Question

Classify each of the following statements as either true or false: (a) A hydrogen atom in the \(n=3\) state can emit light at only two specific wavelengths, (b) a hydrogen atom in the \(n=2\) state is at a lower energy than one in the \(n=1\) state, and (c) the energy of an emitted photon equals the energy difference of the two states involved in the emission.

Short answer

Statement (a) is true because a hydrogen atom in the \(n=3\) state can emit light at two specific wavelengths, approximately 656 nm and 102.6 nm. Statement (b) is false because a hydrogen atom in the \(n=2\) state has a higher energy (-3.4 eV) than one in the \(n=1\) state (-13.6 eV). Statement (c) is true, as the energy of an emitted photon equals the energy difference of the two states involved in the emission, based on the conservation of energy principle.

Step-by-step solution

Statement (a)

A hydrogen atom in the \(n=3\) state can emit light at only two specific wavelengths. First, let's recall the formula for the energy difference between two levels in a hydrogen atom, which is given by: \[ \Delta E = -13.6 \ eV \left( \frac{1}{n_{final}^2} - \frac{1}{n_{initial}^2} \right) \] Here, \(n_{initial}\) and \(n_{final}\) represent the initial and final principal quantum numbers of the electron. In the given statement, the hydrogen atom is in the \(n=3\) state. When an electron in the \(n=3\) state transitions to a lower energy level, it can do so by transitioning to states \(n=2\) or \(n=1\). Let's calculate the energy difference and wavelength for both transitions. 1. For the transition from \(n=3\) to \(n=2\): \[ \Delta E_{3 \to 2} = -13.6 \ eV \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \approx 1.89 \ eV \] The corresponding wavelength can be calculated using the energy-wavelength relationship: \[ \lambda_{3 \to 2} = \frac{hc}{\Delta E_{3 \to 2}} \approx 656 \ nm \] 2. For the transition from \(n=3\) to \(n=1\): \[ \Delta E_{3 \to 1} = -13.6 \ eV \left( \frac{1}{1^2} - \frac{1}{3^2} \right) \approx 12.09 \ eV \] The corresponding wavelength can be calculated using the energy-wavelength relationship: \[ \lambda_{3 \to 1} = \frac{hc}{\Delta E_{3 \to 1}} \approx 102.6 \ nm \] As we see, there are two different possible wavelengths for light emission from the \(n=3\) state. Therefore, statement (a) is true.

Statement (b)

A hydrogen atom in the \(n=2\) state is at a lower energy than one in the \(n=1\) state. This statement can be evaluated by comparing the energy of both states. The energy of a hydrogen atom in any state \(n\) is given by: \[ E_n = -13.6 \ eV \left( \frac{1}{n^2} \right) \] Therefore, for \(n=1\) and \(n=2\), we have: \[ E_1 = -13.6 \ eV \left( \frac{1}{1^2} \right) = -13.6 \ eV \] and \[ E_2 = -13.6 \ eV \left( \frac{1}{2^2} \right) = -3.4 \ eV \] Since the energy of a hydrogen atom in the \(n=2\) state (-3.4 eV) is higher than that in the \(n=1\) state (-13.6 eV), statement (b) is false.

Statement (c)

The energy of an emitted photon equals the energy difference of the two states involved in the emission. This statement can be evaluated by recalling the conservation of energy principle. The energy difference between the initial and final states of the electron equals the energy of the emitted photon. Symbolically, it's expressed as: \[ \Delta E = E_{photon} \] This principle holds true for all energy transitions, including transitions in hydrogen atoms. Therefore, statement (c) is true. In conclusion, statement (a) and statement (c) are true, while statement (b) is false.

Key concepts

Energy Levels

In the context of hydrogen atoms, energy levels refer to the specific energies that an electron in a hydrogen atom can have. These energy levels are quantized, meaning the electrons can only exist at certain, distinct energy levels and not between them. The energy levels are determined by the principal quantum number, represented as \( n \). Each value of \( n \) corresponds to a specific energy level.

For a hydrogen atom, the energy of an electron in a specific level \( n \) is calculated using the formula:\[ E_n = -13.6 \, \text{eV} \left( \frac{1}{n^2} \right) \]- The value of \( n \) can be any positive integer: 1, 2, 3, and so on.- As \( n \) increases, the energy becomes less negative, indicating a higher (less bound) energy state.

For example, when \( n = 1 \), the energy level is at \( -13.6 \, \text{eV} \). When \( n = 2 \), the energy increases to \( -3.4 \, \text{eV} \). This negative sign indicates that energy is required to remove the electron from the atom. Lower values of \( n \) mean the electron is more tightly bound to the nucleus, corresponding to lower (more negative) energies.

Photon Emission

When an electron in a hydrogen atom transitions from a higher energy level to a lower one, a photon is emitted. This photon carries away the energy difference between the two levels. Such transitions are what give rise to the hydrogen spectrum, producing lines of specific wavelengths.

For example, if an electron jumps from level \( n=3 \) to level \( n=1 \), this results in the emission of a photon with the energy equivalent to the difference in energy levels:\[ \Delta E_{3 \to 1} = E_1 - E_3 \]The energy of the photon \( E_{photon} \) is the same as \( \Delta E \), ensuring energy conservation. The emitted photon's wavelength \( \lambda \) can be calculated using the formula:\[ \lambda = \frac{hc}{E_{photon}} \]where:- \( h \) is Planck's constant,- \( c \) is the speed of light.

Photon emission is a crucial process in various applications, such as in lasers and atomic clocks, which rely on precise energy transitions.

Quantum Numbers

Quantum numbers are values used to describe the unique quantum state of an electron in an atom. For hydrogen, the principal quantum number \( n \) is paramount as it determines the electron's energy level. However, in multi-electron atoms, other quantum numbers also become significant.

The four quantum numbers are:- **Principal Quantum Number \( n \)**: Indicates the energy level and size of the orbital.- **Azimuthal Quantum Number \( l \)**: Defines the shape of the orbital, with values from 0 to \( n-1 \).- **Magnetic Quantum Number \( m_l \)**: Denotes the orientation of the orbital in space, with values between \( -l \) and \( +l \).- **Spin Quantum Number \( m_s \)**: Represents the two possible directions of an electron's spin, usually \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

In simpler terms, these quantum numbers act like an address system, pinpointing the exact location and energy state of an electron within the atom. This framework is crucial for understanding the atom’s electronic configuration and predicting chemical behavior.

Conservation of Energy

The principle of conservation of energy states that the total energy of a closed system remains constant. In the context of hydrogen atom transitions, this means that when an electron moves from one energy level to another, the energy must be conserved.

This is why the energy of an emitted photon equals the difference in energy between the initial and final states of the electron. If an electron transitions from level \( n=3 \) to level \( n=2 \), the conservation of energy can be expressed as:\[ \Delta E = E_{n=3} - E_{n=2} = E_{photon} \]- Here, \( \Delta E \) is the actual change in energy, which equals the energy carried by the emitted photon. - This principle ensures that no energy is lost or created; it simply changes from one form to another.

In physics, the conservation of energy is a fundamental concept that applies across all scenarios, ensuring that energy can never just "disappear." It can change forms, such as from potential energy to kinetic energy or from stored energy to electromagnetic radiation, but the total energy remains constant.