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Chapter 8: Problem 110

Show that the volume of a spherical shell of radius \(r\) and thickness \(d r\) is \(4 \pi r^{2} d r .\) [Hint: This exercise requires calculus.]

Short Answer

Expert verified
The volume of a spherical shell of radius \(r\) and thickness \(d r\) is \(4\pi r^2 dr\)

Step by step solution

01

Write down the volume of a sphere

The volume, \(V\) of a sphere with radius \(r\) is given by the equation: \(V=\frac{4}{3}\pi r^3\)
02

Differentiate the volume equation with respect to radius

The derivative of \(V\) with respect to \(r\) is given by \(\frac{dV}{dr}= 4\pi r^2\)
03

Calculate the volume of the shell

The volume of the shell, \(dV\) with thickness \(dr\) is given by: \(dV = \frac{dV}{dr} \cdot dr = 4 \pi r^2 dr\)

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Most popular questions from this chapter

Between which two orbits of the Bohr hydrogen atom must an electron fall to produce light of wavelength \(1876 \mathrm{nm} ?\)

Concerning the electrons in the shells, subshells, and orbitals of an atom, how many can have (a) \(n=4, \ell=2, m_{\ell}=1,\) and \(m_{s}=+\frac{1}{2} ?\) (b) \(n=4, \ell=2,\) and \(m_{\ell}=1 ?\) (c) \(n=4\) and \(\ell=2 ?\) (d) \(n=4 ?\) (e) \(n=4, \ell=2,\) and \(m_{s}=+\frac{1}{2} ?\)

What type of orbital (i.e., \(3 s, 4 p, \ldots)\) is designated by these quantum numbers? (a) \(n=5, \ell=1, m_{\ell}=0\) (b) \(n=4, \ell=2, m_{\ell}=-2\) (c) \(n=2, \ell=0, m_{\ell}=0\)

Describe how the Bohr model of the hydrogen atom appears to violate the Heisenberg uncertainty principle.

What is the expected ground-state electron configuration for each of the following elements? (a) tellurium; (b) cesium; (c) selenium; (d) platinum; (e) osmium; (f) chromium.
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