1.22.5: Volumes: Solutions: Apparent and Partial Molar Volumes: Determination
An aqueous solution is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute. Thus,
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
The density of this solution \(\rho(\mathrm{aq})\) can be accurately measured at the specified temperature and pressure together with the density of the pure solvent, \(\rho_{1}^{*}(\ell)\). The molar mass of the solute is \(\mathrm{M}_{j} \mathrm{~kg mol}^{-1}\). Two equations [1-3] are encountered in the literature depending on the method used to describe the composition of the solution [4]. Molality Scale [1]
\[\phi\left(V_{j}\right)=\left[m_{j} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq}) \nonumber \]
Concentration Scale [2,3]
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell) \nonumber \]
Equation (b) using molalities and (c) using concentrations yield the same property of the solute, namely the apparent molar volume of solute \(j\), \(\phi\left(\mathrm{V}_{j}\right)\). Equations (b) and (c) are exact. The equations are readily distinguished by the difference in the denominators of the last terms. In any event the trick in deriving these equations is to seek an equation having the form, {[Property of solvent] minus [Property of solute]}.
The subject is slightly complicated because the concentration of solute j can be expressed using either the unit ‘\(\mathrm{mol m}^{-3}\)’ or the unit ‘\(\mathrm{mol dm}^{-3}\)’, the latter being the most common. There is also a problem over the unit used for densities. Some authors use the unit ‘\(\mathrm{kg m}^{-3}\)‘ whereas other authors use the unit ‘\(\mathrm{g cm}^{-3}\)‘. The latter practice accounts for the numerical factor \(10^{3}\) which often appears in many published equations of the form shown in equations (b) and (c).
Partial molar and partial molal properties are often identified. The two terms are synonymous in the case of partial molar volumes and partial molal volumes of solutes in aqueous solutions. IUPAC recommends the use of the term ‘partial molar volume’ [5].
Significantly we can never know the absolute value of the chemical potential of a solute in a given solution but we can determine the partial molar volume, the differential dependence of chemical potential on pressure. Indeed the challenge of understanding patterns in partial molar volumes seems less awesome than the task of understanding other thermodynamic properties of solutes.
Equations (b) and (c) do not describe how \(\phi\left(\mathrm{V}_{j}\right)\) for a given solute depends on either \(\mathrm{m}_{j}\) or \(\mathrm{c}_{j}\). This dependence is characteristic of a solute (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) and reflects the role of solute - solute interactions. In many cases where solute \(j\) is a simple neutral solute, \(\phi\left(\mathrm{V}_{j}\right)\) for dilute solutions is often satisfactorily accounted for by an equation in which \(\phi\left(\mathrm{V}_{j}\right)\) is a linear function of \(\mathrm{m}_{j}\). The slope \(\mathrm{S}\) is characteristic of the solute (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) [4d,6].
\[\phi\left(V_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}+\mathrm{S} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \]
In the case of urea(aq) at \(298.2 \mathrm{~K}\) and ambient pressure the dependence of \(\phi\left(\mathrm{V}_{j}\right)\) on \(\mathrm{m}_{j}\) is described by the following quadratic equation [7,8].
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right) / \mathrm{cm}^{3} \mathrm{~mol}^{-1}=44.20+0.126 \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)-0.004 \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \nonumber \]
In general terms therefore \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) {and \(\mathrm{V}_{\mathrm{j}}^{\infty}\) for solute \(j\) in other solvents [9]} characterizes solute - solvent interactions and the dependences of \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) on \(\mathrm{m}_{j}\) characterizes solute - solute interactions. Of course the partial molar volume of solute-\(j\) in solution is not the actual volume of solute-\(j\). Instead \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) measures the differential change in the volume of an aqueous solution when \(\delta \mathrm{n}_{j}\) moles of substance-\(j\) are added. We emphasize the importance of an approach using the molalities of solutes. The reasons are straightforward. If we compare \(\phi\left(\mathrm{V}_{j}\right)\) for a solute in solutions containing \(0.1\) and \(0.01 \mathrm{~mol kg}^{-1}\), in this comparison, the mass of solvent remains the same. If on the other hand we compare \(\phi\left(\mathrm{V}_{j}\right)\) for solute in solutions where \(\mathrm{c}_{j} / \mathrm{~mol dm}^{-3} = 0.1\) and \(0.01\), the amounts of solvent are not defined. Nevertheless many treatments of the properties of solutions examine \(\phi\left(\mathrm{V}_{j}\right)\) as a function of concentration. In fact chemists tend to think in terms of concentrations and hence in terms of distances between solute molecules. So in these terms concentration might be thought of as the 'natural' scale. Just as in life, one is more interested in the distance between two people rather than their mass. No rule forbids one to fit the dependences of \(\phi\left(\mathrm{V}_{j}\right)\) on \(\mathrm{c}_{j}\) using an equation of the following form.
