An electron is revolving in its Bohr orbit having Bohr radius of 0.529 Å, then the radius of third orbit is
An electron is revolving in its Bohr orbit having Bohr radius of 0.529 Å, then the radius of third orbit is
- 4.761 Å
- 4234 nm
- 5125 nm
- 4496 Å
The Correct Option is A
Approach Solution - 1
To determine the radius of the nth orbit in the Bohr model, we use the formula:
\( r_n = r_1 \cdot n^2 \)
Where:
- \( r_n \) is the radius of the nth orbit
- \( r_1 \) is the Bohr radius = \( 0.529 \, \text{Å} \)
- \( n \) is the principal quantum number (orbit number)
For the third orbit (\( n = 3 \)):
\( r_3 = r_1 \cdot 3^2 = 0.529 \, \text{Å} \cdot 9 \)
Calculating the value:
\( r_3 = 4.761 \, \text{Å} \)
Therefore, the radius of the third orbit is: \( \boxed{4.761 \, \text{Å}} \)
Correct Option: (A) 4.761 Å
Approach Solution -2
To find the radius of the nth orbit in the Bohr atomic model, we make use of the following equation:
\( r_n = r_1 \cdot n^2 \)
Explanation of terms:
- \( r_n \): Radius of the nth energy level
- \( r_1 \): Radius of the first orbit (Bohr radius), which is equal to \( 0.529 \, \text{Å} \)
- \( n \): The principal quantum number or orbit number
Let’s calculate for the third orbit, i.e., when \( n = 3 \):
\( r_3 = r_1 \cdot 3^2 = 0.529 \, \text{Å} \cdot 9 \)
Now, performing the multiplication:
\( r_3 = 4.761 \, \text{Å} \)
Hence, the radius of the third orbit comes out to be: \( \boxed{4.761 \, \text{Å}} \)
Correct Answer: (A) 4.761 Å
Approach Solution -3
According to Bohr's model, the radius of the \( n^{th} \) orbit for a hydrogen-like atom is given by the formula: \[ r_n = n^2 a_0 \] where:
- \( r_n \) is the radius of the \( n^{th} \) orbit.
- \( n \) is the principal quantum number (orbit number).
- \( a_0 \) is the Bohr radius, which is the radius of the first orbit (\( n=1 \)) for hydrogen.
(Note: The full formula is \( r_n = \frac{n^2 a_0}{Z} \), where \( Z \) is the atomic number. Since the problem refers to the Bohr radius \( a_0 \) directly, it implies either a hydrogen atom \( (Z=1) \) or that \( a_0 \) is defined as the radius of the first orbit for the specific atom in question. Assuming it's hydrogen or the base Bohr radius is given).
We are given:
- The Bohr radius (radius of the first orbit, \( n=1 \)), \( \mathbf{a_0 = 0.529 \text{ Å}} \).
We need to find the radius of the third orbit, which corresponds to \( \mathbf{n=3} \).
Using the formula for \( n=3 \): \[ r_3 = 3^2 \times a_0 \] \[ r_3 = 9 \times a_0 \]
Substituting the given value of \( a_0 \): \[ r_3 = 9 \times 0.529 \text{ Å} \] \[ \mathbf{r_3 = 4.761 \text{ Å}} \]
Comparing this result with the given options:
- 4.761 Å
- 4234 nm (which is 42340 Å)
- 5125 nm (which is 51250 Å)
- 4496 Å
The calculated radius of the third orbit matches the first option.
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Concepts Used:
Atom Structure Models
The three atomic models are as follows:
Thomson model:
Thomson atomic model was proposed by William Thomson in the year 1900. This model explained the description of an inner structure of the atom theoretically. It was strongly supported by Sir Joseph Thomson, who had discovered the electron earlier.
Thomson assumed that an electron is two thousand times lighter than a proton and believed that an atom is made up of thousands of electrons. In this atomic structure model, he considered atoms surrounded by a cloud having positive as well as negative charges. The demonstration of the ionization of air by X-ray was also done by him together with Rutherford. They were the first to demonstrate it. Thomson’s model of an atom is similar to a plum pudding.
Rutherford’s Alpha Scattering Experiment:
Rutherford’s conducted an experiment by bombarding a thin sheet of gold with α-particles and then studied the trajectory of these particles after their interaction with the gold foil.
Bohr’s Model of an Atom:
Bohr model of the atom was proposed by Neil Bohr in 1915. It came into existence with the modification of Rutherford’s model of an atom. Rutherford’s model introduced the nuclear model of an atom, in which he explained that a nucleus (positively charged) is surrounded by negatively charged electrons.