In the Bohr model of hydrogen, why is the atom 9 times larger in the n = 3 state than in the ground state?

Options

A. The electrons are moving 9 times slower
B. The de Broglie wavelength is 9 times longer
C. There are 9 times as many wavelengths in the orbit
D. The de Broglie wavelength is 3 times longer, and there are 3 times as many wavelengths

Answer

To understand why the hydrogen atom is 9 times larger in the n = 3 state compared to the ground state (n = 1) in the Bohr model, we can follow these steps:

### Step 1: Understand the Bohr Model
The Bohr model describes the hydrogen atom as having electrons in discrete energy levels or orbits around the nucleus. The principal quantum number \( n \) indicates the energy level of the electron.

### Step 2: Identify the Ground State and Excited State
In the ground state, the electron is in the lowest energy level, which is \( n = 1 \). When the electron is in the third energy level, it is in an excited state, represented by \( n = 3 \).

### Step 3: Determine the Radius of the Orbits
According to the Bohr model, the radius of the electron's orbit is given by the formula:
\[ r_n = n^2 \cdot r_1 \]
where \( r_n \) is the radius for the nth orbit and \( r_1 \) is the radius of the ground state (n=1).

### Step 4: Calculate the Radius for n = 3
For \( n = 3 \):
\[ r_3 = 3^2 \cdot r_1 = 9 \cdot r_1 \]
This shows that the radius of the orbit in the n = 3 state is 9 times larger than the radius in the ground state.

### Step 5: Conclusion
Thus, the hydrogen atom is 9 times larger in the n = 3 state than in the ground state because the radius of the electron's orbit increases with the square of the principal quantum number \( n \).

### Summary
In summary, the size of the hydrogen atom in the n = 3 state is 9 times larger than in the ground state due to the relationship between the principal quantum number and the radius of the electron's orbit.

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