\[\phi\left(V_{j}\right)=a_{1}+a_{2} \, c_{j}+a_{3} \, c_{j}^{2}+\ldots \nonumber \]
But if \(\phi\left(\mathrm{V}_{j}\right)\) is a linear function of \(\mathrm{m}_{j}\), \(\phi\left(\mathrm{V}_{j}\right)\) is not a linear function of \(\mathrm{c}_{j}\) [10]. Of course \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\) in both \(\operatorname{limit}\left(\mathrm{m}_{j} \rightarrow 0 \right)\) and \(\operatorname{limit} \left(\mathrm{c}_{j} \rightarrow 0 \right)\).
Granted the outcome of an experiment is the dependence of \(\phi\left(\mathrm{V}_{j}\right)\) on \(\mathrm{m}_{j}\), the partial molar volume \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) of solute \(j\) is readily calculated; equation (h) [11].
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\phi\left(\mathrm{V}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right) \nonumber \]
We note important features in the context of two plots;
-
\(\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1} / \mathrm{kg}=1\right)\) and
- \(\left[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)\right]\) against molality \(\mathrm{m}_{j}\).
Then \(\mathrm{V}_{j}(\mathrm{aq})\) is the gradient of the tangent to the curve in plot (i) at the specified molality; \(\phi\left(\mathrm{V}_{j}\right)\) is the gradient of the line in plot(ii) joining the origin and \(\left[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)\right]\) at the specified molality.
Footnotes
[1] For the solution volume \(\mathrm{V}(\mathrm{aq})\),
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
If the molar mass of the solvent is \(\mathrm{M}_{1} \mathrm{~kg mol}^{-1}\), \(\mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right] / \rho(\mathrm{aq})\) and \(\mathrm{V}_{1}^{*}(\ell)=\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\). From equation (a),
\[\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho(\mathrm{aq})}+\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}=\mathrm{n}_{1} \, \frac{\mathrm{M}_{1}}{\rho_{1}^{*}(\ell)}+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Divide by \(\mathrm{n}_{1} \, \mathrm{~M}_{1}\) and rearrange;
\[\frac{n_{j}}{n_{1} \, M_{1}} \, \phi\left(V_{j}\right)=\frac{1}{\rho(a q)}-\frac{1}{\rho_{1}^{*}(\ell)}+\frac{n_{j} \, M_{j}}{n_{1} \, M_{1} \, \rho(a q)} \nonumber \]
But molality \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\). Then,
\[\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\frac{1}{\rho(\mathrm{aq})}-\frac{1}{\rho_{1}^{*}(\ell)}\right]+\frac{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})} \nonumber \]
or,
\[\phi\left(V_{j}\right)=\left[m_{j} \, \rho(a q) \, \rho_{1}^{*}\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(a q)\right]+\frac{M_{j}}{\rho(a q)} \nonumber \]
Thus,
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\frac{1}{\left[\mathrm{~mol} \mathrm{~kg}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]^{2}}\right] \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]+\frac{\left[\mathrm{kg} \mathrm{mol}^{-1}\right]}{\left[\mathrm{kg} \mathrm{m}^{-3}\right.}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \nonumber \]
[2] At fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{\mathrm{1}} \, \mathrm{V}_{1}^{*}(\mathrm{l})+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \nonumber \]
But, \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{1}\). Then,
\[\mathrm{c}_{\mathrm{j}}=\frac{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)} \nonumber \]
Invert. \(\frac{1}{\mathrm{c}_{\mathrm{j}}}=\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}+\frac{\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}\) But \(\mathrm{n}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}=1\) Then,
\[\frac{1}{\mathrm{c}_{\mathrm{j}}}=\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
or,
\[\frac{1}{\mathrm{~m}_{\mathrm{j}}}=\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
The latter equation links \(\mathrm{m}_{j}\) and \(\mathrm{c}_{j}\).
