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I wondered what that small triangle signifies in the second definition? Thanks!

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- Thread starter etotheipi
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- #1

I wondered what that small triangle signifies in the second definition? Thanks!

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mjc123

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Borek

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It's likely just outdated notation

I am outdated and I have never seen it as well, so more like an obscure notation

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berkeman

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Where is that screenshot from?I wondered what that small triangle signifies in the second definition?

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HAYAO

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I got my PhD in chemistry in 2018 so I believe my knowledge on recent textbooks is rather updated, but me too have never seen that kind of symbol.

That being said, considering that*x*_{A} is a notation used for Mole fraction, I believe μ_{A}^{▽} equals to μ_{A}^{*} (chemical potential of a pure A solvent), and the only difference is that the former notation is not what the IUPAC recommends. This is the chemical potential of the solution based on vapor pressure.

Chemical potential (Gibbs energy) for mixed gas is:

[itex]\mu=\mu^{\ominus}+RTln\frac{p}{p^{\ominus}}[/itex]

where [itex]p^{\ominus}[/itex] is the standard pressure (1 bar) and IUPAC recommends using this symbol. Now turning to solution case, the chemical potential of the vapor pressure of fully pure liquid A (which in equilibrium, this is identical to the chemical potential of the liquid) is:

[itex]\mu _{A}^{*}=\mu _{A}^{\ominus}+RTln \frac{p _{A}^{*}}{p^{\ominus}}[/itex]

If the liquid is not pure A, then:

[itex]\mu _{A}=\mu _{A}^{\ominus}+RTln \frac{p _{A}}{p^{\ominus}}[/itex]

So combining these two equation gives:

[itex]\mu _{A}=\mu _{A}^{*}+RTln\frac{p _{A}}{p _{A}^{*}}[/itex]

We can write this in terms of chemical activity [itex]a_{A}=\frac{p _{A}}{p _{A}^{*}}[/itex], which gives:

[itex]\mu _{A}=\mu _{A}^{*}+RTlna _{A}[/itex]

And using activity coefficient and mole fraction,

[itex]\mu _{A}=\mu _{A}^{*}+RTln\gamma _{A}x_{A}[/itex]

Ideal solution with sufficiently low solute B and pure solvent A (Roult's Law) means that [itex]\gamma _{A}=1[/itex] so,

[itex]\mu _{A}=\mu _{A}^{*}+RTlnx_{A}[/itex]

This is now identical to the second equation of the OP's question.

That being said, considering that

Chemical potential (Gibbs energy) for mixed gas is:

[itex]\mu=\mu^{\ominus}+RTln\frac{p}{p^{\ominus}}[/itex]

where [itex]p^{\ominus}[/itex] is the standard pressure (1 bar) and IUPAC recommends using this symbol. Now turning to solution case, the chemical potential of the vapor pressure of fully pure liquid A (which in equilibrium, this is identical to the chemical potential of the liquid) is:

[itex]\mu _{A}^{*}=\mu _{A}^{\ominus}+RTln \frac{p _{A}^{*}}{p^{\ominus}}[/itex]

If the liquid is not pure A, then:

[itex]\mu _{A}=\mu _{A}^{\ominus}+RTln \frac{p _{A}}{p^{\ominus}}[/itex]

So combining these two equation gives:

[itex]\mu _{A}=\mu _{A}^{*}+RTln\frac{p _{A}}{p _{A}^{*}}[/itex]

We can write this in terms of chemical activity [itex]a_{A}=\frac{p _{A}}{p _{A}^{*}}[/itex], which gives:

[itex]\mu _{A}=\mu _{A}^{*}+RTlna _{A}[/itex]

And using activity coefficient and mole fraction,

[itex]\mu _{A}=\mu _{A}^{*}+RTln\gamma _{A}x_{A}[/itex]

Ideal solution with sufficiently low solute B and pure solvent A (Roult's Law) means that [itex]\gamma _{A}=1[/itex] so,

[itex]\mu _{A}=\mu _{A}^{*}+RTlnx_{A}[/itex]

This is now identical to the second equation of the OP's question.

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