Question

An electron is excited from the \(n=1\) ground state to the \(n=\) 3 state in a hydrogen atom. Which of the following statements is/are true? Correct the false statements to make them true. a. It takes more energy to ionize (completely remove) the electron from \(n=3\) than from the ground state. b. The electron is farther from the nucleus on average in the \(n=3\) state than in the \(n=1\) state. c. The wavelength of light emitted if the electron drops from \(n=3\) to \(n=2\) will be shorter than the wavelength of light emitted if the electron falls from \(n=3\) to \(n=1\) d. The wavelcngth of light cmittcd when the clectron returns to the ground state from \(n=3\) will be the same as the wavelength of light absorbed to go from \(n=1\) to \(n=3\) e. For \(n=3,\) the electron is in the first excited state.

Short answer

a. False. Corrected: It takes less energy to ionize (completely remove) the electron from n=3 than from the ground state. b. True. The electron is farther from the nucleus on average in the n=3 state than in the n=1 state. c. True. The wavelength of light emitted if the electron drops from n=3 to n=2 will be shorter than the wavelength of light emitted if the electron falls from n=3 to n=1. d. True. The wavelength of light emitted when the electron returns to the ground state from n=3 will be the same as the wavelength of light absorbed to go from n=1 to n=3. e. False. Corrected: For n=3, the electron is in the second excited state.

Step-by-step solution

Statement a

It takes more energy to ionize (completely remove) the electron from n=3 than from the ground state. This statement is false. To correct it: It takes less energy to ionize (completely remove) the electron from n=3 than from the ground state. Since the electron in the n=3 state is farther from the nucleus, it experiences less attraction to the positive charge in the nucleus, making it easier to remove.

Statement b

The electron is farther from the nucleus on average in the n=3 state than in the n=1 state. This statement is true. As the principal quantum number n increases, the electron's orbit becomes larger, and it spends more time further away from the nucleus.

Statement c

The wavelength of light emitted if the electron drops from n=3 to n=2 will be shorter than the wavelength of light emitted if the electron falls from n=3 to n=1 This statement is true. The energy difference between the n=3 and n=2 states is smaller than the energy difference between the n=3 and n=1 states. Since the energy of the emitted photon is related to its wavelength by the equation \(E = h\frac{c}{\lambda}\), a larger energy difference corresponds to a shorter wavelength.

Statement d

The wavelength of light emitted when the electron returns to the ground state from n=3 will be the same as the wavelength of light absorbed to go from n=1 to n=3 This statement is true. The energy of the light absorbed during an electron transition is equal to the energy of the light emitted when the electron returns to its initial state, as long as both transitions are between the exact same energy levels. Since the energy of a photon is inversely proportional to its wavelength, this means that the wavelengths are the same for both processes.

Statement e

For n=3, the electron is in the first excited state. This statement is false. To correct it: For n=3, the electron is in the second excited state. The electron's ground state is when n=1 and the first excited state is when n=2.

Key concepts

Electron Excitation

Electron excitation is a phenomenon wherein an electron within an atom gains energy and jumps to a higher energy level, or orbital. This can be observed in the exercise where an electron in a hydrogen atom jumps from its ground state (=1) to the third energy level (=3). This excitation requires energy input, which can come from electromagnetic radiation, such as light or heat. When an electron gets excited and then returns to a lower energy state, it releases a photon with energy equivalent to the difference between the two levels. In the example provided, excitation demonstrates a fundamental atomic process key to understanding phenomena like the emission of light and spectral lines.

Excitation energy, the amount of energy needed for the transition, is specific to each transition and is calculated using the differences in energy levels. In the hydrogen atom, these energy levels are quantified, with the ground state being the most stable and the lowest in energy.

Ionization Energy

Ionization energy is the amount of energy required to completely remove an electron from an atom or ion. In the context of our exercise, this concept is illustrated by correcting the false statement that it takes more energy to ionize an electron from the =3 state than from the ground state (=1). In reality, the further an electron is from the nucleus, the less energy it takes to ionize it due to weaker electrostatic forces of attraction. This is why ionization energy decreases as an electron moves to higher energy states. For the hydrogen atom, removing an electron from its ground state requires the most energy because it is closest to the nucleus where the attractive forces are strongest.

Quantum Numbers

Principal Quantum Number

The exercise mentions quantum numbers through the use of the symbol , which refers to the principal quantum number. It is one of the four quantum numbers used to describe the unique quantum state of an electron. The principal quantum number defines the energy level of an electron in an atom and broadly its distance from the nucleus. The value of starts at 1 (the ground state) and increases with energy, indicating that the electron orbit becomes larger and the electron spends more time further from the nucleus, as mentioned in statement b of the exercise. Quantum numbers are foundational for understanding the structure of atoms and how their electrons are arranged.

Wavelength of Light

Wavelength of light is a key term that connects to the discussion of electron transitions within an atom. It is the distance over which the wave's shape repeats and is commonly denoted by the Greek letter lambda (λ). The relation between the wavelength of emitted light and the energy of an electron transition is given by the formula E = hc/λ, where h is Planck's constant and c is the speed of light. As seen in the exercise's statement c, when an electron drops from =3 to =2, it emits light with a longer wavelength than when it drops from =3 to =1 because the energy change is less. This inverse relationship is important for understanding spectrometry and the color of light associated with various transitions.

Energy Levels

Energy levels are the discrete values of energy that an electron in an atom can have. In a hydrogen atom, these levels can be calculated using a simple formula that includes the principal quantum number and is inversely proportional to ². As the exercise illustrates, different energy levels correspond to different distances from the nucleus. An electron in the =1 level (the ground state) has the lowest energy and is closest to the nucleus, while electrons in higher levels like =3 are at higher energy states and further away. Understanding energy levels helps us predict and explain the electron behavior in atoms, including how they absorb and emit photons to perform transitions between these levels, as described in the exercise.