[3] From [1],
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{1}{\mathrm{~m}_{\mathrm{j}}} \,\left[\frac{1}{\rho(\mathrm{aq})}-\frac{1}{\rho_{1}^{*}(\ell)}\right]+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})} \nonumber \]
Then, from [2],
\[\phi\left(V_{j}\right)=\left[\frac{\rho_{1}^{*}(\ell)}{c_{j}}-\rho_{1}^{*}(\ell) \, \phi\left(V_{j}\right)\right] \,\left[\frac{1}{\rho(\mathrm{aq})}-\frac{1}{\rho_{1}^{*}(\ell)}\right]+\frac{M_{j}}{\rho(\mathrm{aq})} \nonumber \]
Hence,
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}} \, \rho(\mathrm{aq})}-\frac{1}{\mathrm{c}_{\mathrm{j}}}-\frac{\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\rho(\mathrm{aq})}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)+\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq}) \nonumber \]
Then, \(\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \frac{\rho_{1}^{*}(\ell)}{\rho(\mathrm{aq})}=\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}} \, \rho(\mathrm{aq})}-\frac{1}{\mathrm{c}_{\mathrm{j}}}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\)
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{1}{\mathrm{c}_{\mathrm{j}}}-\frac{1}{\mathrm{c}_{\mathrm{j}}} \, \frac{\rho(\mathrm{aq})}{\rho_{1}^{*}(\ell)}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)} \nonumber \]
or,
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell) \nonumber \]
Thus,
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{1}{\left[\mathrm{~mol} \mathrm{~m}^{-3}\right]} \, \frac{1}{\left[\mathrm{~kg} \mathrm{~m}^{-3}\right]^{-1}} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]+\frac{\left[\mathrm{kg} \mathrm{mol}^{-1}\right]}{\left[\mathrm{kg} \mathrm{m}^{-3}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \nonumber \]
[4] The following publications use equation (b) based on the molality composition scale.
- Dipeptides(aq); J. E. Reading, I. D. Watson and G .R. Hedwig, J. Chem. Thermodyn., 1990, 22 , 159.
- N-alkoxyethanols(aq); G. Roux, G. Peron and J. E. Desnoyers, J. Solution Chem.,1978, 7 ,639.
- Cyclic organic chemical substhances in 1-ocytanol; P. Berti, S. Cabani and V. Mollica, Fluid Phase Equilib.,1987, 32 ,195.
- Alkali halides in urea + water; N. Desrosiers, G. Perron, J. G. Mathieson, B. E. Conway and J. E. Desnoyers, J. Solution Chem.,1974,3,789.
- HCl(aq), HBr(aq) and HI(aq); T. M. Herrington, A.D. Pethybridge and M. G. Roffey, J. Chem. Eng. Data,1985, 30 ,264.
- LiOH(aq), NaOH(aq), KOH(aq); T. M. Herrington, A.D. Pethybridge and M. G. Roffey, J. Chem. Eng. Data,1986, 31 ,31.
- \(\mathrm{R}_{4}\mathrm{NI}(\mathrm{aq})\);B. M. Lowe and H. M. Rendall, Trans. Faraday Soc.,1971, 67 ,2318.
- HCl(aq) and \(\mathrm{HClO}_{4}(\mathrm{aq})\); R. Pogue and G. Atkinson, J. Chem. Eng. Data, 1988, 33 ,495.
