Chapter 2: Problem 56
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
Expert verified
Statement 1 is not true.
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
\[ L = \frac{nh}{2\text{π}} \]
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.
\[ r_n = n^2 \frac{r_0}{Z} \]
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.
\[ \text{ΔE} = E_{\text{final}} - E_{\text{initial}} = hu \]
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.
