Question

For an electron in the n^th orbit, which of the following relation is correct if R is the radius and n is the principal quantum number of some n^th orbit.
1. nλ = πr_n
2. \(n\pi =\frac{3}{2}\pi r_n\)
3. nλ = 2πrn
4. nλ = 4πrn

Answer 1

Correct Answer - Option 3 : nλ = 2πrn

CONCEPT:

• According to de Broglie matter has a dual nature of wave-particle.
• The wave associated with each moving particle is called matter waves.
• ​​de Broglie wavelength associated with the particle

\(⇒ λ = \frac{h}{P}\)   

Where, h = Planck's constant and P = Linear momentum of a particle

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.

\(⇒ L = m{v_n}\;{r_n} = n\left( {\frac{h}{{2π }}} \right)\)
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

• The radiation of energy occurs only when an electron jumps from one permitted orbit to another.

​EXPLANATION:

• de Broglie wavelength associated with the particle is given as,

\(⇒ λ_n = \frac{h}{mv_n}\)     -----(1)

Bohr found that the magnitude of the electron's  angular momentum is quantized i.e.

\(⇒ L = m{v_n}\;{r_n} = n\left( {\frac{h}{{2π }}} \right)\)     -----(2)
Where n = 1, 2, 3, ..... each value of n corresponds to a permitted value of the orbit of radius, rn, vn = corresponding speed and h = = Planck's constant

By equation 1 and equation 2,

\(⇒ m{v_n}\;{r_n} = n\left( {\frac{h}{{2π }}} \right)\)

\(⇒ n\left( {\frac{h}{{m{v_n} }}} \right)=2π\;{r_n} \)

⇒ nλ_n = 2πr_n

• Hence, option 3 is correct.

Characteristics of Matter waves:

1. ​​The lighter the particle, the greater is the de Broglie wavelength.
2. The higher the velocity of the particle, the smaller is its de Broglie wavelength.
3. The de Broglie wavelength of a particle is independent of the charge or nature of the particle.
4. The matter waves are not electromagnetic in nature. Only charged particles produce electromagnetic waves.