X-ray diffractometers often use metals that have had their core electrons excited as a source of X-rays. Consider the 2p → 1s transition for copper, which is called the K⍺ transition. Calculate the wavelength of X-rays (in Å) given off by the K⍺ transition if the energy given off by a mole of copper atoms is 7.77⨉10⁵ kJ.(1Å = 10^-10 m)

Verified step by step guidance

1. Identify the energy of the Ka transition for one mole of copper atoms, which is given as 7.77 \times 10^5 \text{ kJ/mol}.
2. Convert the energy from kJ to J by multiplying by 1000, since 1 \text{ kJ} = 1000 \text{ J}.
3. Use Avogadro's number (6.022 \times 10^{23} \text{ atoms/mol}) to find the energy of the transition for a single copper atom by dividing the total energy in joules by Avogadro's number.
4. Apply the energy-wavelength relationship using the formula \( E = \frac{hc}{\lambda} \), where \( E \) is the energy of a single photon, \( h \) is Planck's constant (6.626 \times 10^{-34} \text{ J s}), and \( c \) is the speed of light (3.00 \times 10^8 \text{ m/s}).
5. Solve for the wavelength \( \lambda \) in meters, and then convert it to angstroms by using the conversion factor 1 \text{ Å} = 10^{-10} \text{ m}.

Key Concepts

X-ray Emission

X-ray emission occurs when an electron transitions between energy levels in an atom, releasing energy in the form of X-rays. In the case of copper, the Ka transition refers to the movement of an electron from the 2p orbital to fill a vacancy in the 1s orbital, resulting in the emission of X-rays with a specific energy characteristic of the element.

Energy-Wavelength Relationship

The energy of a photon is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship allows for the calculation of the wavelength of emitted X-rays when the energy is known, highlighting the fundamental connection between energy and electromagnetic radiation.

Mole Concept and Energy Conversion

The mole concept is essential in chemistry for quantifying substances. When dealing with energy given off by a mole of atoms, it is crucial to convert the energy from kJ to joules (1 kJ = 1000 J) to use in calculations. This conversion is necessary to apply the energy-wavelength relationship accurately and derive the wavelength of the emitted X-rays.