- \(\mathrm{MCl}_{2}(\mathrm{aq})\) where M= Mn, Co, Ni, Zn and Cd; T. M. Herrington, M. G. Roffey, and D. P. Smith, J. Chem.Eng. Data,1986, 31 ,221.
- \(\mathrm{NiCl}_{2}(\mathrm{aq}), \mathrm{~Ni}\left(\mathrm{ClO}_{4}\right)_{2}(\mathrm{aq}), \mathrm{~CuCl}_{2}(\mathrm{aq}) \text { and } \mathrm{Cu}\left(\mathrm{ClO}_{4}\right)_{2}(\mathrm{aq})\); R. Pogue and G. Atkinson, J. Chem. Eng. Data, 1988, 33 ,370.
- \(\mathrm{RMe}_{3}\mathrm{NBr}(\mathrm{aq})\); R. De Lisi, S. Milioto and R. Triolo, J. Solution Chem.,1988, 17 ,673.
- \(\mathrm{Ph}_{4}\mathrm{AsCl}(\mathrm{aq})\); F. J. Millero, J. Chem. Eng. Data, 1971, 16 ,229.
-
\(\mathrm{R}_{4}\mathrm{NBr}(\mathrm{aq} + \mathrm{BuOH})\); L. Avedikian, G. Perron and J. E. Desnoyers, J. Solution Chem.,1975,
4
,331.
Applications of equation. (c) include- \(\mathrm{Bu}_{4}\mathrm{N}^{+} \text { carboxylates}(\mathrm{aq})\); P.-A.Leduc, and J. E. Desnoyers, Can. J. Chem.,1973, 51 ,2993.
- N-Alkylamine hydrobromides(aq); P.-A.Leduc, and J. E. Desnoyers, J. Phys. Chem., 1974, 78 , 1217.
- \(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~Cl}^{-} (\mathrm{aq} + \mathrm{~DMSO})\); D. D. Macdonald and J.B.Hyne, Can. J.Chem.,1970, 48 ,2416.
- \(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~Cl}^{-} (\mathrm{aq} + \mathrm{~EtOH})\); I. Lee and J. B. Hyne, Can. J.Chem.,1968, 46 ,2333.
[5] Manual of Symbols and Terminology for Physicochemical Quantities and Units, IUPAC, Pergamon, Oxford, 1979.
[6] F. Franks and H.T. Smith, Trans. Faraday Soc., 1968, 64 , 2962.
[7] D. Hamilton and R. H. Stokes, J. Solution Chem., 1972, 1 , 213.
[8] R. H. Stokes, Aust . J. Chem., 1967, 20 , 2087.
[9]
| Solvent | \(\mathrm{V}^{infty}\) (urea; sln; \(298 \mathrm{~K}\); ambient p)/cr \(\mathrm{mol}^{-1}\) |
| \(\mathrm{H}_{2}\mathrm{O}\) | 44.24 |
| \(\mathrm{CH}_{3}\mathrm{OH}\) | 36.97 |
| \(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH}\) | 40.75 |
| Formamide | 44.34 |
| \(\mathrm{DMF}\) | 39.97 |
| \(\mathrm{DMSO}\) | 41.86 |
[10] From [2], \(\mathrm{m}_{\mathrm{j}}=\mathrm{c}_{\mathrm{j}} /\left[\rho_{1}^{*}(\ell)-\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{c}_{\mathrm{j}}\right]\). If \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{a}_{1}+\mathrm{a}_{2} \, \mathrm{m}_{\mathrm{j}}\) Then,
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{a}_{1}+\left\{\mathrm{a}_{2} / \rho_{1}^{*}(\ell) \,\left[1-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{c}_{\mathrm{j}}\right]\right\} \, \mathrm{c}_{\mathrm{j}} \nonumber \]
i.e. the slope depends on the product \(\phi\left(\mathrm{V}_{j}\right) \, \mathrm{c}_{j}\).
[11] From
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
At constant \(\mathrm{n}_{1}\),
\[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)+\mathrm{n}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{n}_{\mathrm{j}}}\right) \nonumber \]
Or,
\[\mathrm{V}_{\mathrm{j}}=\phi\left(\mathrm{V}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right) \nonumber \]