Use Coulomb's law to calculate the ionization energy in kJ>mol of an atom composed of a proton and an electron separated by 100.00 pm. What wavelength of light has sufficient energy to ionize the atom?

Verified step by step guidance

1. Identify the formula for Coulomb's law: \( F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges of the proton and electron, and \( r \) is the separation distance.
2. Calculate the potential energy \( U \) using the formula \( U = \frac{k \cdot |q_1 \cdot q_2|}{r} \). This energy represents the ionization energy required to separate the electron from the proton.
3. Convert the potential energy from joules to kilojoules per mole by using Avogadro's number \( 6.022 \times 10^{23} \text{ mol}^{-1} \).
4. Use the energy-wavelength relationship \( E = \frac{hc}{\lambda} \) to find the wavelength \( \lambda \) of light that has sufficient energy to ionize the atom, where \( h \) is Planck's constant and \( c \) is the speed of light.
5. Rearrange the formula to solve for \( \lambda \): \( \lambda = \frac{hc}{E} \), and substitute the ionization energy calculated in step 3 to find the wavelength.

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic interaction between charged particles. It states that the force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This principle is crucial for calculating the energy associated with the interaction of an electron and a proton in an atom.

Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an atom or ion in its gaseous state. It is a critical concept in understanding how atoms interact with energy, particularly in the context of light absorption. The ionization energy can be calculated using Coulomb's Law, which helps determine how much energy is needed to overcome the attractive force between the electron and the nucleus.

Wavelength and Energy 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 is essential for determining the wavelength of light that can provide sufficient energy to ionize an atom. By calculating the ionization energy, one can find the corresponding wavelength of light needed for ionization.