Computation of the True Surface of a Pendant Mercury Drop from the Laplace Equation
Abstract Generated abstract
This paper addresses errors introduced in electrochemical calculations when a pendant mercury drop electrode is approximated as a sphere or spherical segment despite gravitational deformation. Using previously computed numerical solutions of the Laplace equation, the authors compile interpolation tables for the true surface area, volume, equatorial diameter, and height of pendant mercury drops on contacts of 0.02 and 0.05 cm diameter over experimentally relevant volume ranges. They compare these values with common empirical formulas and show that the formulas can deviate from the true Laplace-based values, although the spherical-segment approximation gives the closest surface estimates in the tabulated cases. The work provides practical tables and interpolation procedures for improving surface-area estimates in physicochemical and electrochemical studies using hanging mercury drops.
Full Text
CALCULATION OF THE TRUE SURFACE OF A HANGING MERCURY DROP USING THE LAPLACE EQUATION
V. I. Melik-Gaikazyan, A. G. Stromberg, V. F. Yankauskas
(Presented by the Scientific-Methodological Seminar of the KhTF)
An electrode in the form of a hanging mercury drop is widely used in electrochemistry. The importance of the hanging mercury (amalgam) electrode has increased even further in recent years in connection with the development of the method of amalgam polarography with accumulation \[1, 2\]. In calculations connected with the determination of physicochemical quantities, the surface of the hanging mercury electrode is usually taken to be a sphere or a spherical segment. Meanwhile, it is obvious that under the action of gravity the shape of hanging mercury drops will differ from that of a sphere, and therefore the magnitude of their surface will differ from the surface area of a sphere or spherical segment. Inaccuracy in determining the true surface of a hanging mercury drop leads to definite errors in the calculation of physicochemical quantities obtained in experiments using a hanging mercury electrode.
One of us \[3\], using the “Minsk-1” electronic digital computer, carried out a numerical solution of the Laplace equation \[4, 5, 6\] and compiled detailed tables analogous to the tables of Bashforth and Adams \[5\], but, unlike the latter, suitable for calculating the shape of small sessile bubbles or hanging drops. In addition, the tables obtained \[3\] differ from those published earlier \[5, 7\] by the presence in them of information that makes it possible to calculate the area of the curvilinear surface of drops and bubbles from their shape. The solution of the Laplace equation in work \[3\] was undertaken in connection with the consideration of certain questions of flotation theory. At the same time, the resulting tables contained no direct data on the surface of sessile bubbles (or hanging drops).
The purpose of the present work is to use the detailed numerical material obtained with the aid of the electronic digital computer in work \[3\] to compile tables that make it possible to estimate the true surface \(S\) and volume \(V\) of hanging mercury drops for known equatorial diameter \(d\) and drop height \(H\) for two bases of different diameter \(a\) (\(0.02\) and \(0.05\) cm), or, conversely, to estimate the true surface \(S\), equatorial diameter \(d\), and height \(H\) of hanging mercury drops from their volume \(V\) for two different bases\(^{1}\).
\(^{1}\) In practice, the base for hanging mercury drops is usually the end of a platinum or silver wire (contact), ground flush with the end of a glass or polyethylene tube in which this contact is fixed.
Let us briefly explain the course of the calculations performed. The digital material on the machine tapes contains numerical values of the dimensionless quantities \(S/b^2\), \(V/b^3\), \(z/b\), \((x/b)_{90^\circ}\) for various values of the dimensionless parameter \(\beta\) and of the dimensionless contact diameter \(a/b\).
The quantity \(b\), chosen in Tables [3, 5, 7] as the unit of scale, is calculated from the formula:
\[ b=\left(\frac{\sigma\beta}{\Delta g}\right)^{1/2}, \tag{1} \]
where \(\sigma\) is the surface (interfacial) tension at the mercury—solution interface, dyn/cm; \(\Delta\) is the difference between the densities of mercury and the aqueous solution, g/cm\(^3\); \(g\) is the acceleration due to gravity. In the calculations it was assumed that \(\sigma = 400\) dyn/cm and \(\Delta = 12.50\) g/cm\(^3\).
For the selected value of the parameter \(\beta\) and the dimensionless base \(a/b\), the values of the dimensionless quantities \(S/b^2\), \(V/b^3\), \(z/b\), and \((x/b)_{90^\circ}\) were read from the machine tape, and, knowing the value of the scale \(b\) for the given \(\beta\), the numerical values of \(a\), \(S\), \(V\), \(H\), and \(d\) were calculated from formula (1).
Then these numerical values, according to the rules of approximate calculations [6], were interpolated to the selected values of the bases (0.02 and 0.05 cm) and to rounded values of the volumes of mercury drops. The calculations were performed over the range of \(\beta\) values from \(-0.0125\) to \(-0.1500\), corresponding to the sizes of hanging mercury drops and contacts used in electrochemical studies.
The results of these calculations (at intervals of 0.1 mm\(^3\)) in the volume range 0.1–1.7 mm\(^3\) for a contact of diameter 0.02 cm and in the volume range 0.1–1.9 mm\(^3\)\(^1\) for a contact of diameter 0.05 cm are presented in Tables 1 and 2 (columns 1–4). Further condensation of the tables can be carried out with sufficient accuracy by linear interpolation.
Analogous calculations were also performed by us for surface tensions of 360 and 440 dyn/cm. The results of the calculations for all three values of the surface tension coincided with one another to within one unit in the fourth decimal place, and therefore the tables are given for only one value of the surface tension.
It should be noted that all computations were carried out twice, using in the intermediate operations two significant figures more (six significant figures) than in the final numbers (four significant figures) contained in the tables.
