This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
You're on a roll — 3. c) Using Bohr's model, an electron of mass m and charge -e orbits a nucleus of charge +e at a distance r. Step 1: Identify the forces acting on the electron. The electrostatic force of attraction between the electron and the nucleus provides the centripetal force for the electron's circular motion. Electrostatic force (F_e): F_e = (1)/(4_0) (e · e)/(r^2) = (e^2)/(4_0 r^2) Centripetal force (F_c): F_c = (mv^2)/(r) Step 2: Equate the forces to find an expression for mv^2. (mv^2)/(r) = (e^2)/(4_0 r^2) mv^2 = (e^2)/(4_0 r) Step 3: Calculate the kinetic energy (K) of the electron. K = (1)/(2)mv^2 Substitute the expression for mv^2: K = (1)/(2) ( (e^2)/(4_0 r) ) = (e^2)/(8_0 r) Step 4: Calculate the potential energy (U) of the electron. The potential energy of an electron (charge -e) in the electric field of a nucleus (charge +e) at a distance r is: U = (1)/(4_0) ((+e)(-e))/(r) = (-e^2)/(4_0 r) Step 5: Calculate the total energy (E). The total energy is the sum of the kinetic and potential energies: E = K + U E = (e^2)/(8_0 r) + ( (-e^2)/(4_0 r) ) To combine these terms, find a common denominator: E = (e^2)/(8_0 r) - (2e^2)/(8_0 r) E = (e^2 - 2e^2)/(8_0 r) E = (-e^2)/(8_0 r) Thus, the total energy of an electron in a hydrogen atom is E = (-e^2)/(8_0 r). d) Given: Energy of radiation, E = 6.2 eV Planck's constant, h = 6.6 × 10^-34 J s Speed of light, c = 3.0 × 10^8 m s^-1 Conversion factor, 1 eV = 1.6 × 10^-19 J Step 1: Convert the energy from electron volts (eV) to Joules (J). E = 6.2 eV × (1.6 × 10^-19 J/eV) E = 9.92 × 10^-19 J i) Calculate the wavelength () of the emitted radiation. Step 2: Use the formula relating energy, Planck's constant, speed of light, and wavelength: E = (hc)/(). Rearrange to solve for : = (hc)/(E) Step 3: Substitute the values and calculate . = (6.6 × 10^-34 J s) × (3.0 × 10^8 m s^-1)9.92 × 10^-19 J = 19.8 × 10^-269.92 × 10^-19 m ≈ 1.9959677 × 10^-7 m Rounding to two significant figures: ≈ 2.0 × 10^-7 m ii) Calculate the momentum (p) of the photons. Step 4: Use the formula relating momentum, energy, and speed of light: p = (E)/(c). Step 5: Substitute the values and calculate p. p = 9.92 × 10^-19 J3.0 × 10^8 m s^-1 p = 3.3066... × 10^-27 kg m s^-1 Rounding to two significant figures: p ≈ 3.3 × 10^-27 kg m s^-1