Use Coulomb's law to calculate the ionization energy in kJ/mol of an atom composed of a proton and an electron separated by 110.00 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 is the ionization energy for one atom.
3. Convert the ionization energy from joules to kJ/mol by multiplying by Avogadro's number \( 6.022 \times 10^{23} \) and converting from joules to kilojoules.
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. Solve for \( \lambda \) by rearranging the equation to \( \lambda = \frac{hc}{E} \), using the ionization energy calculated in the previous steps.

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 essential for calculating the potential energy of an electron in an atom, which is crucial for determining ionization energy.

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 typically measured in kJ/mol and reflects the strength of the attraction between the electron and the nucleus. Understanding ionization energy is vital for predicting how easily an atom can lose an electron and thus its reactivity.

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 is important for determining the wavelength of light needed to provide sufficient energy to ionize an atom, as it allows us to calculate the wavelength corresponding to the ionization energy obtained from Coulomb's Law.