Question

What wavelength of light, if absorbed by a hydrogen molecule, could cause the molecule to split into the ions \(\mathrm{H}^{+}\) and \(\mathrm{H}^{-}\) ? (The data required are available in this and previous chapters.)

Short answer

Without the ionization energy (I) value, the problem cannot be solved. If I were given, use the equation λ = hc / I to find the wavelength.

Step-by-step solution

Understanding the problem

The problem is asking for the wavelength of light that has enough energy to ionize a hydrogen molecule into its constituent ions. According to the photoelectric effect, the energy of a photon (light) is directly proportional to its frequency and inversely proportional to its wavelength. The relationship between them is given by the equation: Energy (E) = Planck's constant (h) × frequency (v) or Energy (E) = hc / λ, where c is the speed of light and λ is the wavelength.

Find the ionization energy for hydrogen

To solve the problem, we need to know the ionization energy required to split the hydrogen molecule into ions H+ and H-. This information can be found in data tables from previous chapters or scientific literature. Assume the ionization energy for hydrogen is known and represented by the variable 'I'.

Calculate the wavelength of light

With the ionization energy (I) known, we can rearrange the equation (E=hc/λ) to solve for the wavelength (λ). The wavelength λ is equal to Planck's constant (h) multiplied by the speed of light (c), divided by the ionization energy (I): λ = hc / I. Ensure that all units are consistent; typically, h is in J·s, c is in m/s, and I is in J.

Plug in the values and solve for λ

Insert the values for Planck's constant (h=6.626×10^-34 J·s), the speed of light (c=3.00×10^8 m/s), and the ionization energy (I) into the equation λ = hc / I. Calculate the result to find the wavelength in meters. If the ionization energy is not provided, the problem remains unsolvable.

Key concepts

The Photoelectric Effect

The photoelectric effect is a fundamental concept in quantum physics, originally explained by Albert Einstein in 1905. It is the phenomenon where electrons are emitted from a material when it is exposed to electromagnetic radiation of a certain frequency or wavelength.

This effect is critical in understanding how light can cause ionization in atoms and molecules. When a photon with sufficient energy hits an atom, it can transfer its energy to an electron, causing it to be ejected from the atom, thereby ionizing it. The energy (\( E \) of a photon is determined by its frequency (\( u \) or wavelength (\( \text{lambda} \) using the equation: \[ E = h u = \frac{hc}{\text{lambda}} \] where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \text{lambda} \) is the wavelength of the photon.

Understanding the photoelectric effect is crucial when trying to calculate the specific wavelength needed to ionize a hydrogen molecule, as it directly relates light properties to the energy required to cause ionization.

Planck's Constant

Planck's constant (\( h \) is a key quantity in quantum mechanics, representing the proportionality factor between the energy (\( E \) of a photon and the frequency (\( u \) of its associated electromagnetic wave. This constant is fundamental to several equations that describe the behavior of particles at the quantum level.

The value of Planck's constant is approximately \( 6.626 \times 10^{-34} \) Joule seconds (J·s). It sets the scale for actions in the quantum realm and is pivotal in calculations that involve the photoelectric effect and energy quantization.

When we apply Planck's constant to the exercise in question, we use the rearranged form of the energy equation—\( \text{lambda} = \frac{hc}{I} \)—to determine the wavelength necessary to ionize a hydrogen molecule. Since \( h \) is a fixed value, knowing it allows us to proceed with such calculations once we have the required ionization energy (\( I \) of the molecule.

Ionization Energy

Ionization energy is the minimum amount of energy required to remove an electron from an atom or molecule to form a cation. It is a key concept in chemistry and physics when dealing with the reactions and behavior of atoms and molecules.

In the context of our given problem, the ionization energy pertains to the specific amount of energy needed to split a hydrogen molecule into \( \mathrm{H}^{+} \) and \( \mathrm{H}^{-} \) ions. This energy varies depending on the element or molecule in question, and it is generally determined through experimentation or calculated using quantum mechanics.

To calculate the wavelength of light that can ionize the hydrogen molecule, the ionization energy must be known. Once the value of \( I \) is obtained, it can be inserted into the equation \( \text{lambda} = \frac{hc}{I} \), allowing us to solve for the precise wavelength required. This step links the concept of ionization energy directly to the practical application of determining a wavelength consistent with the energy needed for ionization.