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The maximum radial probability in 1s-orb...

The maximum radial probability in `1s`-orbital occures at a distance when : `[r_(0)= "Bohr radius"]`

A

`r=r_(0)`

B

`r=2r_(0)`

C

`r=r_(0)/2`

D

`2 r=r_(0)/2`

Text Solution

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The correct Answer is:
To determine the distance at which the maximum radial probability in the 1s orbital occurs, we can follow these steps: ### Step 1: Understand the Concept of Radial Probability The radial probability density for an electron in an atom is given by the function that describes the likelihood of finding an electron at a certain distance from the nucleus. For the 1s orbital, this is influenced by the quantum number \( n \) and the radial wave function. ### Step 2: Identify the Quantum Numbers For the 1s orbital, the principal quantum number \( n = 1 \). The number of nodes in the radial wave function is given by \( n - 1 \). Since \( n = 1 \), the number of nodes is: \[ n - 1 = 1 - 1 = 0 \] This indicates that the 1s orbital has no nodes. ### Step 3: Analyze the Radial Probability Distribution The radial probability distribution for the 1s orbital can be expressed mathematically. The maximum radial probability occurs at a specific distance from the nucleus, which is related to the Bohr radius \( r_0 \). ### Step 4: Relate to the Bohr Radius The Bohr radius \( r_0 \) is a fundamental physical constant that represents the most probable distance of the electron from the nucleus in the hydrogen atom. For the 1s orbital, the maximum radial probability is found at this distance: \[ r = r_0 \] ### Step 5: Conclusion Thus, the maximum radial probability in the 1s orbital occurs at a distance equal to the Bohr radius \( r_0 \). ### Final Answer The maximum radial probability in the 1s orbital occurs at a distance when \( r = r_0 \) (Bohr radius). ---
To determine the distance at which the maximum radial probability in the 1s orbital occurs, we can follow these steps: ### Step 1: Understand the Concept of Radial Probability The radial probability density for an electron in an atom is given by the function that describes the likelihood of finding an electron at a certain distance from the nucleus. For the 1s orbital, this is influenced by the quantum number \( n \) and the radial wave function. ### Step 2: Identify the Quantum Numbers For the 1s orbital, the principal quantum number \( n = 1 \). The number of nodes in the radial wave function is given by \( n - 1 \). Since \( n = 1 \), the number of nodes is: \[ ...
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