Question

The average and the most probable distance from the nucleus for 1 s electron in hydrogen atom are, respectively \(\left(a_{\mathrm{o}}\right.\) is the first Bohr radius.), (a) \(a_{\mathrm{o}}, a_{\mathrm{o}}\) (b) \(a_{\mathrm{o}}, 1.5 a_{\mathrm{o}}\) (c) \(1.5 a_{0}, a_{\mathrm{o}}\) (d) \(1.5 a_{0}, 1.5 a_{\mathrm{o}}\)

Short answer

The average distance is \(1.5 a_{0}\) and the most probable distance is \(a_{0}\), hence the correct answer is (c).

Step-by-step solution

Understand the Concepts of Average and Most Probable Distances

The average distance of an electron in a 1s orbital from the nucleus is calculated using the expectation value of the radial distance in spherical coordinates. The most probable distance, on the other hand, is the radius at which the probability density is maximum for the electron.

Recall the Formulas for Hydrogen-like Atoms

The probability density for an electron in the 1s orbital is highest at the Bohr radius, therefore the most probable distance for a 1s electron is indeed \(a_{0}\). The average distance is given by the expectation value \( \left\langle r \right\rangle =\frac{3}{2} a_{0}\) for the 1s orbital.

Choose the Correct Option

Comparing the derived results with the options given, the average distance is \(1.5 a_{0}\) and the most probable distance is \(a_{0}\), matching with option (c).

Key concepts

Bohr Radius

The Bohr radius, symbolized as \(a_0\), is a fundamental constant in the study of atomic physics, particularly the hydrogen atom. It represents the average distance between the proton (nucleus) and the electron in the simplest Bohr model of the hydrogen atom. The exact value is approximately \(0.529 \times 10^{-10}\) meters or \(52.9 \, pm\). This concept is critical for understanding the size and scale of atoms as well as the calculations involving electron orbits.

Even though the Bohr model has been succeeded by the more accurate quantum mechanical model of the atom, the Bohr radius remains an important quantity for describing hydrogen-like atoms and ions. It serves as a scale factor in the size of orbitals and provides a point of comparison for more complex calculations of atomic structure.

Probability Density

Probability density in quantum mechanics refers to the likelihood of finding an electron in a particular location around the nucleus. For an electron in the 1s orbital of a hydrogen atom, this is described by a probability density function that depends on the distance from the nucleus. The function peaks at a particular value of radius, indicating the most probable distance to find the electron.

Mathematically, the probability density function for the 1s orbital has the form \(\rho(r) = |\psi(r)|^2\), where \(\psi(r)\) is the wave function of the electron. When graphed, the probability density for the 1s orbital is a sphere centered around the nucleus with the density decreasing as the distance from the nucleus increases. It's essential to visualize these concepts to better understand the quantum behavior of electrons in atoms.

Expectation Value

The expectation value represents the average outcome of a measurement performed on a quantum system. In the context of electron distances, the expectation value of the radial distance \(\langle r \rangle\) gives the average distance of an electron from the nucleus over many measurements. It is calculated by integrating the product of the probability density function with the radial variable \(r\) over all space.

For a hydrogen atom's 1s orbital, the expectation value of the radial distance can be expressed as \(\langle r \rangle = \int r \rho(r) \, dV\), where \(\rho(r)\) is the probability density and \(dV\) signifies the volume element in spherical coordinates. The result for a 1s orbital is \(1.5 a_{0}\), meaning that on average, the electron will be found 1.5 times the Bohr radius away from the nucleus.

Spherical Coordinates

Spherical coordinates are a system for representing points in three-dimensional space using three parameters: the radial distance \(r\), the polar angle \(\theta\), and the azimuthal angle \(\phi\). This system is particularly useful in quantum mechanics and atomic physics for describing the position of electrons in atoms.

Since electrons move in a three-dimensional space around the nucleus, spherical coordinates simplify the mathematical description of the quantum states. For example, when calculating the expectation value or the probability density for the 1s orbital of the hydrogen atom, spherical coordinates enable analysts to integrate over the entire volume of space around the nucleus. This provides a clearer and mathematically accurate depiction of atomic structures and behaviors.

1s Orbital

The 1s orbital is the simplest and the lowest energy electron orbital in an atom. It is spherically symmetrical around the nucleus, meaning it looks like a shell or a cloud surrounding the nucleus uniformly in all directions. Electrons in this orbital are in their ground state, which is the most stable and the least energy-intensive configuration.

The 1s orbital serves as a key model for discussing foundational principles of quantum mechanics, such as energy quantization, orbital shapes, and electron probabilities. A basic comprehension of the 1s orbital provides insights into the electron configurations of more complex atoms, as well as the underlying reasons for chemical bonding and the periodic table arrangement.