Consider two atoms: Hydrogen (Z = 1) and Carbon (Z = 6). If the Bohr radius of ground state electron orbit in hydrogen atom be R, what would be the Bohr radius of the single ground-state electron in the carbon atom? 6R R6 R/62 R/6
Consider two atoms: Hydrogen (Z = 1) and Carbon (Z = 6). If the Bohr radius of ground state electron orbit in hydrogen atom be R, what would be the Bohr radius of the single ground-state electron in the carbon atom? 6R R6 R/62 R/6
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![### Bohr Radius in Hydrogen and Carbon Atoms
**Problem Statement:**
Consider two atoms: Hydrogen (Z = 1) and Carbon (Z = 6). If the Bohr radius of the ground state electron orbit in a hydrogen atom is \( R \), what would be the Bohr radius of the single ground-state electron in the carbon atom?
**Multiple Choice Options:**
- \( 6R \)
- \( R^6 \)
- \( \frac{R}{6^2} \)
- \( \frac{R}{6} \)
To understand the relationship between the Bohr radius of different atoms, we use the formula for the Bohr radius of a hydrogen-like atom:
\[ r_n = \frac{n^2 \hbar^2}{Z k e^2 m_e} \]
Where:
- \( r_n \) is the radius of the orbit.
- \( n \) is the principal quantum number.
- \( \hbar \) is the reduced Planck's constant.
- \( e \) is the electron charge.
- \( k \) is Coulomb's constant.
- \( m_e \) is the electron mass.
- \( Z \) is the atomic number.
For the ground state (\( n = 1 \)):
\[ r = \frac{\hbar^2}{Z k e^2 m_e} \]
Given that for hydrogen (Z = 1), the Bohr radius \( R \) is:
\[ R = \frac{\hbar^2}{k e^2 m_e} \]
For the carbon atom (\( Z = 6 \)):
\[ r_{carbon} = \frac{\hbar^2}{6 k e^2 m_e} = \frac{R}{6} \]
Therefore, the correct answer is:
- \( \frac{R}{6} \)
This question tests the understanding of the Bohr model and how the atomic number \( Z \) affects the radius of the electron orbit in different atoms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c939cff-bc46-4066-b22a-d4e55e2c89ab%2F32e81f2f-56ac-4e29-aceb-008c54c99aef%2Fvdwpvi_processed.png&w=3840&q=75)
Transcribed Image Text:### Bohr Radius in Hydrogen and Carbon Atoms
**Problem Statement:**
Consider two atoms: Hydrogen (Z = 1) and Carbon (Z = 6). If the Bohr radius of the ground state electron orbit in a hydrogen atom is \( R \), what would be the Bohr radius of the single ground-state electron in the carbon atom?
**Multiple Choice Options:**
- \( 6R \)
- \( R^6 \)
- \( \frac{R}{6^2} \)
- \( \frac{R}{6} \)
To understand the relationship between the Bohr radius of different atoms, we use the formula for the Bohr radius of a hydrogen-like atom:
\[ r_n = \frac{n^2 \hbar^2}{Z k e^2 m_e} \]
Where:
- \( r_n \) is the radius of the orbit.
- \( n \) is the principal quantum number.
- \( \hbar \) is the reduced Planck's constant.
- \( e \) is the electron charge.
- \( k \) is Coulomb's constant.
- \( m_e \) is the electron mass.
- \( Z \) is the atomic number.
For the ground state (\( n = 1 \)):
\[ r = \frac{\hbar^2}{Z k e^2 m_e} \]
Given that for hydrogen (Z = 1), the Bohr radius \( R \) is:
\[ R = \frac{\hbar^2}{k e^2 m_e} \]
For the carbon atom (\( Z = 6 \)):
\[ r_{carbon} = \frac{\hbar^2}{6 k e^2 m_e} = \frac{R}{6} \]
Therefore, the correct answer is:
- \( \frac{R}{6} \)
This question tests the understanding of the Bohr model and how the atomic number \( Z \) affects the radius of the electron orbit in different atoms.
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