Given that, y = x + 11.1% of x
\(=\left(x+\frac{11.1}{100} x\right)=x+0.111 x\)
Y = 1.111 x
mass number = 37
number of protons + number of neutrons = 37
(x -1) + 1.111x  = 37
x+1.111x = 38
2.111x  = 38
\(x=\frac{38}{2.11}\)
x = 18.009
x 18 (whole number)
ஃ  Atomic number = x- 1 = 18 - 1 = 17
Mass number = 37
Symbol of the ion \({ _{ 17 }^{ 37 }{ Cl } }^{ - }\)

Erwin Schrodinger expressed the wave nature of electron in terms of a differential equation. This equation determines the change of wave function in space depending on the field of force in which the electron moves. The time independent Schrodinger equation can be expressed as,
\(\overset { \wedge }{ H } \psi =E\psi \)          .........(1)
Where \(\overset { \wedge }{ H } \) is called Hamiltonian operator, \(\psi \) is the wave function and is a function of position coordinates of the particle and is denoted as \(\psi \) (x, y, z) E is the energy of the system
\(\overset { \wedge }{ H } =\left[ \frac { { -h }^{ 2 } }{ 8{ \pi }^{ 2 } } \left( \frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } \right) +V \right] \)
can be written as
\(\left[ \frac { { -h }^{ 2 } }{ 8{ \pi }^{ 2 }m } \left( \frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } \right) +V\Psi \right] =E\Psi \)
Multiply by \(\frac { 8{ \pi }^{ 2 }m}{ { -h }^{ 2 } } \)and rearranging
\(\frac { { \partial }^{ 2 }\psi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial y }^{ 2 } } +\frac { { \partial }^{ 2 }\psi }{ { \partial z }^{ 2 } } +\frac { 8{ \pi }^{ 2 }m }{ { -h }^{ 2 } } (E-V)\Psi =0\)          ........(2)
The above Schrodinger wave equation does not contain time as a variable and is referred to as time independent Schrodinger wave equation. This equation can be solved only for certain values of E, the total energy. i.e. the energy of the system is quantised. The permitted total energy values are called eigen values and corresponding wave functions represent the atomic orbitals.

(i) The energy of photon is sufficient to disturb a sub-atomic particle so that there is uncertainty in the measurement of position and momentum of the sub-atomic particle.
(ii) However, the energy is insufficient to disturb a macroscopic object.

From de Broglie equation, wavelength \(\lambda =\frac { h }{ mv } \)
For same wavelength with two different particles, (ie) electron and proton m₁v₁ = m₂v₂ (h is constant) Lesser the mass of the particle, greater will be the velocity.
Hence electron will have higher velocity

The word Aufbau in German means 'building up'. In the ground state of the atoms, the orbitals are filled in the order of their increasing energies. That is the electrons first occupy the lowest energy orbital available to them.
Once the lower energy orbitals are completely filled, then the electrons enter the next higher energy orbitals. The order of filling of various orbitals as per the Aufbau principle which is in accordance with (n + l) rule.

Statement : "No two electrons in an atom can have the same set of values of all four quantum numbers"
Explanation : It means that, each electron must have unique values for the four quantum numbers (n, l, m and s).
For the lone electron present in hydrogen atom, the four quantum numbers are: n = 1; l = 0; m = 0 and s = +1/2. For the two electrons present in helium, one electron has the quantum numbers same as the electron of hydrogen atom, n = 1.
l = 0, m = 0 and s = +1/2. For other electron, the fourth quantum number is different i.e., n = 1, l = 0, m = 0 and s = -1/2.
As we know that the spin quantum number can have only two values +1/2 and - 1/2, only two electrons can be accommodated in a given orbital in accordance with pauli exclusion principle.

Applying Bohr's postulates to a hydrogen like atom (one electron species such as H, He⁺ and Li²⁺ etc...)the radius of the n^th orbit and the energy of the electron revolving in the n^th orbit were derived. The results are as follows:
r_n = \(\frac { (0.529){ n }^{ 2 } }{ x } \mathring { A } \)  ...(1)
E_n = \(\frac { -13.6({ z }^{ 2 }) }{ { n }^{ 2 } } ev\quad { atom }^{ -1 }\)   or ...  (2) or
E_n = \(\frac { (-1312.8){ z }^{ 2 } }{ { n }^{ 2 } } kJ\quad { mol }^{ -1 }\) .....(3)

Bohr's atom is based on the following assumptions:
(a) The energies of electrons are quantised
(b) The electron is revolving around the nucleus in a certain fixed circular path called stationary orbit.
(c) Electron can revolve, only in those orbits in which the angular momentum (mvr) of the electron must be equal to an integral multiple of h/2π i.e mvr =nh/2π
where n = 1,2,3 ...etc...
As long as an electron revolves in the fixed stationary orbit, it doesn't lose its energy.
However, when an electron jumps from higher energy state (E₂)to a lower energy state (E₁)the excess energy is emitted as radiation. The frequency of the emitted radiation is
E2 = E1 = hv and 
\(v=\frac { ({ E }_{ 2 }-{ E }_{ 1 }) }{ h } \)
Conversely, when suitable energy is supplied to an electron, it will jump from lower energy orbit to a higher energy orbit.

(i) Atoms consist of huge positively charged centers called nuclei.
(ii) Most of the space inside the atom is empty.

According to this theory, atom was assumed to consist of a sphere of uniform distribution of about 10^-10 m positive charge with electrons embedded in it such that the number of electrons equal to the number of positive charges and the atom as a whole is electrically neutral.

a) Principal quantum number defines energy and size of an orbital.
b) Azimuthal quantum number defines shape of an orbital
c) Magnetic quantum number defines spatial orientation (direction) of an orbital.