Chapter 41: Q. 54 (page 1209)

Prove that the most probable distance from the proton of an electron in the $2 s$ state of hydrogen is $5.236 a_{B}$ .

Short Answer

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The most probable distance from the proton of an electron in the $2 s$ state of hydrogen is $5.236 a_{B}$ .

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Given information

We need to prove that the most probable distance from the proton of an electron in the $2 s$ state of hydrogen is $5.236 a_{B}$ .

Explanation

We need to prove that $r = 5.236 a_{B}$ where $r$ is most probable distance.

For that we will find radial probability density for $2 s$ state:

role="math" localid="1650381765979" $P r (r) = 4 {πr}^{2} {|R_{2 s} (r)|}^{2} \Pr (r) = 4 π {A_{2 s}}^{2} r^{2} {(1 - \frac{r}{2 a_{B}})}^{2} e \frac{- r}{a_{B}}$ .

We can treat as constant $4 {πA}_{2 s}^{2} r^{2} = C$ and rearranging the equation on right hand side ,

$P r (r) = C (r^{2} - \frac{r^{3}}{a_{B}} + \frac{r^{4}}{4 {a^{2}}_{B}}) e \frac{- r}{a_{B}}$ .

Now we will differentiate it with respect to $r$ $\frac{d P_{r}}{d r} = C [(2 r - \frac{3 r^{2}}{a_{B}} + \frac{r^{3}}{{a_{B}}^{2}}) e^{\frac{- r}{a_{B}}} - \frac{1}{a_{B}} (r^{2} - \frac{r^{3}}{a_{B}} + \frac{r^{4}}{4 {a_{B}}^{2}}) e^{\frac{- r}{a_{B}}}] \frac{d P_{r}}{d r} = C [2 - \frac{4 r}{a_{B}} + \frac{2 r^{2}}{a_{B}} - \frac{r^{3}}{{4 a_{B}}^{2}}] r e^{\frac{- r}{a_{B}}}$

Rearranging the terms , we get

$0 = [2 - 4 (\frac{r}{a_{B}}) + 2 {(\frac{2}{a_{B}})}^{2} - \frac{1}{4} {(\frac{r}{a_{B}})}^{3}] 0 = 2 - 4 (5.236) + 2 {(5.236)}^{2} - \frac{1}{4}$

$r = 5.236 a_{B}$

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Prove that the most probable distance from the proton of an electron in the $2 s$ state of hydrogen is $5.236 a_{B}$ .

Short Answer

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Explanation

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Prove that the most probable distance from the proton of an electron in the2sstate of hydrogen is 5.236aB.

Short Answer

Step by step solution

Given information

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

Torque and Rotational Motion

Magnetism and Electromagnetic Induction

Electricity and Magnetism

Geometrical and Physical Optics

Space Physics

Study anywhere. Anytime. Across all devices.

Prove that the most probable distance from the proton of an electron in the $2 s$ state of hydrogen is $5.236 a_{B}$ .