Question

Which of the following concerning Bohr's model is not true? (1) It predicts that probability of an electron near nucleus is more. (2) Angular momentum of electron in \(n\) th orbit is given by \(n \mathrm{~h} / 2 \pi\). (3) The radius of an orbit is proportional to \(\frac{n^{2}}{Z}\). (4) When an electron jump from \(N\) to \(K\) shell, energy is released.

Short answer

Statement 1 is not true.

Step-by-step solution

Understanding Bohr's model statements

Examine each given statement and recall the key principles of Bohr's model of the atom. Bohr's model describes electrons in discrete orbits around the nucleus with specific energies.

Evaluate Statement 1

Statement 1: It predicts that the probability of an electron near the nucleus is more. This is not true in Bohr's model; the probability distribution of an electron is not addressed. This is more accurately described by quantum mechanics.

Evaluate Statement 2

Statement 2: Angular momentum of an electron in the nth orbit is given by \(\frac{nh}{2\pi}\). This is a core aspect of Bohr's model where the angular momentum is quantized.

Evaluate Statement 3

Statement 3: The radius of an orbit is proportional to \(\frac{n^{2}}{Z}\). Bohr's model states that the radius of an electron's orbit in a hydrogen-like atom is given by \(r_n = n^2 \frac{r_0}{Z}\), which agrees with this statement.

Evaluate Statement 4

Statement 4: When an electron jumps from the N to the K shell, energy is released. According to Bohr's model, moving to a lower energy level releases energy in the form of photons.

Conclusion

Compare all statements against Bohr's model. Statement 1 is the only one that does not match as it relates more to quantum mechanics than Bohr’s model.

Key concepts

Quantized Angular Momentum

In Bohr's model of the atom, one of the pivotal concepts is the idea of quantized angular momentum. This means that an electron can only occupy certain allowed orbits around the nucleus, each with a specific, fixed value of angular momentum. Bohr introduced the formula:
\[ L = \frac{nh}{2\text{π}} \]
where
• L is the angular momentum,
• n is the principal quantum number (which can be 1, 2, 3, ...), and
• h is Planck's constant.

By using this formula, Bohr highlighted that the angular momentum of the electron is not arbitrary but comes in discrete multiples of \frac{h}{2\text{π}}. This solved the problem of why electrons do not spiral into the nucleus and collapse the atom.

Electron Orbit Radius

Bohr's model also describes how the radius of an electron's orbit varies. Specifically, the radius is directly proportional to the square of the principal quantum number, n, and inversely proportional to the atomic number, Z. The formula is given by:
\[ r_n = n^2 \frac{r_0}{Z} \]
Where
• r_n is the radius of the nth orbit,
• r_0 is a constant for the radius of the first orbit (also known as the Bohr radius), and
• Z is the atomic number of the element.

This indicates that as the principal quantum number increases, the orbit radius grows rapidly (since it's proportional to n^2), and for atoms with higher atomic numbers, the orbit radius decreases due to the stronger nuclear attraction.

Energy Level Transitions

Another crucial aspect of Bohr's model is the concept of energy level transitions. Electrons can move between fixed orbits, but to do so, they must absorb or emit a specific amount of energy in the form of photons. When an electron moves from a higher energy orbit (e.g., N shell) to a lower energy orbit (e.g., K shell), the transition releases energy:
\[ \text{ΔE} = E_{\text{final}} - E_{\text{initial}} = hu \]
Here,
• ΔE is the change in energy,
• h is Planck's constant, and
• ν is the frequency of the emitted or absorbed photon.

Bohr's model successfully explains the spectral lines of hydrogen by attributing them to these energy transitions. Different elements have distinct energy levels and thus unique spectral lines.