Chapter 41: Q. 30 (page 1207)

Calculate (a) the radial wave function and (b) the radial probability density at $r = \frac{1}{2} a_{B}$ for an electron in the $1 s$ state of hydrogen. Give your answers in terms of $a_{B}$

Short Answer

Expert verified

(a) The radial wave function for an electron at $r = \frac{1}{2} a_{B}$ in the $1 s$ state of hydrogen is $R_{1 s} (r) = 0.34 {a_{B}}^{- 3 / 2}$ .

(b) The radial probability density at $r = \frac{1}{2} a_{B}$ in the $1 s$ state of hydrogen is $P_{r} (r) = 0.37 {a_{B}}^{- 1}$ .

Step by step solution

Part (a) Step 1 : Given Information

We have find the radial wave function for an electron in the $1 s$ state of hydrogen at $r = \frac{1}{2} a_{B}$

Part (a) Step 2 : Simplification

The radial wave function of hydrogen in the $1 s$ state is

$R_{1 s} (r) = \frac{1}{\sqrt{π {a_{B}}^{3}}} e^{- r / a_{B}}$ ( $a_{B}$ is Bohr radius and $r$ is distance of electron from nuclei)

localid="1650366257896" $\Rightarrow R_{1 s} (\frac{1}{2} a_{B}) = \frac{1}{\sqrt{π {a_{B}}^{3}}} e^{- 1 / 2} = 0.34 {a_{B}}^{- 3 / 2}$

Part (b) Step 1 : Given Information

We have find the probability density for an electron in the $1 s$ state of hydrogen at $r = \frac{1}{2} a_{B}$

Part (b) Step 2 : Simplification

The probability density is

$P_{r} (r) = 4 π r^{2} {|R_{n l} (r)|}^{2}$ ( $a_{B}$ is Bohr radius and $r$ is distance the distance of electron from nuclei)

localid="1650366757646" $\Rightarrow P_{r} (\frac{1}{2} a_{B}) = 4 π {(\frac{a_{B}}{2})}^{2} {|\frac{1}{\sqrt{π {a_{B}}^{3}}} e^{- 1 / 2}|}^{2} = 0.37 a_{_{B}}^{- 1}$

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Calculate (a) the radial wave function and (b) the radial probability density at $r = \frac{1}{2} a_{B}$ for an electron in the $1 s$ state of hydrogen. Give your answers in terms of $a_{B}$

Short Answer

Step by step solution

Part (a) Step 1 : Given Information

Part (a) Step 2 : Simplification

Part (b) Step 1 : Given Information

Part (b) Step 2 : Simplification

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Calculate (a) the radial wave function and (b) the radial probability density at r=12aBfor an electron in the 1s state of hydrogen. Give your answers in terms ofaB

Short Answer

Step by step solution

Part (a) Step 1 : Given Information

Part (a) Step 2 : Simplification

Part (b) Step 1 : Given Information

Part (b) Step 2 : Simplification

One App. One Place for Learning.

Most popular questions from this chapter

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Calculate (a) the radial wave function and (b) the radial probability density at $r = \frac{1}{2} a_{B}$ for an electron in the $1 s$ state of hydrogen. Give your answers in terms of $a_{B}$