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Chapter 7: Problem 126

An electron in a hydrogen atom is excited from the ground state to the \(n=4\) state. Comment on the correctness of the following statements (true or false). (a) \(n=4\) is the first excited state. (b) It takes more energy to ionize (remove) the electron from \(n=4\) than from the ground state. (c) The electron is farther from the nucleus (on average) in \(n=4\) than in the ground state. (d) The wavelength of light emitted when the electron drops from \(n=4\) to \(n=1\) is longer than that from \(n=4\) to \(n=2\). (e) The wavelength the atom absorbs in going from \(n=1\) to \(n=4\) is the same as that emitted as it goes from \(n=4\) to \(n=1\)

Short Answer

Expert verified
(a) False; (b) False; (c) True; (d) False; (e) True.

Step by step solution

01

Analyzing statement (a)

Statement: 'n=4' is the first excited state. The ground state is denoted by 'n=1', and any state above this is considered an excited state. Therefore, 'n=4' would not be the first excited state. In terms of energy levels, the first excited state would be 'n=2'. Thus, this statement is 'false'.
02

Analyzing statement (b)

Statement: It takes more energy to ionize the electron from 'n=4' than from the ground state. Ionization energy is the energy required to remove an electron from an atom. It's easier (requires less energy) to remove electron from a higher energy level (further from the nucleus) than from a lower energy level (closer to the nucleus). Therefore, this statement is 'false'.
03

Analyzing statement (c)

Statement: The electron is farther from the nucleus (on average) in 'n=4' than in the ground state. According to the Bohr model of an atom, electrons in higher energy levels are likely to be found further from the nucleus than those in lower energy levels. Thus, this statement is 'true'.
04

Analyzing statement (d)

Statement: The wavelength of light emitted when the electron drops from 'n=4' to 'n=1' is longer than that from 'n=4' to 'n=2'. The energy of a photon is inversely proportional to its wavelength. An electron transitioning from 'n=4' to 'n=1' will release more energy (shorter wavelength photon) than an electron transitioning from 'n=4' to 'n=2'. So, this statement is 'false'.
05

Analyzing statement (e)

Statement: The wavelength the atom absorbs in going from 'n=1' to 'n=4' is the same as that emitted as it goes from 'n=4' to 'n=1'. The energy difference between these two states is specific. The same amount of energy will need to be either absorbed to move from 'n=1' to 'n=4' or emitted to move from 'n=4' to 'n=1'. Since the energy of a photon equals the difference in energy levels and is intrinsically tied to wavelength, this statement is 'true'.

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Most popular questions from this chapter

Consider the following energy levels of a hypothetical atom: \(E_{4}\) \(-1.0 \times 10^{-19} \mathrm{~J}\) \(E_{3}\) \(--5.0 \times 10^{-19} \mathrm{~J}\) \(E_{2}\) \(--10 \times 10^{-19} \mathrm{~J}\) \(E_{1}\) \(-15 \times 10^{-19} \mathrm{~J}\) (a) What is the wavelength of the photon needed to excite an electron from \(E_{1}\) to \(E_{4} ?\) (b) What is the energy (in joules) a photon must have in order to excite an electron from \(E_{2}\) to \(E_{3} ?\) (c) When an electron drops from the \(E_{3}\) level to the \(E_{1}\) level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process.

Photodissociation of water \(\mathrm{H}_{2} \mathrm{O}(l)+h \nu \longrightarrow \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)\) has been suggested as a source of hydrogen. The \(\Delta H_{\mathrm{xn}}^{\circ}\) for the reaction, calculated from thermochemical data, is \(285.8 \mathrm{~kJ}\) per mole of water decomposed. Calculate the maximum wavelength (in \(\mathrm{nm}\) ) that would provide the necessary energy. In principle, is it feasible to use sunlight as a source of energy for this process?

Draw the shapes (boundary surfaces) of the following orbitals: (a) \(2 p_{y},\) (b) \(3 d_{z^{2}}\) (c) \(3 d_{x^{2}-y^{2}}\), (Show coordinate axes in your sketches.)

Determine the maximum number of electrons that can be found in each of the following subshells: \(3 s\), \(3 d, 4 p, 4 f, 5 f\)

$$ \begin{array}{lccccc} \lambda(\mathrm{nm}) & 405 & 435.8 & 480 & 520 & 577.7 \\ \hline \mathrm{KE}(\mathrm{J}) & 2.360 \times & 2.029 \times & 1.643 \times & 1.417 \times & 1.067 \times \\ & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} \end{array} $$ A ruby laser produces radiation of wavelength \(633 \mathrm{nm}\) in pulses whose duration is \(1.00 \times 10^{-9} \mathrm{~s}\). (a) If the laser produces \(0.376 \mathrm{~J}\) of energy per pulse, how many photons are produced in each pulse? (b) Calculate the power (in watts) delivered by the laser per pulse. \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s})\)
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