### Entropy Change in Mixing of Two Gases

What is the change in entropy when 0.7 m3 of CO2 and 0.3 m3 of N2 each at 1 bar and 25oC blend to form a homogenous mixture at the same conditions. Assume ideal gases.

Calculations:

Since there is no change in temperature after mixing, internal energy change of individual gases is zero.

For a reversible process, dU = TdS - PdV

And for constant internal energy process dU = 0. Therefore TdS = PdV

dS = PdV/T → 1

For an ideal gas, PV = nRT

Therefore, P/T = nR/V

Substituting for P/T in Equn.1,

dS = nR dV/V

Integrating the above equation,

ΔS = nR ln(V2/V1) → 2

Since mixing is an irreversible process, we can calculate the entropy change of individual gases by assuming the equivalent reversible process given below:

(a) Reversible expansion of gas from its initial volume to the final volume of 1 m3. Entropy change for this process is given by Equn.2

(b) Mixing two gases at identical conditions. (each at a volume of 1 m3). Since there is no change in thermodynamic conditions in this process, entropy change is zero for this step. (Entropy is a state function)

We shall calculate the entropy change of individual gases and total entropy change of the system is the sum of the entropy changes of two gases.

For CO2:

Moles of CO2 = 105 x 0.7 / (R x 298) = 234.9 / R

ΔS = 234.9 ln (1/0.7) = 83.78 J/oC

For N2:

Moles of N2 = 105 x 0.3 / (R x 298) = 100.7 / R

ΔS = 100.7 ln (1/0.3) = 121.24 J/oC

Entropy change of the system = 83.78 + 121.24 = 205.02 J/oC