Introduction:
The average distance of an electron from the nucleus in the ground state of a hydrogen atom can be calculated using the Bohr model of the atom. In this model, the electron is assumed to move in circular orbits around the nucleus, and the radius of the orbit is determined by the electron's energy.

Bohr Model:
1. In the Bohr model, the electron is considered to be in a stable orbit around the nucleus without emitting or absorbing energy.
2. The ground state of the hydrogen atom corresponds to the lowest energy level, known as the n=1 level.
3. According to the Bohr model, the radius of the electron's orbit in the ground state can be calculated using the formula:
r = (0.529 Å) * n^2 / Z
where r is the radius of the orbit, n is the principal quantum number (1 for the ground state), and Z is the atomic number of the nucleus (1 for hydrogen).
4. Plugging in the values, we get:
r = (0.529 Å) * 1^2 / 1 = 0.529 Å

Average Distance:
1. The average distance of the electron from the nucleus can be determined by considering the probability distribution of the electron's position.
2. In the ground state, the electron is most likely to be found near the nucleus, but there is also a small probability of finding it at larger distances.
3. The probability distribution is described by the wave function of the electron, which is a mathematical function that gives the probability of finding the electron at different positions.
4. The square of the wave function, known as the electron density, gives the probability density of finding the electron at a particular position.
5. By integrating the electron density over all possible positions, we can calculate the average distance of the electron from the nucleus.
6. For the ground state of the hydrogen atom, the average distance is found to be approximately 1.5 times the Bohr radius (0.529 Å).
Average distance = 1.5 * 0.529 Å = 0.794 Å ≈ 1.5

Conclusion:
The average distance of an electron from the nucleus in the ground state of a hydrogen atom is approximately 1.5 times the Bohr radius. This is determined by considering the probability distribution of the electron's position, which is described by the wave function of the electron. The wave function gives the probability density of finding the electron at different positions, and by integrating over all possible positions, the average distance can be calculated.