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Chapter 6: Problem 59

A certain orbital of the hydrogen atom has \(n=4\) and \(l=3\). (a) What are the possible values of \(m_{l}\) for this orbital? (b) What are the possible values of \(m_{s}\) for the orbital?

Short Answer

Expert verified
For a hydrogen atom's orbital with n = 4 and l = 3: (a) The possible values of the magnetic quantum number \(m_{l}\) are -3, -2, -1, 0, 1, 2, and 3. (b) The possible values of the electron spin quantum number \(m_{s}\) are \(+\frac{1}{2}\) and \(-\frac{1}{2}\).

Step by step solution

01

Determine the possible values of m_l

Since the given value for the azimuthal quantum number (l) is 3, the magnetic quantum number (m_l) can have integer values ranging from -3 to 3. We will write all integers from -3 to 3, inclusive: \[m_{l} = -3, -2, -1, 0, 1, 2, 3\]
02

Determine the possible values of m_s

For the electron spin quantum number (m_s), there are two possible values which describe the two possible orientations of the electron spin. These values are +1/2 and -1/2. We can write these values as: \[m_{s} = +\frac{1}{2}, -\frac{1}{2}\] In conclusion, for a hydrogen atom's orbital with n = 4 and l = 3: (a) The possible values of the magnetic quantum number (m_l) are -3, -2, -1, 0, 1, 2, and 3. (b) The possible values of the electron spin quantum number (m_s) are +1/2 and -1/2.

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Most popular questions from this chapter

Label each of the following statements as true or false. For those that are false, correct the statement. (a) Visible light is a form of electromagnetic radiation. (b) Ultraviolet light has longer wavelengths than visible light. (c) X rays travel faster than microwaves. (d) Electromagnetic radiation and sound waves travel at the same speed.

(a) Using Equation 6.5 , calculate the energy of an electron in the hydrogen atom when \(n=3\) and when \(n=6\). Calculate the wavelength of the radiation released when an electron moves from \(n=6\) to \(n=3 .(\mathbf{b})\) Is this line in the visible region of the electromagnetic spectrum?

For orbitals that are symmetric but not spherical, the contour representations (as in Figures 6.23 and 6.24 ) suggest where nodal planes exist (that is, where the electron density is zero). For example, the \(p_{x}\) orbital has a node wherever \(x=0\). This equation is satisfied by all points on the \(y z\) plane, so this plane is called a nodal plane of the \(p_{x}\) orbital. (a) Determine the nodal plane of the \(p_{z}\) orbital. (b) What are the two nodal planes of the \(d_{x y}\) orbital? (c) What are the two nodal planes of the \(d_{x^{2}-y^{2}}\) orbital?

Einstein's 1905 paper on the photoelectric effect was the first important application of Planck's quantum hypothesis. Describe Planck's original hypothesis, and explain how Einstein made use of it in his theory of the photoelectric effect.

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