Entropy & its Physical Significance | Physical Chemistry PDF Download

Entropy:  Let

then entropy change for a finite change of state of a system at constant temperature is given by

Unit of entropy is JK^–1.
Entropy is an extensive property. Its value depends upon the amount of the substance involved.
Entropy change for an ideal gas: According to first law, we have dqrev = dV + PdV
Dividing T; we have

                                 


or

This is the expression when both the volume and temperature or pressure and temperature of an ideal gas are changed.
For an isothermal process, the change is given by the relation

∵ T₂ = T₁

_and 

For the change of temperature at constant volume

For the change of temperature at constant pressure

Entropy change in a few typical cases.

Entropy change in a reversible phase transformat ion.

Since the reversible phase transformation takes place at constant equilibrium temperature then entropy.

where q_rev is the heat invo lved in the phase transformat ion.
Examples   

Entropy change in an irreversible phase transition.
A → B
then  
 

Problem.  The phase transformation of one mole of liquid water at –10°C (T₁) to solid water –10°C can be calculated following the paths given below: 

(i)       ΔC_Pl = 75.312 JK^-1 mol^-1
(ii)          ΔHf = 6088.2 J mol^-1

(iii)           C_Ps = 36.401 JK^-1 mol^-1

Then changes of entropy can be calculated 
 Sol. (i)  

ΔS = -1.358 JK mol ^-1

then, the change of the entropy of the process 

is given by 

Entropy change when two solid at different temperatures are brought together: Let one at higher temperature Th and other at lower temperature Tc. When two bodies are brought together then
Heat lost = heat gain                         

(by conservat ion law)

i.e. Heat lost by hot body = Heat gained by cold body
       Cp(T_h – T) = C_p (T – T_c)
or     

Entropy changes of two bodies are bodies are:

The total change of entropy is   ΔS_total = ΔS_h + ΔS_c 

Problem. 5.0 gm ice at 273 K is added to 30 gm of water at 323 K in a thermally insulated container.
 What is the final temperature? 
ΔH_fus of ice at 273 K = 334.72 J gm^-1
 C_p of water = 4.184 JK^-1 gm^-1 
ΔH_vap of water at 373 K = 2.259 J gm^-1 
 

Sol. Heat require to convert 5 gm of ice at 273 K to 5 gm water at 273 K
= (5 g) × (334.72 J gm^-1) = 1673.6 J

Final temperature after mixing
Heat gained by ice = Heat lost by hot water

(5 gm) (334.72 J gm^–1) + (5 g) (4.184 J gm^–1 K^–1) (T – 273 K)
= (30 g) (4.184 JK^–1 gm^–1) (323 K – T)

Solving for T, we get
T = 304.43 K
 

Entropy of mixing: Entropy of mixing is defined as the difference between the entropy of the mixture of gas and the sum of the entropy of the separate gases, each at a pressure P.
For a mole of an ideal gas,

S = C_V ln T + R ln V + S₀

where S₀ is the integration constant.
We know that,   
 

, we get S = C_P ln T – R ln P + R ln R + S₀ 

                        = C_P ln T – R ln P + S₀’ …(1)

Where         S₀ = (R ln R + S₀) is another constant.
If n₁, n₂, …. etc. are the number of moles of the various gases present in the mixture and p₁, p₂, … etc. are their partial pressures, then the entropy of the mixture is given by

S = n₁ (Cp ln T –R ln P₁ + S0’) + n₂(Cp ln T – R ln P₂ + S₀’) + ……

       ...............(2)

The partial pressure (p) of an ideal gas is given by P = xp where x is the mole fract ion and P is the total pressure.
Putting this value we get

 
 ...............(3)

Equation (3) gives the entropy of a mixture of ideal gases. “Entropy of mixing is difference between the entropy of the mixture of gases and the sum of the entropies of the separate gases”
Thus 

where n_i and n_x are number of moles and mole fraction of each constituent of the mixture.
If n is the total number of moles then
n = n₁ + n₂ ……..
then

then molar entropy of gases are

Problem.  Calculate the entropy of mixing of one mole of nitrogen gas and three moles of hydrogen gas, assuming that no chemical reaction occurs. i.e. mixing is ideal.
 Sol.
n₁ = 1,  n₂ = 3

The mole fract ion are:

form equation

Physical Significance of Entropy: 

(1)  Entropy as a measure of the disorder of the system: Spontaneous process are accompanied by increase in entropy as well as increase in the disorder of the system.
Thus, entropy is a measure of the disorder of a system.

(2)  Entropy as a measure of probability: All spontaneous process lead to increase in entropy and also to increase in disorder. It appears that there is a close relation between ‘entropy S and the probability (W).

This relationship was expressed by Boltzmann as
S = k ln (Probabilit y) = kln (W)

S = k ln W

where k is Boltzmann constant
S = k ln W = k ln (arrangement)N_A
= k N_A ln (arrangement)
S = R ln (arrangement)