Test: Bohr’s Model of Atom - JEE MCQ

We know that
1/λ​=R(1/n²​₁​−1/n²₂​​)
1/λ=R(1/2²) − (1/3²​)⇒R(1/4)−(1/9​)
1/λ​=(9−4​/36)R=5R​/36⇒λ=36​/5R

The speed of revolving electron in nth state of hydrogen atom is:
v=​e²/2nhϵ₀ ​
For n=1,
v= (1.6×10⁻¹⁹)^2​/2(1)(6.6×10⁻³⁴)(8.85×10⁻¹²)
v=2.56×10^−38​/116.82×10⁻⁴⁶
 
v=0.0219×10⁸ms⁻¹
The speed of light is 3×10⁸
Hence, 
v/c​=0.0219×10^8​/3×10⁸
v/c​=1/137
 

K.E. of nth orbit
=> (1/k) Ze²/2r
For H atom,
K.E.=(1/4πε) x (e²/2r)

The angular momentum L =m_e​vr is on integer multiple of h​/2π
mvr= nh​/2π
For, n=1
mvr= h​/2π
The correct answer is option B.

r=(0.529Å)n²/7
r ∝n²

Statement i. Radius of Bohr's orbit of hydrogen atom is given by
r= n2h2​/4π2mKze2
or, r=(0.59A˚)(n2​/z)
So, from expression we found r∝n2
Hence the 1st statement is correct.
Statement ii. 
We know that
En=-13.6 x z2/n2
So, En ∝1/n2
Hence the 2nd statement is wrong.
Statement iii.Bohr defined these stable orbits in his second postulate. According to this postulate:

• An electron revolves around the nucleus in orbits
• The angular momentum of revolution is an integral multiple of h/2π – where Planck’s constant [h = 6.6 x 10-34 J-s].
• Hence, the angular momentum (L) of the orbiting electron is: L = nh/2 π

 Hence the 3rd statement is correct.
Statement iv.According to Bohr's theory
Angular momentum of electron in an orbit will be Integral multiple of (h/2π)
Magnitude of potential energy is twice of kinetic energy of electron in an orbit
∣P.E∣=2∣K.E∣
K.E=(13.6ev)( z2/n2)​
Hence, The 4th statement is correct.

Bohr used conservation of angular momentum.
For stationary orbits, Angular momentum Iω=nh2π
where n=1,2,3,...etc

• In the Bohr model, electrons orbit the nucleus in specific energy levels.
• When an electron jumps from a higher energy level (orbit) to a lower one, it emits energy in the form of radiation.
• This emitted radiation corresponds to a photon, which is the light we observe.
• Conversely, if an electron moves to a higher orbit, it absorbs energy instead.

The ratio of the volume of the atom and the volume of the nucleus is 10¹⁵
The radius of an atomic nucleus is of the order of 10⁻¹³cm or 10⁻¹⁵m or one Fermi unit.
On the other hand, the radius of an atom is of the order of 10⁻⁸cm or 10⁻¹⁰m or one angstrom unit.
Note:
The radius of nucleus is much smaller than atomic radius.
The ratio of atomic radius to radius of nucleus is 10⁻¹⁰m /10⁻¹⁵m ​=10⁵
Volume is proportional to cube of radius.
The ratio of atomic radius to radius of nucleus is (10⁵)³=10¹⁵