CHEM 101 Lecture 7: 1s Orbital Wave Function

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Questions and Answers

What is the wave function for the 1s orbital of hydrogen at a distance of $2a_0$?

$N_{1s} e^{-3}$

$N_{1s} 0.135$ (correct)

$N_{1s} 0.0497$

$N_{1s} e^{-2}$

Which wave function corresponds to the 2s orbital of hydrogen?

$N_{2s}(r^2)e^{-r/2a_0}$

$N_{2s}(2 + r/a_0)e^{-r/2a_0}$

$N_{2s}(2 - r/a_0)e^{-r/2a_0}$ (correct)

$N_{2s} e^{-r/a_0}$

Which statement about the 1s and 2s orbitals is true?

Both orbitals have one radial node.

1s has no radial nodes, 2s has one radial node. (correct)

Both orbitals have the same number of radial nodes.

1s has two radial nodes, 2s has none.

The 2p orbital possesses which characteristic feature?

1 radial node and 1 angular node. Signup and view all the answers

What equation represents the radial part of the wave function for the 3s orbital?

$\psi_{3s} = N_{3s}(27 - 18(r/a_0) + 2(r/a_0)^2)e^{-r/3a_0}$ Signup and view all the answers

For which orbital is the following statement true: 'has no radial nodes'?

1s Signup and view all the answers

What is the characteristic of the wave function for the 2s and 2p orbitals in terms of nodes?

2p has more angular nodes than 2s. Signup and view all the answers

What does the term 'radial node' refer to in atomic orbitals?

A point where the wave function is zero. Signup and view all the answers

Which wave function is used for the 2p orbital at $r = 5a_0$?

$N_{2p}(4)e^{-2.5}$ Signup and view all the answers

What is the equation for the volume of a shell of radius $r$ and thickness $\delta r$?

$4\pi r^2 \cdot \delta r$ Signup and view all the answers

The radial distribution function (RDF) for the 1s orbital is expressed as which of the following?

$[\psi_{1s}(r)]^2 \cdot 4\pi r^2$ Signup and view all the answers

If $\delta r$ is kept constant in the expression for RDF, which two terms does the RDF depend on?

$[\psi_{1s}(r)]^2$ and $r^2$ Signup and view all the answers

For the 2s orbital, how is the radial distribution function denoted?

$P_{2s}(r) = [R_{2s}(r)]^2 \cdot r^2$ Signup and view all the answers

What does the parameter $r_{mp}$ typically represent for orbitals?

Most probable radius of an electron Signup and view all the answers

Which radial distribution function relates to the 3p orbital?

$P_{3p}(r) = [\psi_{3p}(r)]^2 \cdot r^2$ Signup and view all the answers

Which expression correctly represents the probability of finding an electron in a volume $\delta V$?

$[\psi_{1s}(r)]^2 \cdot 4\pi r^2$ Signup and view all the answers

Which of the following statements regarding radial distribution functions of ns orbitals is correct?

P3s has two radial nodes. Signup and view all the answers

Study Notes

Wave Functions of Hydrogen Atom Orbitals

1s Orbital: Given by the wave function ( \psi_{1s}(r) = N_{1s} e^{-r/a_0} )
2s Orbital: Described by ( \psi_{2s}(r) = N_{2s}(2 - r/a_0)e^{-r/2a_0} )
3s Orbital: Defined as ( \psi_{3s}(r) = N_{3s}{27 - 18(r/a_0) + 2(r/a_0)^2} e^{-r/3a_0} )

Properties of Radial Distribution Functions (RDF)

RDF Formula: ( P_{n, l}(r) = |R_{n,l}(r)|^2 \cdot r^2 ) where ( R_{n,l}(r) ) is the radial part of the wave function.
1s RDF: ( P_{1s}(r) = |R_{1s}(r)|^2 \cdot r^2 )
2s RDF: ( P_{2s}(r) = |R_{2s}(r)|^2 \cdot r^2 )
3s RDF: ( P_{3s}(r) = |R_{3s}(r)|^2 \cdot r^2 )

Nodes in Orbitals

Radial Nodes: Points where the probability density ( P(r) = 0 ).
1s Orbital: Contains zero radial nodes.
2s Orbital: Contains one radial node.
3s Orbital: Contains two radial nodes.

Evaluation Points of Wave Functions

Specific evaluations for ( \psi_{1s}(r) ) include:
- ( r = a_0 ): ( \psi_{1s} = N_{1s} e^{-1} \approx N_{1s} 0.368 )
- ( r = 2a_0 ): ( \psi_{1s} = N_{1s} e^{-2} \approx N_{1s} 0.135 )
Specific evaluations for ( \psi_{2s}(r) ):
- ( r = a_0 ): ( \psi_{2s} \approx N_{2s} 0.606 )
- ( r = 2a_0 ): ( \psi_{2s} = 0 ) (Radial node)
- ( r = 3a_0 ): ( \psi_{2s} \approx -N_{2s} 0.222 )

Angular Node and Nodal Planes

Angular Nodes: Regions where probability density is zero due to angular momentum; for example, 2p and 3p orbitals exhibit nodal planes.

Radial Distribution of Orbitals

1s, 2s, 3s Orbital RDFs:
- Maximum radial distances occur at ( r_{mp} = a_0 ) for 1s, ~5a0 for 2s, and ~11a0 for 3s.
Comparative RDFs: The shapes of the radial distribution functions highlight the increasing complexity and number of nodes in higher energy states.

Conclusion

The hydrogen atom's orbitals have defined wave functions characterized by radial functions that govern electron distribution and probability. Understanding these properties is crucial for deeper insights into atomic structure and behavior in quantum chemistry.

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Description

This quiz covers the key concepts related to the wave function for the 1s orbital of hydrogen as presented in CHEM 101 Lecture 7. Focus on understanding the mathematical expressions and their implications for quantum chemistry and atomic structure.