Q. 54 Prove that the most probable dis... [FREE SOLUTION]

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 $5.236 a_{B}$ .

Step by step solution

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" $P r (r) = 4 {蟺谤}^{2} {|R_{2 s} (r)|}^{2} \Pr (r) = 4 蟺 {A_{2 s}}^{2} r^{2} {(1 - \frac{r}{2 a_{B}})}^{2} e \frac{- r}{a_{B}}$ .

We can treat as constant $4 {蟺础}_{2 s}^{2} r^{2} = C$ 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

Step by step solution

Given information

Explanation

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91影视

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

Rotational Dynamics

Electric Charge Field and Potential

Atoms and Radioactivity

Torque and Rotational Motion

Conservation of Energy and Momentum

Work Energy and Power

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 $5.236 a_{B}$ .