What Variables Must X̅ Have in Order to be Considered a Partial Molar Quantity?

[physicisttobe]

Homework Statement

partial molar quantities

Relevant Equations

...

Hi everyone!

It's about the following task.

Partial molar quantities
a) How are partial molar quantities defined in general?
b) If X is an extensive state variable and X̅ is the associated partial variable, what types of variables must X̅ have?
c) Is the chemical potential of component i in a mixture a partial molar quantity? Why?
d) Is the chemical potential of component i in a mixture a partial molar quantity? Why? Justification?

I have some difficulties in answering some questions, especially the question b) If X is an extensive state variable and X̅ is the associated partial variable, what types of variables must X̅ have?
Unfortunately, I don't have a clue how to answer it. I only know that partial molar quantities are intensive variables like pressure, temperature and so on. But what types of variables must X̅ have? Can you explain me that please?

 

[Chestermiller]

Homework Statement: partial molar quantities
Relevant Equations: ...

Hi everyone!

It's about the following task.

Partial molar quantities

Partial molar properties apply to solid, liquid, and gaseous solutions of chemical constituents.

a) How are partial molar quantities defined in general?

A partial molar property for species "i" is defined as $$\bar{X}_i=\left(\frac{\partial X}{\partial n_i}\right)_{T,P,n_j,\ all\ j\neq i}$$where the n's are numbers of moles of the various species in the solution.

b) If X is an extensive state variable and X̅ is the associated partial variable, what types of variables must X̅ have?

$\bar{X}_i$ is an intensive property that depends on T, P, and the mole fractions of all chemical constituents

c) Is the chemical potential of component i in a mixture a partial molar quantity? Why?

The partial molar Gibbs free energy of a chemical species is defined as its chemical potential.

 

[physicisttobe]

@Chestermiller, thank you so much for your reply. So the answer to that question "If X is an extensive state variable and X̅ is the associated partial variable, what types of variables must X̅ have" is: the variables are T,p, and the mole fraction ni because they are all intensive?

 

[physicisttobe]

And I noticed that I forgot to post all the relevant equations for that task.
The complete task looks like this (see below).
The question d) is false because V and S are not constant, they are not intensive, V and S are extensive properties. Therefore, they can not be considered as constant. Is my consideration correct?

 

[Chestermiller]

@Chestermiller, thank you so much for your reply. So the answer to that question "If X is an extensive state variable and X̅ is the associated partial variable, what types of variables must X̅ have" is: the variables are T,p, and the mole fraction ni because they are all intensive?

I don't quite understand this question. Can you rephrase it? The phase rule tells us that for a single phase system containing N chemical components, the number of intensive variables required to specify the system at chemical equilibrium is N +1: T, P, and N-1 mole fractions.

 

[physicisttobe]

The question is: If X is an extensive state quantity and X̅ is the associated partial quantity, what types of variables must X̅ have?
I think we should explain the properties of partial molar quantities. Which characteristics do they have? We shoul count the different variables of these partial quantities.

 

[Chestermiller]

And I noticed that I forgot to post all the relevant equations for that task.
The complete task looks like this (see below).
The question d) is false because V and S are not constant, they are not intensive, V and S are extensive properties. Therefore, they can not be considered as constant. Is my consideration correct?

Consider this. The definition of G is $$G=U+PV-TS$$So we have $$dG=dU+PdV+VdP-TdS-SdT$$In addition, we have: $$dG=-SdT+VdP+\sum{\mu_i dn_i}$$What do you get if you eliminate dG from these equations and solve for dU?

 

[physicisttobe]

I think dU= TdS - PdV + sum of ... dni ?
But what does that equation have to do with the question above? Should we not explain what types of variables X̅ must have? Please apologize, I'm a lil bit confused.

 

[Chestermiller]

I think dU= TdS - PdV + sum of ... dni ?
But what does that equation have to do with the question above? Should we not explain what types of variables X̅ must have? Please apologize, I'm a lil bit confused.

Sorry. I have no idea what "what types of variables X̅ must have" means. Can you provide an example of what you mean.