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## Main Question or Discussion Point

<mentor: change title>

In thermodynamics, there is a function which, for the three variables x, y, and z, can be given as

##G = xG_x+yG_y+zG_z + N[x\ln(x) + y\ln(y)+ z\ln(z)]+E(x,y,z)##

where G_x, G_y, G_z, and N are some constants and E is some arbitrarily complicated term.

There is a derivative of G called the chemical potential [itex]\mu[/itex] which is useful because

##G=x\mu_x+y\mu_y+z\mu_z## [Equation A]

I am trying to understand how this is satisfied, but I am having trouble even when E=0. To my mind, it should be the case that

##\mu_x = \frac{d}{dx}G = G_x+N(\ln(x)+1)+\frac{d}{dx}E## [Equation B]

but if this is true then Equation A doesn't make sense unless the value "1" in Equation B is removed. There must be something basic that I am not getting because I am also failing to satisfy Equation A for any function for E.. The problem is probably related to the fact that [itex]\mu_x[/itex] is actually

##\mu_x = \frac{d}{dn_x}G = ?????##

so that

##x = n_x/(n_x+n_y+n_z)##

but I don't get how to evaluate the derivatives even with this "change".

Thanks

In thermodynamics, there is a function which, for the three variables x, y, and z, can be given as

##G = xG_x+yG_y+zG_z + N[x\ln(x) + y\ln(y)+ z\ln(z)]+E(x,y,z)##

where G_x, G_y, G_z, and N are some constants and E is some arbitrarily complicated term.

There is a derivative of G called the chemical potential [itex]\mu[/itex] which is useful because

##G=x\mu_x+y\mu_y+z\mu_z## [Equation A]

I am trying to understand how this is satisfied, but I am having trouble even when E=0. To my mind, it should be the case that

##\mu_x = \frac{d}{dx}G = G_x+N(\ln(x)+1)+\frac{d}{dx}E## [Equation B]

but if this is true then Equation A doesn't make sense unless the value "1" in Equation B is removed. There must be something basic that I am not getting because I am also failing to satisfy Equation A for any function for E.. The problem is probably related to the fact that [itex]\mu_x[/itex] is actually

##\mu_x = \frac{d}{dn_x}G = ?????##

so that

##x = n_x/(n_x+n_y+n_z)##

but I don't get how to evaluate the derivatives even with this "change".

Thanks

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