Question

For a hydrogen atom in the ground state, determine (a) the most probable location at which to find the electron and (b) the most probable radius at which to find the electron. (c) Comment on the relationship between your answers in parts (a) and (b).

Short answer

The most probable location of the electron in a hydrogen atom in its ground state is at the nucleus (\(r = 0\)), while the most probable radius is at \(r = a_{0}\), where \(a_{0}\) is the Bohr radius. The discrepancy arises from the probability distribution of finding the electron at different distances from the nucleus.

Step-by-step solution

Understanding a Hydrogen Atom

For a hydrogen atom in the ground state, the electron is in the first energy level (n=1), specifically in the 1s orbital. The probability density of finding the electron at a particular location is given by \[^2_r \]\(\psi^{2}(r) = \frac{4}{a_{0}^{3}} e^{-2r/a_{0}}\). Here, \(\psi(r)\) is the wave function of the electron, \(a_{0}\) is the Bohr radius, and \(r\) is the distance from the nucleus.

Finding the Most Probable Location and Radius

According to the radial probability distribution, the electron has the highest probability of being found at the nucleus of the atom (r=0) in the ground state. This is the most probable location. However, the most probable radius (distance from the nucleus where electron is most likely to be found) is at the radius where the probability density function is maximum. We have to differentiate the probability density function with respect to \(r\) to locate the maxima, and then equate it to zero, i.e., \(\frac{d(\frac{4}{a_{0}^{3}} e^{-2r/a_{0}})}{dr} = 0\). Solving this will give \(r = a_{0}\), which is the most probable radius.

Comment on the Relationship

The most probable location and most probable radius are different due to the nature of the electron's wave function. The location is where the electron is most likely to be, but the most probable radius describes the most probable distance from the nucleus, accounting for the different probabilities at different distances due to the nature of the orbital shape. In short, the electron is likely to be at the nucleus, but it could also be found with a high probability at a radius equal to \(a_{0}\)

Key concepts

Ground State

The ground state of an atom refers to the lowest energy state that the electron can occupy. In the context of a hydrogen atom, this means that the electron is in the n=1 energy level, which is the closest to the nucleus. When an electron is in the ground state, it is more stable and less likely to undergo energy changes unless acted upon by outside forces.

For hydrogen, the electron occupying the ground state is found in the 1s orbital. This state is crucial because it sets the baseline for understanding how electrons behave in different energy states.

In the ground state, the electron's energy and the probability density distribution are defined by specific quantum mechanical equations. This gives us a precise understanding of how the electron is distributed around the proton in a hydrogen atom.

Probability Density

Probability density provides insights into where an electron is most likely to be found relative to the nucleus. This concept is essential in quantum mechanics, as the exact location of an electron cannot be defined; rather, we talk about the likelihood of finding it in a certain region.

For a hydrogen atom in the ground state, the probability density is described by the wave function \(\psi(r)\) squared, which translates to \(\psi^{2}(r) = \frac{4}{a_{0}^{3}} e^{-2r/a_{0}}\). This gives a mathematical probability for an electron to be found at a particular distance \(r\) from the nucleus.

The exponential nature of this function indicates that the maximum probability does not occur at zero distance from the nucleus. Instead, there's a more complex interplay of forces, which leads to a non-zero most probable distance where the electron is most likely found.

Bohr Radius

The Bohr radius, denoted as \(a_0\), represents a fundamental constant in atomic physics. It defines the most probable distance from the nucleus where the electron in a hydrogen atom's ground state might be found.

Mathematically, this radius corresponds to the distance at which the probability density function peaks. In simpler terms, it is \(a_0\) where the electron is most likely to be when it is in the ground state. This means the electron spends a lot of its time around this radius, appearing to orbit the nucleus at this distance.

The calculated value of \(a_0\) is approximately 0.529 × 10^-10 meters, a crucial metric which helps in comparing atomic sizes and understanding the structure of atoms more deeply. This distance, though small on human scales, is significant in defining atomic scales.

1s Orbital

The 1s orbital is a spherical region around the nucleus where an electron is likely found when in the n=1 energy level of a hydrogen atom. This is the simplest type of orbital and is associated with the ground state.

Unlike higher orbitals, the 1s orbital does not have nodes—regions within the orbital where the probability density drops to zero. Consequently, the probability density decreases steadily as you move away from the nucleus, reflecting the electron radiation pattern's changing likelihood.

Electrons in the 1s orbital have the highest probability of being found near the nucleus due to their low energy state, characterized by the wave function's expanse. This is where the balance of attractive and repulsive forces achieves a steady state, allowing the electron to reside optimally as per quantum mechanical principles.