Chapter 7: Q53E (page 282)

An electron is in the 3d state of a hydrogen atom. The most probable distance of the electron from the proton is $9 a_{o}$ . What is the probability that the electron would be found between $8 a_{o} a n d 10 a_{o}$ ?

Short Answer

Expert verified

The probability of finding the electron between $8 a_{o} a n d 10 a_{o} i s 0.212 .$

Step by step solution

Given data

The state of an electron – 3d.

Concept

The space around the proton in a hydrogen atom, in which the probability of finding the electron is at least $90 %$ , is known as the orbital.

Solution

The radial function is given as-

$R_{3, 2} (r) = \frac{1}{{(3 a_{o})}^{3 / 2}} \frac{(2 \sqrt{2 r^{2}})}{(27 \sqrt{5 a_{o}^{2}})} e^{- r / (3 a_{o})}$

The probability to find the electron between and

$P = \int_{8 a_{o}}^{10 a_{o}} {R_{3, 2}}^{2} (r) r^{2} d r = \int_{8 a_{o}}^{10 a_{o}} {(\frac{1}{{(3 a_{°})}^{\frac{3}{2}}} \frac{2 \sqrt{2} r^{2}}{27 {\sqrt{5}}_{0}^{2}} e^{- r / 3 a_{°}})}^{2} r^{2} d r = \frac{1}{{(3 a_{°})}^{3}} \frac{8}{27^{2} \times 5 a_{0}^{4}} \int_{8 a_{o}}^{10 a_{o}} r^{6} e^{- r / 3 a_{°}} d r$

Let $\frac{r}{a_{o}} = x$ , then the equation becomes-

$P = \frac{8}{27^{3} \times 5} \int_{8 a_{o}}^{10 a_{o}} X^{6} e^{2 x / 3} d r = 0.212$

This integral is found by the calculator.

Therefore, the probability of finding the electron between $8 a_{o} a n d 10 a_{o} i s 0.212 .$

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An electron is in the 3d state of a hydrogen atom. The most probable distance of the electron from the proton is $9 a_{o}$ . What is the probability that the electron would be found between $8 a_{o} a n d 10 a_{o}$ ?

Short Answer

Step by step solution

Given data

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An electron is in the 3d state of a hydrogen atom. The most probable distance of the electron from the proton is9ao. What is the probability that the electron would be found between8aoand10ao?

Short Answer

Step by step solution

Given data

Concept

Solution

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An electron is in the 3d state of a hydrogen atom. The most probable distance of the electron from the proton is $9 a_{o}$ . What is the probability that the electron would be found between $8 a_{o} a n d 10 a_{o}$ ?