need help if this equation follows math rules: XA・dμA+XB・dμB=0

[echa]

need help if this equation follows math rules: XA・dμA+XB・dμB=0

Trying to work on ideal solution equation and was wondering if this equation follows math rules.

X_A・dμ_A+X_B・dμ_B=0

dμ_B = -(X_A/X_B)RTlnX_A where dμ(A)=RT*lnX_A

dμ_B=-(X_A/X_B)RT(dX_A/X_A) where lnX_A=dX_A/X_A

dμ_B=-(X_A/X_A)RT(dX_A/X_B) what happened here is I've switched the denomenator X_A and X_B which I'm wondering is it along the math rules because I'm going to integrate dμ_B to obtain chemical potential μ_B

 

[Deleted member 4993]

Trying to work on ideal solution equation and was wondering if this equation follows math rules.

X_A・dμ_A+X_B・dμ_B=0

dμ_B = -(X_A/X_B)RTlnX_A where dμ(A)=RT*lnX_A

dμ_B=-(X_A/X_B)RT(dX_A/X_A) where lnX_A=dX_A/X_A

dμ_B=-(X_A/X_A)RT(dX_A/X_B) what happened here is I've switched the denomenator X_A and X_B which I'm wondering is it along the math rules because I'm going to integrate dμ_B to obtain chemical potential μ_B

I do not understand these substitutions. You are equating differential quantities with non-differential quantities.

 

[echa]

Sorry for your troubles. Turns out i tackled the question wrong.
Anyway the question was about the chemical potential,μ of a component i in an ideal solution
μ_i =μ_i^o + RTlnX_i where X_i is the mole fraction
is the free energy before mixing the solution
is the change in free energy after mixing
for 2 components i=A,B, if
μ_A=μ_A^o + RTlnX_A -------1
show that
μ_B=μ_B^o + RTlnX_B----------2all i had to was differentiate equation 1 into dμ_A and substitute it into
X_A・dμ_A+X_B・dμ_B=0
and integrate it after simplifying it.

 

[Ishuda]

Trying to work on ideal solution equation and was wondering if this equation follows math rules.

X_A・dμ_A+X_B・dμ_B=0

dμ_B = -(X_A/X_B)RTlnX_A where dμ(A)=RT*lnX_A

dμ_B=-(X_A/X_B)RT(dX_A/X_A) where lnX_A=dX_A/X_A

dμ_B=-(X_A/X_A)RT(dX_A/X_B) what happened here is I've switched the denomenator X_A and X_B which I'm wondering is it along the math rules because I'm going to integrate dμ_B to obtain chemical potential μ_B

I would assume that \(\displaystyle \mu_A\) and \(\displaystyle \mu_B\) are the independent variables in this formulation but what are the dependent variables and what do they depend on? For example, does x_A depend only on \(\displaystyle \mu_A\)?

And you need a differential in some of those equations, i.e.
d(lnX_A)=d(X_A)/X_A for example.

Oh, and what does RT stand for?