1.7.10: Compressions- Desnoyers - Philip Equation
In terms of isentropic and isothermal compressibilities the Desnoyers-Philip Equation is important. A key equation expresses the difference between two apparent properties, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}\) and \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) [1]. We develop the proof in the general case starting from equation (a).
\[\delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}=\mathrm{T} \,\left(\alpha_{\mathrm{p}}\right)^{2} / \sigma \nonumber \]
\[\text { Hence, for an aqueous solution, } \delta(\mathrm{aq})=T \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq}) \nonumber \]
\[\text { For water }(\ell) \text { at the same } \mathrm{T} \text { and } \mathrm{p}, \delta_{1}^{*}(\ell)=\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell) \nonumber \]
We formulate an equation for the difference, \(\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\)
\[\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=\mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell) \nonumber \]
We add and subtract the same term. With some slight reorganisation,
\[\begin{aligned}
\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma(\mathrm{aq}) \\
&-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)+\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma(\mathrm{aq})
\end{aligned} \nonumber \]
Or,
\[\begin{aligned}
\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,[\sigma(\mathrm{aq})]^{-1} \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}-\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}\right\} \\
&-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} \,\left[\frac{1}{\sigma_{1}^{*}(\ell)}-\frac{1}{\sigma(\mathrm{aq})}\right]
\end{aligned} \nonumber \]
We identify the term \(\left\{\left[\alpha_{p}(a q)\right]^{2}-\left[\alpha_{p 1}^{*}(\ell)\right]^{2}\right\}\) as ‘a square minus a square’.
\[\begin{aligned}
\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,[\sigma(\mathrm{aq})]^{-1} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \\
&-\mathrm{T} \, \frac{\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\sigma(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]
\end{aligned} \nonumber \]
We use equations (b) for \(\delta(\mathrm{aq})\) and (c) for \(\delta_{1}^{*}(\ell)\) to remove explicit reference to temperature in equation (g).
\[\begin{aligned}
&\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)= \\
&\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{-2} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right\} \\
&-\delta_{1}^{*}(\ell) \,[\sigma(\mathrm{aq})]^{-1} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]
\end{aligned} \nonumber \]
\[\text { But } \phi\left(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(\mathrm{c}_{\mathrm{j}}\right)^{-1} \,\left[\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\right]+\delta_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
We insert equation (h) for the difference \(\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\) into equation (i).
\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \\
&\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}} \, \frac{\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\mathrm{c}_{\mathrm{j}}} \\
&-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \frac{\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]}{\mathrm{c}_{\mathrm{j}}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell)
\end{aligned} \nonumber \]
\[\text { But, } \phi\left(E_{p j}\right)=\left[c_{j}\right]^{-1} \,\left[\alpha_{p}(a q)-\alpha_{p l}^{*}(\ell)\right]+\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right) \nonumber \]
We identify the difference \(\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right]\).
\[\text { Then } \phi\left(E_{p j}\right)-\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right)=\left[c_{j}\right]^{-1} \,\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right] \nonumber \]
\[\text { Similarly } \phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right] \nonumber \]
Then from equations (k), (l), (m) and (n),
\[\begin{gathered}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\
-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell)
\end{gathered} \nonumber \]
We collect the \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) terms.
\[\begin{gathered}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)= \\
\frac{\delta(\mathrm{aq})}{\alpha_{\mathrm{p}}(\mathrm{aq})} \,\left\{1+\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\
+\left\{-\frac{\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}-\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}}+\frac{\delta_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}+\delta_{1}^{*}(\ell)\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{gathered} \nonumber \]
We note that in the second {----} bracket, the product term of \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). By using equations (b) for \(\delta(\mathrm{aq})\) and (c) for \(\delta_{1}^{*}(\ell)\) the second and third terms are together equal to zero.
Hence
\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \\
&\qquad \frac{\delta(\mathrm{aq})}{\alpha_{\mathrm{p}}(\mathrm{aq})} \,\left\{1+\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\
&+\left\{\delta_{1}^{*}(\ell)-\frac{\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
The latter is the full Desnoyers–Philip equation [1]. But
\[\begin{gathered}
\operatorname{limit}\left(c_{j} \rightarrow 0\right) \alpha_{p}(a q)=\alpha_{p l}^{*}(\ell) \\
\phi\left(E_{p j}\right)=\phi\left(E_{p j}\right)^{\infty}, \\
\phi\left(C_{p j}\right)=\phi\left(C_{p j}\right)^{\infty}, \delta(a q)=\delta_{1}^{*}(\ell) \\
\text { and } \sigma(a q)=\sigma_{1}^{*}(\ell)
\end{gathered} \nonumber \]
\[\text { Then } \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}=\delta_{1}^{*}(\ell) \,\left\{\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\right\} \nonumber \]
Footnotes
[1] J. E. Desnoyers and P. R. Philip, Can. J. Chem, 1972, 50 ,1094.
[2] M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc. Rev., 2001, 30 , 8.