Question

Ionization energy is the energy required to remove an electron from an atom in the gas phase. The ionization energy of gold is 890.1 kJ/mol. Is light with a wavelength of 225 nm capable of ionizing a gold atom (removing an electron) in the gas phase?

Short answer

The energy of light with a wavelength of 225 nm is calculated as 531.65 kJ/mol, which is less than the ionization energy of gold (890.1 kJ/mol). Therefore, light with a wavelength of 225 nm is not capable of ionizing a gold atom in the gas phase.

Step-by-step solution

Identify given values and constants

We are given the following information: - Ionization energy of Gold (E_ionization): 890.1 kJ/mol - Wavelength of Light (λ): 225 nm Additionally, we'll need the following constants: - Planck's constant (h): 6.626 x 10^{-34} J s - Speed of light in a vacuum (c): 3 x 10^8 m/s - Avogadro's number (N_A): 6.022 x 10^{23} atoms/mol

Convert wavelength to frequency

Using the speed of light (c) and the wavelength (λ), we can calculate the frequency (ν) of the light using the following equation: ν = c / λ First, convert the given wavelength to meters: λ = 225 nm = 225 x 10^{-9} m Now, calculate the frequency: ν = (3 x 10^8 m/s) / (225 x 10^{-9} m) = 1.333 x 10^{15} s^{-1}

Calculate the energy of light

Now, we can calculate the energy (E) of the light using Planck's equation: E = h × ν Inserting the values, we have: E = (6.626 x 10^{-34} J s) × (1.333 x 10^{15} s^{-1}) = 8.834 x 10^{-19} J

Convert energy per photon to energy per mole

To compare this energy to the ionization energy of gold, we need to convert the energy per photon to energy per mole by multiplying it by Avogadro's number: E_per_mole = E × N_A = (8.834 x 10^{-19} J) × (6.022 x 10^{23} atoms/mol) = 531.65 kJ/mol

Compare the energy of light with ionization energy

Finally, we'll compare the energy of light (E_per_mole) with the ionization energy of gold (E_ionization) to determine if it is sufficient to ionize a gold atom: E_per_mole = 531.65 kJ/mol E_ionization = 890.1 kJ/mol Since the energy of light (531.65 kJ/mol) is less than the ionization energy of gold (890.1 kJ/mol), it is not enough to ionize a gold atom by removing an electron in the gas phase.

Key concepts

Photon Energy Calculation

Photon energy is an essential concept in understanding how light interacts with atoms. Photons are particles of light, and they carry energy that can be calculated using Planck's equation. This equation is expressed as:\[E = h \times u\]Where:

• \(E\) is the energy of the photon
• \(h\) is Planck's constant (\(6.626 \times 10^{-34}\) Joule-seconds)
• \(u\) is the frequency of the light

Planck's equation illustrates that photons with higher frequency have more energy. To solve problems involving photon energy, it is often required to know the frequency of the light, which might involve converting the wavelength to frequency first. Knowing the individual photon energy allows you to calculate more complex interactions, such as whether the photon can ionize an atom.

Wavelength to Frequency Conversion

In physics, the properties of waves such as light are frequently analyzed through their wavelengths and frequencies. The wavelength is the distance between two peaks of a wave, and the frequency is the number of wave cycles per second. The equation that relates these two is:\[u = \frac{c}{\lambda}\]Where:

• \(u\) is the frequency of the wave
• \(c\) is the speed of light in a vacuum (approximately \(3 \times 10^8\) meters per second)
• \(\lambda\) is the wavelength

To convert a given wavelength into frequency, first make sure the wavelength is in meters by multiplying the value in nanometers by \(10^{-9}\). Once converted, you can use the formula to find the frequency. This frequency value is essential for calculating photon energy using Planck's equation.

Planck's Constant

Planck's constant is a fundamental constant in physics that has profound implications in quantum mechanics. Its value is \(6.626 \times 10^{-34}\) Joule-seconds, and it plays a crucial role in equations describing energy and frequency relationships for photons. Planck's constant helps us understand phenomena at atomic and subatomic levels, such as how energy is emitted or absorbed by matter.

• Planck's constant is part of the relationship that shows energy is quantized.
• The energy of each photon is directly proportional to its frequency: higher frequency means higher energy.
• Planck's constant symbolises the smallest possible unit of energy exchange.

Quantum mechanics, the domain where Planck's constant is most commonly applied, revolutionized our understanding of light, matter, and energy by explaining their wave-particle duality. Understanding this constant is essential for grasping how light behaves at the smallest scales.