Question

As the electron in Bohr is orbit of hydrogen atom Passes from state \(\mathrm{n}=2\) to \(\mathrm{n}=1\), the \(\mathrm{K} . \mathrm{E}\). and Potential energy changes as (A) Two fold, also two fold (B) four fold, two fold (C) four fold, also four fold (D) two fold, four fold

Short answer

As the electron in the hydrogen atom transitions from n = 2 to n = 1, both the kinetic energy and potential energy change by a factor of 4. Therefore, the correct answer is (C) four fold, also four fold.

Step-by-step solution

1. Recall the formula for kinetic and potential energy in Bohr's model.

In Bohr's model, the kinetic energy (KE) and potential energy (PE) of an electron in the hydrogen atom, at a particular energy level n, are given by: KE = \(\frac{KZe^2}{2\cdot a_0\cdot n^2}\) and PE = \(\frac{-KZe^2}{a_0\cdot n^2}\) Here, K is Coulomb's constant, Z is the atomic number (Z = 1 for hydrogen), e is the elementary charge, and \(a_0\) is the Bohr's radius.

2. Calculate the ratio of kinetic energies and potential energies at the two principal quantum numbers n = 1 and n = 2.

We will find the ratio of the kinetic energy at n = 2 to the kinetic energy at n = 1, and similarly for the potential energy. \(\frac{KE_2}{KE_1}\) = \(\frac{\frac{KZe^2}{2\cdot a_0\cdot 4}}{\frac{KZe^2}{2\cdot a_0}}\) = \(\frac{1}{4}\) \(\frac{PE_2}{PE_1}\) = \(\frac{\frac{-KZe^2}{a_0 \cdot 4}}{\frac{-KZe^2}{a_0}}\) = \(\frac{1}{4}\)

3. Analyze the changes in kinetic and potential energy.

From the ratios calculated in step 2, we can see that as the electron transitions from n = 2 to n = 1, both the kinetic energy and potential energy change four-fold (reduced by a factor of 4). So, the correct answer is: (C) four fold, also four fold.

Key concepts

Kinetic Energy

Kinetic energy is the energy that an object has due to its motion. In the context of Bohr's model of the hydrogen atom, this energy refers to the motion of the electron as it orbits the nucleus. In Bohr's model, the kinetic energy (appellation{KE}alp) of an electron in orbit is given by the formula:
\[KE = \frac{KZe^2}{2 \times a_0 \times n^2}\]
Where:

• \(K\) is Coulomb's constant.
• \(Z\) is the atomic number.
• \(e\) is the elementary charge.
• \(a_0\) is the Bohr radius.
• \(n\) is the principal quantum number indicating the orbit level.

As the electron transitions between orbits, its kinetic energy changes according to the inverse square of the quantum number. For instance, as an electron moves from \(n=2\) to \(n=1\), the kinetic energy increases by a factor of four.

Potential Energy

Potential energy in Bohr's model of the hydrogen atom refers to the energy due to the electrostatic interaction between the positively charged nucleus and the negatively charged electron. It can be visualized as the energy stored in the position of the electron with respect to the nucleus. The potential energy (appellation{PE}alp) is calculated as:
\[PE = \frac{-KZe^2}{a_0 \times n^2}\]
Notice the negative sign, indicating an attractive force between the opposite charges.

• As the quantum number \(n\) decreases, the potential energy magnitude increases (or becomes more negative), indicating the electron is at a state of lower energy.
• This change correlates with the electron moving closer to the nucleus, reducing the separation distance and therefore releasing energy.

As the electron falls from one energy level to another, the potential energy changes by the same factor as kinetic energy, which is four times when the transition is from \(n=2\) to \(n=1\).

Hydrogen Atom

The hydrogen atom, being the simplest possible atomic structure, consists of a single proton and a single electron. This simplicity permits a deep understanding of atomic models and quantum mechanics.

• Bohr's model approximates the hydrogen atom with an electron moving in a circular orbit around the proton.
• This model introduces quantized energy levels, where each orbit corresponds to a different energy level denoted by the principal quantum number \(n\).

When Bohr analyzed the hydrogen spectra, he found that electrons jump between these quantized orbits, emitting or absorbing light. The emitted light has a frequency that matches the energy difference between the levels thus providing experimental evidence for quantized orbits.

Quantum Numbers

Quantum numbers are a set of values that provide solutions to the Schrödinger equation for the hydrogen atom and define the properties of atomic orbitals and the properties of electrons within those orbitals.

• The principal quantum number \(n\) indicates the size and energy level of the orbital.
• The angular momentum quantum number \(l\) describes the shape of the orbital.
• The magnetic quantum number \(m_l\) indicates the orientation of the orbital in space.
• The spin quantum number \(m_s\) specifies the orientation of the electron’s spin.

In Bohr's model, only the principal quantum number \(n\) is relevant, as it determines the energy level of the electron's orbit. The transition of an electron between these levels requires or releases energy, manifesting as spectral lines observed in experiments involving hydrogen.