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Chapter 7: Problem 47

It takes \(7.21 \times 10^{-19} \mathrm{~J}\) of energy to remove an electron from an iron atom. What is the maximum wavelength of light that can do this?

Short Answer

Expert verified
The maximum wavelength of light capable of removing an electron from the iron atom is approximately \(275.6 \mathrm{~nm}\).

Step by step solution

01

Identify the relevant equations and constants

For this problem, we need the following equation: Planck's equation: \(E = h \times f\) where: - \(E\) is the energy of the light (in Joules), - \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{~J \cdot s}\)), - \(f\) is the frequency of the light (in Hertz). Also, we need the relation between frequency and wavelength of light: \(c = \lambda \times f\) where: - \(c\) is the speed of light (\(3.0 \times 10^8 \mathrm{~m/s}\)), - \(\lambda\) is the wavelength of the light (in meters).
02

Express the frequency in terms of energy

Now we will solve the Planck's equation for the frequency: \(\Rightarrow f = \frac{E}{h}\)
03

Replace the energy value in the equation

Now replace the given energy value in the equation: \(f = \frac{7.21 \times 10^{-19} \mathrm{~J}}{6.626 \times 10^{-34} \mathrm{~J \cdot s}}\)
04

Calculate the frequency

Now calculate the frequency, giving: \(f \approx 1.088 \times 10^{15} \mathrm{~Hz}\)
05

Relate frequency and wavelength

Now we will use the relation between frequency and wavelength: \(\lambda = \frac{c}{f}\)
06

Compute the maximum wavelength

Now substitute the frequency value and speed of light into the equation to find the maximum wavelength: \(\lambda = \frac{3.0 \times 10^8 \mathrm{~m/s}}{1.088 \times 10^{15} \mathrm{~Hz}}\) \(\lambda \approx 2.756 \times 10^{-7} \mathrm{~m}\) Since the wavelength is often expressed in nanometers, we can convert the result: \(\lambda = 2.756 \times 10^{-7} \mathrm{~m} \times \frac{10^9 \mathrm{~nm}}{1\mathrm{~m}} \approx 275.6 \mathrm{~nm}\) Therefore, the maximum wavelength of light capable of removing an electron from the iron atom is approximately \(275.6 \mathrm{~nm}\).

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Most popular questions from this chapter

Consider the following ionization energies for aluminum: $$ \begin{aligned} \mathrm{Al}(g) \longrightarrow \mathrm{Al}^{+}(g)+\mathrm{e}^{-} & I_{1}=580 \mathrm{~kJ} / \mathrm{mol} \\ \mathrm{Al}^{+}(g) \longrightarrow \mathrm{Al}^{2+}(g)+\mathrm{e}^{-} & I_{2}=1815 \mathrm{~kJ} / \mathrm{mol} \end{aligned} $$ $$ \begin{array}{ll} \mathrm{Al}^{2+}(g) \longrightarrow \mathrm{Al}^{3+}(g)+\mathrm{e}^{-} & I_{3}=2740 \mathrm{~kJ} / \mathrm{mol} \\ \mathrm{Al}^{3+}(g) \longrightarrow \mathrm{Al}^{4+}(g)+\mathrm{e}^{-} & I_{4}=11,600 \mathrm{~kJ} / \mathrm{mol} \end{array} $$ a. Account for the trend in the values of the ionization energies. b. Explain the large increase between \(I_{3}\) and \(I_{4}\). c. Which one of the four ions has the greatest electron affinity? Explain. d. List the four aluminum ions given in order of increasing size, and explain your ordering. (Hint: Remember that most of the size of an atom or ion is due to its electrons.)

Arrange the following groups of atoms in order of increasing size. a. \(\mathrm{Te}, \mathrm{S}, \mathrm{Se}\) b. \(\mathrm{K}, \mathrm{Br}, \mathrm{Ni}\) c. \(\mathrm{Ba}, \mathrm{Si}, \mathrm{F}\)

Calculate the wavelength of light emitted when each of the following transitions occur in the hydrogen atom. What type of electromagnetic radiation is emitted in each transition? a. \(n=3 \rightarrow n=2\) b. \(n=4 \rightarrow n=2\) c. \(n=2 \rightarrow n=1\)

Are the following statements true for the hydrogen atom only, true for all atoms, or not true for any atoms? a. The principal quantum number completely determines the energy of a given electron. b. The angular momentum quantum number, \(\ell\), determines the shapes of the atomic orbitals. c. The magnetic quantum number, \(m_{\ell}\), determines the direction that the atomic orbitals point in space.

A carbon-oxygen double bond in a certain organic molecule absorbs radiation that has a frequency of \(6.0 \times 10^{13} \mathrm{~s}^{-1}\). a. What is the wavelength of this radiation? b. To what region of the spectrum does this radiation belong? c. What is the energy of this radiation per photon? per mole of photons? d. A carbon-oxygen bond in a different molecule absorbs radiation with frequency equal to \(5.4 \times 10^{13} \mathrm{~s}^{-1}\). Is this radiation more or less energetic?
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