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I know how to get the equation from the Clapeyron equation but I have a question regarding a the integration along a phase boundary and a small step in the derivation that I will make clear when I reach that step. Firstly, the Clapeyron equation:

$$\frac{\mathrm{d}p}{\mathrm{d}T}=\frac{\Delta S}{\Delta V}$$

Or alternatively, by recognizing that $\Delta G=0$ when two phases are in equilibrium; $\Delta G =\Delta H - T\Delta S$ can be rearranged to give:

$$\Delta S = \frac{\Delta H}{T}$$

Substituting this into to the the first equation gives:

$$\frac{\mathrm{d}p}{\mathrm{d}T}=\frac{\Delta H}{T\Delta V}$$

At a solid/gas or liquid/gas phase phase boundary, it is a reasonable to approximate that $\Delta V \approx V_{m,\text{gas}}$ as $\Delta V = V_{m,\text{gas}}-V_{m,\text{condensed}}$ since $V_{m,\text{gas}} \gg V_{m,\text{condensed}}$

Consequently, using the equation of state for a perfect gas and substituting for $\Delta V$ in the second form of the Clapeyron equation, the following result is obtained:

$$\frac1p \frac{\mathrm{d}p}{\mathrm{d}T}=\frac{\mathrm{d}\ln p}{\mathrm{d}T}=\frac{\Delta H}{RT^2}$$

However, why is $\frac1p \frac{\mathrm{d}p}{\mathrm{d}T}=\frac{\mathrm{d}\ln p}{\mathrm{d}T}$?

The following step is the step that I don’t understand In my lecture handout this step is shown as:

$$\int_{p_1}^{p_2}\frac1p\mathrm{d}p=\int_{T_1}^{T_2}\frac{\Delta H}{RT^2}\mathrm{d}T$$

Why can the left hand side be integrated with respect to $p$, yet the right hand side be integrated with respect to $T$. I cannot make sense of this. Other instances in thermodynamics when integration is used like this, both sides are integrated with respect to the same variable. For example, finding how $H$ varies with $p$. I will quickly go through what I mean without much verbal explanation:

$$\mathrm{d}H=T\mathrm{d}S+V\mathrm{d}p=\left(\frac{\partial H}{\partial S}\right)_p \mathrm{d}S+\left(\frac{\partial H}{\partial p}\right)_S \mathrm{d}p$$

Consequently, for one mole of a perfect gas:

$$V=\left(\frac{\partial H}{\partial p}\right)_S=\frac{RT}p$$

Thus:

$$\int_{p_1}^{p_2}\left(\frac{\partial H}{\partial p}\right)_S\mathrm{d}p=\int_{p_1}^{p_2}\frac{RT}p\mathrm{d}p$$

Since $\left(\frac{\partial H}{\partial p}\right)_S\mathrm{d}p=\mathrm{d}H$ at a constant $T$ (I think this is right – please clarify)

$$\int_{p_1}^{p_2}\mathrm{d}H=RT\int_{p_1}^{p_2}\frac{1}p\mathrm{d}p$$

This can obviously be integrated quite easily but this only illustrates my point regarding the variable that the integration is carried out with respect to. Surely it must be the same on both the RHS and the LHS of the equation?

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See Integration by substitution, see also this question and another question at math.SE. This is the rule we have to apply here:

$$\int_{x_0}^{x_1} f(u(x))u'(x)\,\mathrm dx = \int_{u(x_0)}^{u(x_1)} f(u)\,\mathrm du $$

So we go from this equation: $$\frac1p \frac{\mathrm dp}{\mathrm dT}=\frac{\mathrm d\ln p}{\mathrm dT}=\frac{\Delta H}{RT^2},$$ and integrate over temperature: $$\int_{T_1}^{T_2}\frac1p \frac{\mathrm dp}{\mathrm dT}\,\mathrm dT = \int_{T_1}^{T_2} \frac{\Delta H}{RT^2}\,\mathrm dT,$$ Now we use the first equation, with $f=\frac{1}{p}$, $p=u$, and $x=T$: $$\int_{T_1}^{T_2}\frac1p \frac{\mathrm dp}{\mathrm dT}\,\mathrm dT = \int_{p_1(T_1)}^{p_2(T_2)}\frac1p\,\mathrm dp=\int_{T_1}^{T_2}\frac{\Delta H}{RT^2}\,\mathrm dT$$

Bonus question: Its just the application of the chain rule, namely $$\frac {\mathrm d}{\mathrm dx}z = \frac {\mathrm dz}{\mathrm dy} \frac {\mathrm dy}{\mathrm dx} $$ which, if applied leads to $$\frac{\mathrm d}{\mathrm dT}\ln p = \frac{\mathrm d\ln(p)}{\mathrm dp} \frac{\mathrm dp}{\mathrm dT} = \frac{1}{p}\frac{\mathrm dp}{\mathrm dT} .$$

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    $\begingroup$ Thank you very much for your answer. I would, however, like to dispute the assertion that my question is off topic due to the fact it requires maths to answer it. Chemistry requires a high level of maths and it is, in fact, integral in topics such as thermodynamics. If I had posted it on the Maths section of this site, I would expect a response complaining that it's content is largely Chemistry based. I believe questions like this should be more welcomed. After all, the theory behind this question was taught in my first term at Oxford - studying Chemistry. However, Thanks again for the answer. $\endgroup$ – RobChem Jan 2 '15 at 17:23
  • $\begingroup$ Well, I simply wasn't sure about it being off-topic or not - Im new to chemistry.SE, I'm not that acustomed to the etiquette here. And yes, on math.SE it may be the other way around with this question. $\endgroup$ – John H. K. Jan 2 '15 at 17:31
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    $\begingroup$ I completely understand; my comment was more of a general comment that I hope will be seen by the Chemistry SE community rather than any personal attack at you - I remain very grateful for your answer. I just believe that sometimes some perfectly appropriate questions are not well received on this website, I was simply making an effort to change that. $\endgroup$ – RobChem Jan 2 '15 at 17:43
  • $\begingroup$ Apologies if you were offended. $\endgroup$ – RobChem Jan 2 '15 at 17:44
  • $\begingroup$ No offense taken! $\endgroup$ – John H. K. Jan 2 '15 at 19:40

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