What Is the Change in Entropy When Mixing N2 and O2 Gases?

[mooncrater]

Homework Statement

The question is says:
Two vessels divided by a partition contain 1 mol of N₂ and 2 mol of O₂ gas. If the partition is removed and gases ate mixed isothermally, then find the change in entropy due to mixing assuming initial and final pressure are same .

Homework Equations

ΔS=q_rev/T [the one I know]
ΔS_sys=-RΣn_iln x_i
Where n_i = number of moles of gas.
x_i= mole fraction of gas.[ the foreigner equation]

The Attempt at a Solution

As written here in the relevant equations part, the second equation is the one which is new to me. I don't know how it works, what it does and from where it came from. It looks like boltzmann's entropy formula but I don't think that both of them are same. The question can be easily solved by this equation, but this equation is the one creating problem.

 

[MostlyHarmless]

What is R?

 

[MostlyHarmless]

Also, this at least looks related to the formula for change in entropy of an ideal gas.

That is, taking the entropy of each which I believe, in general, is $S= Nk[\log(Ω)]$ where Ω is multiplicity. So taking the difference of two entropies gives you a ration of multiplicities inside the log. I don't know what formulas you have for multiplicity, but it seems like this may be where your equation comes from. Though, someone else can likely explain it better, or correct me.

 

[mooncrater]

R is the universal gas constant.
What is N here?

 

[Raghav Gupta]

R is the universal gas constant.
What is N here?

Number of atoms.

 

[Chestermiller]

Smith and van Ness, Introduction to Chemical Engineering Thermodynamics, page 300:

A ideal gas is a model gas comprised of imaginary molecules of zero volume that do not interact. Each chemical species in an ideal gas mixture therefore has its own private properties, uninfluenced by the presence of other species. This is the basis of Gibb's theorem:

A total thermodynamic property (U, H, S, A, G) of an ideal gas mixture is the sum of the total properties of the individual species, each evaluated at the mixture temperature but at is own partial pressure.

Chet

 

[Raghav Gupta]

ΔS_sys=-RΣn_iln x_i
Where n_i = number of moles of gas.
x_i= mole fraction of gas.

What would be the proof for this?
Is it a complicated one involving partial derivatives?

 

[Chestermiller]

ΔS_sys=-RΣn_iln x_i
Where n_i = number of moles of gas.
x_i= mole fraction of gas.

What would be the proof for this?
Is it a complicated one involving partial derivatives?

No. From the information I gave in my previous post, you should be able to derive the equation for the entropy change for mixing ideal gases, going from the pure components at pressure P to the mixture at the same pressure P.

Chet

 

[Raghav Gupta]

A total thermodynamic property (U, H, S, A, G) of an ideal gas mixture is the sum of the total properties of the individual species, each evaluated at the mixture temperature but at is own partial pressure.

Chet

So S = V + P + T ?
ΔS= ΔV + ΔP + ΔT ?

 

[Chestermiller]

So S = V + P + T ?
ΔS= ΔV + ΔP + ΔT ?

Do you know how to determine the change in entropy of a pure species ideal gas if its pressure changes at constant temperature?

 

[Raghav Gupta]

No,
Actually entropy is to me degree of randomness and also have formula,
ΔS = Δq/T

 

[Chestermiller]

No,
Actually entropy is to me degree of randomness and also have formula,
ΔS = Δq/T

OK. Before you can do the derivation of the ideal gas entropy of mixing, you need to develop some background understanding of entropy. Did I send you my write up on the first and second laws of thermo?

Chet

 

[Raghav Gupta]

No, you till now have not send it. Why that question?
I think you may be thinking that we had a discussion in some other thread about it?
Or are you asking can I send you the write - up?

 

[Chestermiller]

No, you till now have not send it. Why that question?
I think you may be thinking that we had a discussion in some other thread about it?
Or are you asking can I send you the write - up?

No. I'll send you a link to it. It is for students who are struggling with some of the concepts because the concepts are presented so poorly in textbooks.

 

[Chestermiller]

See this thread: https://www.physicsforums.com/threa...n-adiabatic-process-and-heat-capacity.796304/

Post #12, starting with the tutorial section

If you have any questions, I'll be glad to answer them. Pay particular attention to how to go about determining the entropy change between two thermodynamic equilibrium states.

Chet