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Chapter 6: Problem 38

(a) Calculate the energies of an electron in the hydrogen atom for \(n=1\) and for \(n=\infty .\) How much energy does it require to move the electron out of the atom completely (from \(n=1\) to \(n=\infty),\) according to Bohr? Put your answer in \(\mathrm{kJ} / \mathrm{mol}\). (b) The energy for the process \(\mathrm{H}+\) energy \(\rightarrow \mathrm{H}^{+}+\mathrm{e}^{-}\) is called the ionization energy of hydrogen. The experimentally determined value for the ionization energy of hydrogen is \(1310 \mathrm{~kJ} / \mathrm{mol}\). How does this compare to your calculation?

Short Answer

Expert verified
The energy levels of an electron in the hydrogen atom for $n=1$ and $n=\infty$ are -13.6 eV and 0 respectively. The energy required to move the electron out of the atom completely (from $n=1$ to $n=\infty$) is 13.6 eV, which is equivalent to 1310 kJ/mol. This calculated ionization energy of hydrogen matches the given experimental value of 1310 kJ/mol.

Step by step solution

01

Understanding Bohr's formula for energy levels

According to Bohr's model, the energy of an electron in the hydrogen atom at a given energy level (n) can be calculated using the formula: \[E_n=-\frac{2\pi^2e^4m_eZ^2}{h^2n^2}\] Where \(E_n\) is the energy of the electron at level n, e is the elementary charge, \(m_e\) is the mass of electron, Z is the atomic number (1 for hydrogen), h is the Planck's constant, and n is the energy level.
02

Calculate the energy for n=1

We will plug in the known values (elementary charge, Planck's constant, electron mass, and atomic number) into the formula and calculate the energy for n=1: \[E_1=-\frac{2\pi^2e^4m_e(1)^2}{h^2(1)^2}\] \[E_1=-13.6 \text{ eV}\]
03

Calculate the energy for n=∞

Now we calculate the energy for n=∞: \[E_{\infty}=-\frac{2\pi^2e^4m_e(1)^2}{h^2(\infty)^2}\] As n approaches infinity, the energy level will be zero: \[E_{\infty}=0\]
04

Calculate the energy difference between n=1 and n=∞

The energy required to ionize the electron completely (from n=1 to n=∞) is the difference between the energies for n=1 and n=∞: \[\Delta E=E_{\infty} - E_1\] \[\Delta E=0 - (-13.6 \text{ eV})\] \[\Delta E=13.6 \text{ eV}\]
05

Convert energy to kJ/mol

To convert the energy difference to kJ/mol, first, multiply by the electron charge to obtain joules: \[\Delta E=13.6 \text{ eV} * 1.6 \times 10^{-19} \text{ J/eV} = 2.176 \times 10^{-18} \text{ J}\] Next, multiply by Avogadro's number to convert from energy per atom to energy per mol: \[\Delta E=2.176 \times 10^{-18} \text{ J/atom} * 6.022 \times 10^{23} \text{ atoms/mol}=1310 \text{ kJ/mol}\]
06

Compare calculated value with experimental value

According to our calculation, the ionization energy of hydrogen is 1310 kJ/mol, which is the same as the experimental value (1310 kJ/mol) given in the exercise. Therefore, our calculated value of the ionization energy of hydrogen is consistent with the experimental data.

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Most popular questions from this chapter

Determine which of the following statements are false and correct them. (a) The frequency of radiation increases as the wavelength increases. (b) Electromagnetic radiation travels through a vacuum at a constant speed, regardless of wavelength. (c) Infrared light has higher frequencies than visible light. (d) The glow from a fireplace, the energy within a microwave oven, and a foghorn blast are all forms of electromagnetic radiation.

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