Chapter 7: Q54E (page 282)

Using the functions given in Table 7.4, verify that for the more circular electron orbit in hydrogen $(i . e ., l = n - 1)$ , the radial probability is of the form
$P (r) \propto r^{2} n e^{- 2 r / n a_{o}}$
Show that the most probable radius is given by
$r_{m o s t p r o b a b l e} = n^{2} a_{o}$

Short Answer

Expert verified

The most probable radius is $r = n^{2} a_{0}$ .

Step by step solution

Given data

The radial function for $l = n - 1 i s,$

role="math" localid="1659182273339" $R_{n, n - 1} \propto r^{n - 1} e^{- r / (n a_{o})}$

Concept

The most probable radius is the radius of the orbit in which the probability of finding the electron is maximum.

The probability density is

$P (r) = r^{2} {R^{2}}_{n, n - 1}$

Solution

$P (r) = r^{2} {R^{2}}_{n, n - 1} = r^{2} (r^{n - 1} e^{- r / (n a_{o})}) = r^{2} r^{2 n - 2} e^{- r / (n a_{o})} = r^{2 n} e^{- r / (n a_{o})}$

To find the most probable location, take the derivative of the probability density,

$\frac{d P (r)}{d r} = \frac{d}{d r} r^{2 n} e^{- 2 r l (n a_{°})} = 2 n r^{2 n - 1} e^{- 2 r l (n a_{°})} + r^{2 n} (\frac{- 2}{n a_{°}}) e^{- 2 r l (n a_{°})} = e^{- 2 r l (n a_{°})} r^{2 n - 1} (2 n + \frac{- 2 r}{n a_{°}})$

For the most probable location, the derivative should be equal to zero.

role="math" localid="1659182748704" $= e^{- 2 r l (n a_{°})} r^{2 n - 1} (2 n + \frac{- 2 r}{n a_{°}})$

Hence,

$2 n + \frac{- 2 r}{n a_{°}} = 0 r = n^{2} a_{°}$

The most probable radius is $r = n^{2} a_{°}$ .

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Using the functions given in Table 7.4, verify that for the more circular electron orbit in hydrogen $(i . e ., l = n - 1)$ , the radial probability is of the form
$P (r) \propto r^{2} n e^{- 2 r / n a_{o}}$
Show that the most probable radius is given by
$r_{m o s t p r o b a b l e} = n^{2} a_{o}$

Short Answer

Step by step solution

Given data

Concept

Solution

One App. One Place for Learning.

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Using the functions given in Table 7.4, verify that for the more circular electron orbit in hydrogen (i.e.,l=n-1), the radial probability is of the form P(r)∝r2ne-2r/nao Show that the most probable radius is given by rmostprobable=n2ao

Short Answer

Step by step solution

Given data

Concept

Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Conservation of Energy and Momentum

Wave Optics

Astrophysics

Fluids

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Study anywhere. Anytime. Across all devices.

Using the functions given in Table 7.4, verify that for the more circular electron orbit in hydrogen $(i . e ., l = n - 1)$ , the radial probability is of the form
$P (r) \propto r^{2} n e^{- 2 r / n a_{o}}$
Show that the most probable radius is given by
$r_{m o s t p r o b a b l e} = n^{2} a_{o}$