Question

In a hydrogen-like atom, an electron is orbiting in an orbit having quantum number n. Its frequency of revolution is found to be 13.2×10¹⁵ Hz. Energy required to move this electron from the atom to the above orbit is 54.4 eV. In a time of 7 nano second the electron jumps back to orbit having quantum number n/2. If τ be the average torque acted on the electron during the above process, then τ=(10+x)×10²⁷ in Nm. Find the value of x.

Given: h/λ=2.1×10⁻³⁴ J−s, frequency of revolution of electron in the ground state of H atom v0=6.6×10¹⁵ and ionization energy of H atom E0=13.6 eV

Answer 1

To find x, calculate the initial energy level using ΔE, determine angular frequencies, the change in angular momentum, and use Newton's second law for rotational motion to calculate the average torque.

Step-by-step explanation:

To determine the value of x in the average torque acted on the electron, use the energy levels of the hydrogen-like atom as well as the energy of a photon emitted during the transition:

ΔE = hf = E1 - E2

For a hydrogen-like atom, the energy levels can be represented by the formula En = (-13.6eV/n²). In this case, when the electron transitions from n to n/2, the energy difference ΔE can be calculated. Since we are given ΔE as the energy needed to move an electron to the orbit with quantum number n, which is 54.4 eV, we can use that information to find the quantum number n initially. Remembering that the ionization energy of the H atom, E0, is 13.6 eV, if n were 1, we would need exactly 13.6 eV to ionize the atom. However, ΔE is 54.4 eV, therefore the initial energy level n must be such that moving to a free state (zero energy) would require exactly this amount of energy.

Once n is found, we can calculate the angular frequency (ω = 2πf) for both orbits and then determine the change in angular momentum ΔL = L1 - L2, which during the transition should equal the angular impulse (the product of torque and time). By isolating for the torque value τ, we have τ = ΔL /Δt, where Δt is given as 7 nanoseconds.

Finally, substituting the known values and solving for τ, we can determine the missing coefficient x for the torque equation.