Easy radial probability question

In summary, to determine the most probable radius of an electron for a 2s hydrogenic atom, you can use the equation r*=a0/Z where a0 is the Bohr radius. The book provides the answer of ~5.2a0/Z, but it may be necessary to use the derivative of the wavefunction to find the coefficient of 5.2. To find the most probable value, you can use the equation P(r+dr)=|\Psi_{200}|^24\pi r^2dr and look for the value of r that makes P(r+dr) a maximum. This condition involves taking the derivative of P(r) with respect to r.
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Homework Statement



Determine the most probable radius of an electron for a 2s hydrogenic atom.

Homework Equations



r*=a0/Z where a0 is Bohr radius

The Attempt at a Solution



Well, I know that the answer is ~5.2a0/Z. The book tells me that the most probable radius for hydrogen (EDIT - a 2s hydrogen shell I mean) is 5.2a0 - thus I know that the answer is simply 5.2a0/Z (275pm/Z) for the series of hydrogenic atoms. My only problem is this: I don't know how to get the coefficient of 5.2, and presumably simply getting it out of the book is not the right way to do it. I think it may have to do with taking the derivative of the wavefunction but I'm unsure...Any help would be appreciated, thanks!
 
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  • #2
The probability for an electron to be found in the radial range r+dr can be determined using the wavefunction for the 2S state:

[tex]P(r+dr)=|\Psi_{200}|^24\pi r^2dr[/tex]

Now if you want to find the most probable value, your looking for the value of r, such that P(r+dr) is a maximum. Think back to calculus. What is the condition for a function, P(r) to be maximum?

HINT: The condition involves [tex]\frac{dP}{dr}[/tex]
 

1. What is a radial probability question?

A radial probability question is a type of question that relates to the likelihood of finding an electron in a particular region of an atom's electron cloud. It is used to understand the distribution of electrons in an atom and their probability of being found at a specific distance from the nucleus.

2. How do you calculate radial probability?

Radial probability is calculated using the Schrödinger equation, which takes into account the wave function of an electron and the potential energy of the atom. The equation is solved numerically to determine the probability of finding an electron at different distances from the nucleus.

3. What does the radial probability graph show?

The radial probability graph shows the probability of finding an electron at different distances from the nucleus. It is a plot of the radial probability function, which is a function of the electron's radial distance from the nucleus. The graph typically has a peak at the most probable distance from the nucleus and decreases as the distance increases.

4. How does the radial probability change with different atomic orbitals?

The radial probability changes with different atomic orbitals because each orbital has a different shape and size, which affects the distribution of electrons around the nucleus. For example, the 1s orbital has a spherical shape and a higher probability of finding an electron closer to the nucleus compared to the 2p orbital, which has a dumbbell shape and a higher probability of finding an electron at a larger distance from the nucleus.

5. What is the significance of the radial probability question in chemistry?

The radial probability question is significant in chemistry because it helps us understand the behavior of electrons in atoms. The distribution of electrons in an atom determines its chemical properties and reactivity. The radial probability also helps in predicting the locations of chemical bonds and the energy levels of electrons in an atom.

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