This is one of those "note to self" posts where I try to get my head around a concept, this time fugacity and activity for a gas.

If the gas is ideal, i.e. for one mole $V=RT/p$, then

$$G(p)=G^\circ+RT\int_{p^\circ}^{p}\frac{dp}{p}=G^\circ+RT\ln\frac{p}{p^\circ}$$

For $A\rightleftharpoons B$

$$G(p_B)-G(p_A)=0 \implies \frac{p_B}{p_A}=e^{-\Delta G^\circ/RT}$$

What about a real gas where $V\neq RT/p$? We introduce the fugacity $(f)$ for which $V=RT/f$ so that

$$G(p)=G^\circ+RT\ln\frac{f}{p^\circ} \text{ and } \frac{f_B}{f_A}=e^{-\Delta G^\circ/RT}$$

To determine $f$:

$$\int_{p'}^{p} (V-V_{ideal})dp=RT\ln\left(\frac{f}{f'}\cdot \frac{p'}{p}\right) = RT\ln\left(\frac{f}{p}\cdot \frac{p'}{f'}\right) $$

Gases approach ideality at low pressure: $f'/p'\rightarrow 1$ as $p\rightarrow 0$ so:

$$\ln\left(\frac{f}{p}\right)=\ln(\phi)=\frac{1}{RT}\int_{0}^{p} (V-V_{ideal})dp$$

So for sticky non-ideal gases for which $V<V_{ideal}$ the fugacity coefficient $\phi$ is less than 1.

Finally the relationship between fugacity and activity ($a$) is

$$a=\frac{f}{p^\circ}$$.

This work is licensed under a Creative Commons Attribution 3.0 Unported License.

$dG=Vdp-SdT \implies dG=Vdp \text{ if } dT=0$

$$G(p)=G^\circ+\int_{p^\circ}^{p}Vdp$$If the gas is ideal, i.e. for one mole $V=RT/p$, then

$$G(p)=G^\circ+RT\int_{p^\circ}^{p}\frac{dp}{p}=G^\circ+RT\ln\frac{p}{p^\circ}$$

For $A\rightleftharpoons B$

$$G(p_B)-G(p_A)=0 \implies \frac{p_B}{p_A}=e^{-\Delta G^\circ/RT}$$

What about a real gas where $V\neq RT/p$? We introduce the fugacity $(f)$ for which $V=RT/f$ so that

$$G(p)=G^\circ+RT\ln\frac{f}{p^\circ} \text{ and } \frac{f_B}{f_A}=e^{-\Delta G^\circ/RT}$$

To determine $f$:

$$\int_{p'}^{p} (V-V_{ideal})dp=RT\ln\left(\frac{f}{f'}\cdot \frac{p'}{p}\right) = RT\ln\left(\frac{f}{p}\cdot \frac{p'}{f'}\right) $$

Gases approach ideality at low pressure: $f'/p'\rightarrow 1$ as $p\rightarrow 0$ so:

$$\ln\left(\frac{f}{p}\right)=\ln(\phi)=\frac{1}{RT}\int_{0}^{p} (V-V_{ideal})dp$$

So for sticky non-ideal gases for which $V<V_{ideal}$ the fugacity coefficient $\phi$ is less than 1.

**So even though**$V=RT/f$**don't confuse**$f$**with**$p_{ideal}: f<p<p_{ideal}$ for a given number of gas particles.Finally the relationship between fugacity and activity ($a$) is

$$a=\frac{f}{p^\circ}$$.

This work is licensed under a Creative Commons Attribution 3.0 Unported License.

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