Question

Assume an imaginary world, where angular momentum is quantized to even multiple $\hslash$. Find the longest possible wavelength emitted by Hydrogen in the visible spectrum.

• A. 700nm
• B. 484 nm
• C. 600nm
• D. 584 nm

Answer 1

Answer: (B)

484 nm

Answer 2

Mvr = 2 n$\hslash$

or mv = $\frac{2n\hslash}{r}$

mv² = $\frac{m^{2}v^{2}}{n}$ = $\frac{(2n\hslash)^{2}}{mr^{2}}$

$\frac{Ze^{2}}{4\pi\varepsilon_{0}r^{2}}$ = $\frac{mv^{2}}{r}$

or $\frac{Ze^{2}}{4\pi\varepsilon_{0}r^{2}}$ = $\frac{(2n\hslash)}{mr^{2}(r)}$

or r = $\frac{(2n\hslash)24\pi\varepsilon_{0}}{mZe^{2}}$

be = k + e = $\frac{–Ze^{2}}{8\pi\varepsilon_{0}r}$ = $\frac{–Z^{2}e^{4}m}{8\pi\varepsilon_{0}(2n\hslash)^{2}4\pi\varepsilon_{0}}$

= $\frac{–Z^{2}e^{4}m}{32\varepsilon_{0}n^{2}h^{2}}$

BE = $\frac{–3.4}{n^{2}}$eV for Hydrogen. To find longest wavelength hn = 3.4 $\left\lbrack 1–\frac{1}{4} \right\rbrack$

= 3.4 × $\frac{3}{4}$ = 2.55

l (nm) = $\frac{1250}{2.55}$ = 484 nm