative charge and the proton has a positive charge. This attractive force between oppositely charged particles is what keeps the electron in orbit around the proton, stabilizing the hydrogen atom. If the energy was positive, it would mean that the electron and proton are repelling each other, which would not be a stable configuration for the hydrogen atom. In conclusion, the electric potential energy of the hydrogen atom is approximately \(-4.28 \times 10^{-18} \, \mathrm{J}\), and the negative sign shows that the attractive force between the proton and electron is keeping the atom stable.