(i) The value of the entropy of a system of atoms and molecules in a thermodynamic system is a measure of the disorder in the arrangements of its particles.

(ii) In solids, which are typically ordered on the molecular scale, usually have smaller entropy than liquids, and liquids have smaller entropy than gases and colder gases have smaller entropy than hotter gases.

(iii) Moreover, according to the third law of thermodynamics, at absolute zero temperature, crystalline structures are approximated to have perfect order and zero entropy.

(iv) Entropy and disorder also have associations with equilibrium. Entropy, from this perspective, is defined as a thermodynamic property which serves as a measure of how close a system is to equilibrium.

(v) In a stretched out piece of rubber, the arrangement of the molecules of its structure has an ordered distribution and has zero entropy, while the disordered kinky distribution of the atoms and molecules in the rubber in the non-stretched state has positive entropy

(vi) In a gas, the order is perfect and the measure of entropy of the system has its lowest value when all the molecules are in one place, whereas when more points are occupied the gas is all the more disorderly and the measure of the entropy of the system has higher value.

(vii) The mathematical basis with respect to the association entropy has with order and disorder is given by famous Boltzmann formula:

`S = k ln W`

Which relates entropy `S` to the number of possible states `W` in which a system can be found

(viii) It is obvious that entropy is a measure of order or, most likely, disorder in the system. Second law of thermodynamics, as famously enunciated by Clausius in `1865`, states that : The entropy of the universe tends to a maximum.

(ix) Entropy is also a measure of the tendency of a process, such as a chemical reaction, to be entropically favored, or to proceed in a particular direction.

One can think entropy as a measure of the degree of randomness or disorder in a system. The greater the disorder, in a system, the higher is the entropy.

Prediction of sign of `Delta S` using the concept of Randomness.

(i) With change in temperature at constant `V`

As `T uparrowes S uparrow es => Delta S_text(sys) > 0`

(ii) Change in volume at constant `T`

As `V uparrowes S uparrowes => DeltaS_text(sys) > 0`

(iii) For phase change

`S_text(solid) < S_text(liquid) < S_text(gas)`

(iv) In chemical reaction, entropy change (`Delta S`)

(a) Involving only solids and liquids entropy change will be small

eg. entropy of graphite > diamond (only when we know the structure or any other property).

(b) Involving gases

If `(Delta n)_g > 0 => (DeltaS) > 0`
If `(Delta n)_g < 0 => (Delta S) < 0`
If `(Delta n)_g = 0 => (Delta S) ne 0`

(v) As atomicity `uparrow`es disorder `uparrow`es

`1 mol. quad quad 1mol.`

`S_[NO(g)] < S_[NO_2(g)]`

`S_(CH_4) < S_(C_2H_2)`

(vi) For the molecules having same atomicity, entropy will be more for the substance having more molecular mass.

(vii) In an irreversible process entropy of universe increases but it remains constant in a reversible process.

`Delta S_text(sys) + Delta S_text(surr) =0` for rev. process

`Delta S_text(sys) + Delta S_text(surr) > 0` for irrev. process

`Delta S_text(sys) + Delta S_text(surr) >= 0` ( ln general )

`text[Entropy change in isolated system (isolated system = sys + surr)]`

Consider a system taken state `A` to state `B` by an irreversible path and returned to state `A` by a reversible path. Since one of the step is irreversible, according to classius inequality, sum of `q//T` over the cycle must be less than zero(See fig.). Hence

`sum_(A->B)q_text(irr)/T + sum_(B->A)q_text(rev)/T <= 0 => sum_(A->B)q_text(irr)/T <= - sum_(B->A)q_text(rev)/T`

But `sum_(B-> A) q_text(rev)/T = sum_(A-> B) q_text(rev)/T` since the process is reversible

for infinitesimally small change

`[dq_text(rev)]/T_(A-> B) = dS_[text(sys) A-> B]`

`dS_text(system) *[(dq)/T]_(A-> B) > 0`

`Delta S_text(Total isolated sys) > 0`

Entropy calculation in process involving ideal gases.

From First law

`dq =dU + PdV`

`=> [dq_text(rev)]/T = dU/T + (PdV)/T`

But for ideal gas

`(dU)/T = (nC_v dT)/T`

`dS_text(sys) = (nC_v dT)/T + (nR)/V dV`

Integration gives

`Delta S = nC_v ln T_2/T_1 + nR ln (V_2/V_1)`

General Expression, for any process

`Delta S = nC_v ln T_2/T_1 + nR ln V_2/V_1 = nC_p ln T_2/T_1 + nR ln P_1/P_2`

(i) `text(Isothermal process)` :

Reversible & irreversible isothermal expansion and contraction of an ideal gas.

`Delta S_text(sys) = nR In (V_2/V_1)`

For surroundings

`Delta S_text(surr) = - Delta S_text(sys)` (for riversible)

`Delta S_text(surr) = q_text(surr)/T` (for irreversible)

(ii) `text(Isobaric process)` :

`DeltaU = C_V DeltaT`
`DeltaH = C_p DeltaT = q_p`

For surroundings

`DeltaS_text(surr) = - DeltaS_text(sys)` (for reversible)
`DeltaS_text(surr) = q_text(surr)/T` (for irreversible)

(iii) `text(lsochoric process)` :

`DeltaU =C_v DeltaT = q_v`

`DeltaH = C_pDeltaT`

`DeltaS_text(sys) = nC_v In (T_2/T_1)`

For surrounding

`DeltaS_text(surr) =- DeltaS_text(sys)` (for reversible)

`DeltaS_text(surr) = q_text(surr)/T` (for irreversible)


(iv) `text(Adiabatic process)`

`DeltaU = C_v DeltaT`

`DeltaH = C_p DeltaT`

`DeltaS_text(sys) = nC_v In T_2/T_1 + nR In V_2/V_1` for irreversible process

`DeltaS_text(sys) = 0` for reversible adiabatic compression and expansion.

For surroundings

`DeltaS_text(surr)= q_text(surr)/T` (for irreversible)