Chapter 41: Q. 10 (page 1206)

The hydrogen atom $1_{s}$ wave function is a maximum at $r = 0$ . But the $1_{s}$ radial probability density, shown peaks at $r = a_{B}$ and is zero at $r = 0$ . Explain this paradox.

Short Answer

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Given Information

We need to prove peaks at $r = a_{B}$ and is zero at $r = 0$ .

Explanation

The radial probability density $P_{r} (r) = 4 π r^{2} |R_{n l} (r)|$ is the probability density for finding the electron at a distance from the nucleus. The radial wave function $R_{l s} (r)$ is a maximum at the origin. the origin is the point for finding the electron. The probability density $P (r)$ is smaller at any one point $r = a_{B}$ than the origin. Since there are many points having $r = a_{B}$ only a single point $r = 0 m$ The increased number of points more than compensates for the decreased probability per point. As a result, finding the electron at a distance $r = a_{B}$ So, it is at a distance $r = 0 m$ .

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The hydrogen atom $1_{s}$ wave function is a maximum at $r = 0$ . But the $1_{s}$ radial probability density, shown peaks at $r = a_{B}$ and is zero at $r = 0$ . Explain this paradox.

Short Answer

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The hydrogen atom1swave function is a maximum atr=0. But the1s radial probability density, shown peaks at r=aB and is zero atr=0. Explain this paradox.

Short Answer

Step by step solution

Given Information

Explanation

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The hydrogen atom $1_{s}$ wave function is a maximum at $r = 0$ . But the $1_{s}$ radial probability density, shown peaks at $r = a_{B}$ and is zero at $r = 0$ . Explain this paradox.