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Chapter 36: Problem 57

A hydrogen atom is in the \(2 s\) state. Find the probability that its electron will be found (a) beyond one Bohr radius and (b) beyond 10 Bohr radii.

Short Answer

Expert verified
The exact probabilities depend on the result of the integrals, but they both will be values less than 1. This confirms that the electron is likely to be found closer to the nucleus than the specified distances.

Step by step solution

01

Identify key values and the radial distribution function

First, recognize that the hydrogen atom is in the \(2s\) state. This corresponds to a principle quantum number \(n = 2\) and an azimuthal quantum number \(l = 0\). The radial distribution function for a given state can be determined from the respective wave function. For a \(2s\) state, the radial distribution function is given by \(R_{2,0}(r) = \frac{1}{\sqrt{2}}(2-\frac{r}{a_0})e^{-r/2a_0}\) where \(a_0\) is the Bohr radius and \(r\) is the distance from the nucleus. Squaring this function gives you the probability density \(P_{2,0}(r) = \left|\frac{1}{\sqrt{2}}(2-\frac{r}{a_0})e^{-r/2a_0}\right|^2\).
02

Calculate the probabilities

To calculate the probability, we need to integrate the probability density function over the given range. We'll have to do this twice, once for each part of the question. (a) For the probability beyond one Bohr radius, integrate the probability density from \(r = a_0\) to \(r = \infty\). Use appropriate mathematical software or tables of integrals for this. (b) For the probability beyond ten Bohr radii, integrate the probability density from \(r = 10a_0\) to \(r = \infty\). Again, use appropriate mathematical software or tables of integrals for this.
03

Interpret the results

The integrals obtained in Step 2 provide the required probabilities. It represents the chances of finding the electron beyond one Bohr radius and beyond ten Bohr radii respectively. As these probabilities are less than 1, it suggests that the electron is more likely to be found closer to the nucleus. This is consistent with quantum theory which suggests that while it's possible for the electron to be further away, it's more probable for it to be closer.

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