1. Equation for wavelengths of radiation emitted by the hydrogen atom.
1/λ = R [1/n² - 1/m²]
where R = 1.09737*10^7 m-1.
n and m are integers with m>n.
2. Velocity of an electron when it is in a stationary orbit represented by n which is an integer
v = Ze²/2 ε0hn
3. Radius of a stationary orbit based on n which is an integer
r = ε0h²n²/πmZe²
4. Kinetic energy when electron is in nth orbit is
K = ½ mv² = mZ²e4/8 ε0²h²n²
5. The potential energy of the atom is
V = - Ze²/4π ε0r = -mZ²e4/4ε0²h²n²
The expression for potential energy is obtained by assuming the potential energy to be zero when the nucleus and the electron are widely separated.
6. The total energy of the atom is
E = K+V = - mZ²e4/8 ε0²h²n²
7. For a hydrogen like ion with Z protons in the nucleus,
rn = radius of ‘n’ th orbit = n²a0/Z
8. Energy of hydrogen atom when the single electron is in the nth orbit.
En = E1/n² = -13.6/n² eV
-13.6 eV is the energy when the electron is in the n = 1 orbit.
Note that the energy is expressed in negative units, so that larger magnitude means lower energy.
9. If an electron jumps from mth orbit to nth orbit (m>n) of a hydrogen like ion, the energy of the atom gets reduced from Em to En. The wavelength of the emitted radiation will be
1/ λ = (Em – En)/hc = RZ²{1/n² - 1/m²]
10. The wave function of the electron ψ(r,t) is obtained from the Schrodinger’s equation
-(h²/8π²m) [∂²ψ /∂x² + ∂²ψ /∂y² + ∂²ψ/∂z²] - Ze²ψ/4πε0r = E ψ
where
(x.y,z ) refers to a point with the nucleus as the origin and r is the distance of this point from the nucleus.
E refers to the energy.
Z is the number of protons.
11. The energy of the wave function of characterized by n,l, and ml depends only on n and may be written as
En = - mZ²e4/8 ε0²h²n²
12. When n = 1, the wave function of the hydrogen atom is
ψ(r,) = ψ100 = √(Z³/ π a0²) *(e-r/ a0)
where
ψ100 denotes that n =1, l = 0 and ml = 0
a0 = Bohr radius
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