Correct Answer - Option 2 :
\(\frac{h}{4\pi}\)
CONCEPT:
Bohr's Atomic Model:
- Bohr proposed a model for hydrogen atom which is also applicable for some lighter atoms in which a single electron revolves around a stationary nucleus of positive charge Ze (called hydrogen-like atom).
Bohr's model is based on the following postulates:
- He postulated that an electron in an atom can move around the nucleus in certain circular stable orbits without emitting radiations.
- Bohr found that the magnitude of the electron's angular momentum is quantized i.e.
\(⇒ L = m{v_n}\;{r_n} = n\left( {\frac{h}{{2\pi }}} \right)\)
Where n = 1, 2, 3, ..... each value of n corresponds to a permitted value of the orbit radius, rn = Radius of nth orbit, vn = corresponding speed, and h = Planck's constant
EXPLANATION:
According to Bohr's atomic model,
- The magnitude of the electron's angular momentum is quantized and is given as,
\(⇒ L = n\left( {\frac{h}{{2\pi }}} \right)\)
For option 1,
\(⇒ n\left( {\frac{h}{{2\pi }}} \right)=\frac{h}{\pi}\)
⇒ n = 2 -----(1)
For option 2,
\(⇒ n\left( {\frac{h}{{2\pi }}} \right)=\frac{h}{4\pi}\)
⇒ n = 0.5 -----(2)
For option 3,
\(⇒ n\left( {\frac{h}{{2\pi }}} \right)=\frac{2h}{\pi}\)
⇒ n = 4 -----(3)
For option 4,
\(⇒ n\left( {\frac{h}{{2\pi }}} \right)=\frac{3h}{\pi}\)
⇒ n = 6 -----(4)
- Since the angular momentum is quantized so the value of n can only be an integer.
- For \(\frac{h}{4\pi}\), the value of n is a decimal number so \(\frac{h}{4\pi}\) can't be the value of the angular momentum of the electron.
- Hence, option 2 is correct.
