Question

At what value of \(Z\) would the innermost orbit be expected to be pulled inside a nucleus of radius \(1.0 \times 10^{-13} \mathrm{~cm}\), assuming simple Bohr theory and Bohr radius \(=0.5292 \times 10^{-8} \mathrm{~cm} ?\) (Assume \(\left.\mathrm{N}=1 .\right)\)

Short answer

The atomic number Z is approximately equal to \(5.292 \times 10^4\) when the innermost orbit (n=1) is expected to be pulled inside the nucleus with the given radius of \(1.0 \times 10^{-13} cm\).

Step-by-step solution

Rewrite the formula for Z

From the formula for effective Bohr radius, solve for Z by rewriting the formula as follows: \[Z = \frac{n^2r_{0}}{r_{n}}\]

Plug in the given values

We are given that n=1, \(r_0 = 0.5292 \times 10^{-8} cm\), and \(r_n = 1.0 \times 10^{-13} cm\). Plug these values into the formula derived in Step 1: \[Z = \frac{(1)^2\left(0.5292\times 10^{-8}\right)}{1.0\times 10^{-13}}\]

Perform the calculations

Now, calculate Z using the given values: \[Z = \frac{0.5292\times 10^{-8}}{1.0\times 10^{-13}}\] \[Z = 0.5292 \times 10^5\]

Find the value of Z

Finally, find the value of Z: \[Z \approx 5.292 \times 10^4\] We find that the atomic number Z is approximately equal to \(5.292 \times 10^4\) when the innermost orbit is expected to be pulled inside the nucleus with the given radius.

Key concepts

Understanding the Bohr Radius

The Bohr radius, denoted as \( r_0 \), is a fundamental physical constant that represents the most likely distance between the nucleus and the electron in the ground state (the lowest energy level) of a hydrogen atom. It is a key element in the Bohr model of the atom, where electrons orbit the nucleus in well-defined circular paths. The Bohr radius is approximately \( 0.529 \times 10^{-10} \) meters, or \( 0.5292 \times 10^{-8} \) centimeters.

In simple terms, you can think of the Bohr radius as the 'ideal' radius of the first orbit an electron would take around the nucleus in a hydrogen atom. When students work with Bohr radius calculations, understanding this concept is critical, as it forms the basis of predicting the behavior of electrons in other elements when their atomic number, represented by \( Z \), changes.

Often presented in equation form, the effective radius of an electron orbit in the \( n \)-th level for any atom other than hydrogen can be calculated using \( r_n = \frac{n^2r_{0}}{Z} \), where \( n \) is the principal quantum number. This relationship helps explain why the size of electron orbits changes with different atomic numbers, implying that the size of the atom itself changes as well.

The Atomic Number and its Role

The atomic number, represented as \( Z \), is the number of protons present in the nucleus of an atom. This number is fundamental as it defines the identity of an element: each element in the periodic table has a unique atomic number. For instance, hydrogen has an atomic number of 1, helium has 2, and so on.

This number isn't just a label; it significantly affects the atom's properties and behavior. A larger atomic number means more protons in the nucleus, which leads to a stronger attraction to the electrons that are orbiting and thus, correlates to a smaller Bohr radius for any given electron shell level. Students should note that when dealing with problems related to the Bohr model, the value of \( Z \) directly impacts various calculations, including that of the electron orbits and the energies associated with different electron transitions.

Nucleus Radius and its Significance

Understanding the nucleus radius is essential when studying atomic structure. The nucleus radius refers to the size of the atomic nucleus and is typically measured in femtometers (\( 1 \text{ fm} = 10^{-15} \text{ m} \)). While the Bohr model primarily concerns electron orbits, the nucleus radius can provide insight into the limits of the model, as it reflects the physical space that the nucleus occupies within the atom.

The exercise given uses the term 'nucleus radius' to understand up to which atomic number, considering the limitations of the Bohr model, an electron's orbit might be physically constrained by the size of the nucleus itself. In the context of the problem, increasing the atomic number while maintaining a constant nucleus radius eventually leads to a scenario where the innermost orbit would be pulled inside the nucleus' physical region.

In real-world scenarios, while the nucleus of an atom is extremely dense, its radius is much smaller than the radius of the atom itself, highlighting why the electron, under normal circumstances, wouldn't fall into the nucleus. The example provided is theoretical, demonstrating the Bohr model's limits rather than an actual physical occurrence.