Schrodinger’s Equation

Quantum mechanics describes the spectra in a much better way than Bohr’s model.

Electron has a wave character as well as a particle character. The wave function of the electron ψ(r,t ) is obtained by solving Schrodinger’s wave equation. The probability of finding an electron is high where | ψ(r,t )|² is greater. Not only the information about the electron’s position but information about all the properties including energy etc. that we calculated using the Bohr’s postulates are contained in the wave function of ψ(r,t).

Quantum Mechanics of the Hydrogen Atom

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.

There are infinite number of functions ψ(r,t) which satisfy the equations.

These functions may be characterized by three parameters n,l, and m_l.

For each combination of n,l, and m_l there is an associated unique value of E of the atom of the ion.

The energy of the wave function of characterized by n,l, and m_l depends only on n and may be written as


En = - mZ²e⁴/8 ε₀²h²n²

These energies are identical with Bohr’s model energies.

The paramer n is called the principal quantum number, l the orbital angular momentum quantum number and m_l. The magnetic quantum number.

When n = 1, the wave function of the hydrogen atom is

ψ(r) = ψ₁₀₀ = √(Z³/ π a₀²) *(e^{-r/ a₀})


ψ₁₀₀ denotes that n =1, l = 0 and m_l = 0

a₀ = Bohr radius

In quantum mechanics, the idea of orbit is invalid. At any instant the wve function is spread over large distances in space, and wherever ψ≠ 0, the presence of electron may be felt.

The probability of finding the electron in a small volume dV is | ψ(r)| ² dV
We can calculate the probability p(r)dr of finding the electron at a distance between r and r+dr from the nucleus.

In the ground state for hydrogen atom it comes out to be

P(r) = (4/ a₀)r²e ^{-2r/ a₀}

The plot of P(r) versus r shows that P(r) is maximum at r = a₀ Which the Bohr’s radius.

But when we put n =2, the maximum probability comes at two radii one near r = a₀ and the other at r = 5.4 a₀. According to Bohr model all electrons should be at r = 4 a₀.

Carboxylic Acids - Physical Properties - Revision Points

a. Physical state and smell

The first three members are colourless liquids and have pungent smell. The next six members are oily liquids with a faint unpleasant odour.

Still higher acids are colourless waxy solids.

Benzoic acids and its homologues are colourless solids.

b. Boiling points

They have higher boiling points than the corresponding alcohols of comparable molecular masses.

Carboxylic acids have higher boiling points due to the presence of intramolecular hydrogen bonding. Due to the hydrogen bonding, carboxylic acids exist as dimers.

c. Melting point

In the case of first ten carboxylic acids, the melting points of acids containing even number of carbon atoms is higher than the next lower and higher member containing odd number of carbon atoms.

The melting and boiling points of aromatic acids are usually higher than those of aliphatic acids of comparable molecular masses.

d. Solubility in water

The first four members of aliphatic carboxylic acids are very soluble in water. The solubility in water decreases gradually with rise in molecular mass. All are soluble in alcohol or ether.

Benzoic acid is sparingly soluble in cold water but is soluble in hot water, alcohol and ether.