Usually, in electrochemical studies and in studies by the APTN method, the volume of a hanging mercury drop is known with sufficient accuracy, since the hanging mercury drop on the contact is obtained electrolytically from a solution of a mercury salt (the amount of electrodeposited mercury being calculated by Faraday’s law), or several drops that have flowed out of a polarographic capillary (whose weight is known) are suspended on the contact. In this case, by interpolation from Tables 1 and 2, for the known contact diameter and drop volume, one can directly find the true surface of the hanging mercury drop.
Until now, the most commonly used formulas for calculating the volume and surface of hanging mercury drops have been the following empirical formulas:
\[ V_1=\frac{\pi}{6}\left(\frac{d+H}{2}\right)^3, \tag{2} \]
\[ V_2=\frac{\pi}{6}d^2H, \tag{3} \]
\(^1\) The theoretically maximum possible volume of a hanging mercury drop on a contact of diameter 0.02 cm is 1.7 mm\(^3\), and on a contact of diameter 0.05 cm it is 3.4 mm\(^3\).
Table 1
Volume, surface, equatorial diameter, and height of a hanging mercury drop. Contact diameter \(a = 0.05\) cm
| 1 а) \(V \times 10^{+3}\), cm³ |
2 б) \(S \times 10^{+2}\), cm² |
3 в) \(d \times 10^{+1}\), cm |
4 г) \(H \times 10^{+1}\), cm |
5 д) \(V_1 \times 10^{+3}\), cm³ |
6 е) \(V_2 \times 10^{+3}\), cm³ |
7 ж) \(S_1 \times 10^{+2}\), cm² |
8 з) \(S_2 \times 10^{+2}\), cm² |
9 и) \(S_3 \times 10^{+2}\), cm² |
10 к) \(d_k \times 10^{+1}\), cm |
11 л) \(S_0 \times 10^{+2}\), cm² |
|---|---|---|---|---|---|---|---|---|---|---|
| 0,1 | 0,883 | 0,600 | 0,469 | 0,080 | 0,088 | 0,898 | 0,884 | 1,042 | 0,580 | 0,882 |
| 0,2 | 1,482 | 0,733 | 0,645 | 0,171 | 0,182 | 1,491 | 1,485 | 1,654 | 0,726 | 1,481 |
| 0,3 | 1,990 | 0,832 | 0,766 | 0,267 | 0,278 | 2,006 | 2,002 | 2,167 | 0,831 | 1,989 |
| 0,4 | 2,445 | 0,911 | 0,862 | 0,365 | 0,375 | 2,469 | 2,472 | 2,625 | 0,914 | 2,444 |
| 0,5 | 2,864 | 0,978 | 0,944 | 0,465 | 0,473 | 2,901 | 2,901 | 3,047 | 0,985 | 2,863 |
| 0,6 | 3,257 | 1,037 | 1,016 | 0,566 | 0,571 | 3,310 | 3,310 | 3,440 | 1,046 | 3,256 |
| 0,7 | 3,629 | 1,089 | 1,081 | 0,668 | 0,671 | 3,698 | 3,700 | 3,813 | 1,102 | 3,627 |
| 0,8 | 3,984 | 1,136 | 1,140 | 0,772 | 0,771 | 4,069 | 4,069 | 4,168 | 1,152 | 3,981 |
| 0,9 | 4,325 | 1,180 | 1,196 | 0,877 | 0,871 | 4,433 | 4,434 | 4,508 | 1,198 | 4,320 |
| 1,0 | 4,654 | 1,220 | 1,248 | 0,984 | 0,972 | 4,784 | 4,783 | 4,836 | 1,241 | 4,648 |
| 1,1 | 4,969 | 1,257 | 1,298 | 1,091 | 1,074 | 5,127 | 5,126 | 5,153 | 1,281 | 4,965 |
| 1,2 | 5,281 | 1,292 | 1,345 | 1,200 | 1,176 | 5,461 | 5,459 | 5,461 | 1,318 | 5,272 |
| 1,3 | 5,582 | 1,325 | 1,391 | 1,310 | 1,278 | 5,794 | 5,790 | 5,760 | 1,354 | 5,571 |
| 1,4 | 5,876 | 1,356 | 1,435 | 1,422 | 1,381 | 6,118 | 6,133 | 6,052 | 1,388 | 5,862 |
| 1,5 | 6,163 | 1,386 | 1,477 | 1,535 | 1,485 | 6,438 | 6,431 | 6,337 | 1,420 | 6,147 |
| 1,6 | 6,445 | 1,414 | 1,518 | 1,649 | 1,589 | 6,752 | 6,743 | 6,616 | 1,451 | 6,425 |
| 1,7 | 6,721 | 1,441 | 1,559 | 1,765 | 1,693 | 7,069 | 7,044 | 6,888 | 1,481 | |
| 1,8 | 6,992 | 1,466 | 1,598 | 1,883 | 1,799 | 7,374 | 7,360 | 7,154 | 1,502 | |
| 1,9 | 7,258 | 1,491 | 1,636 | 2,001 | 1,904 | 7,680 | 7,663 | 7,419 | 1,537 | |
| 2,0 | 7,520 | 1,515 | 1,674 | 2,123 | 2,012 | 7,987 | 7,968 | 7,677 | 1,563 |
Continuation of Table 1
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|
| 2,1 | 7,779 | 1,538 | 1,711 | 2,245 | 2,119 | 8,291 | 8,257 | 7,931 | 1,589 | |
| 2,2 | 8,034 | 1,560 | 1,748 | 2,369 | 2,227 | 8,595 | 8,567 | 8,180 | 1,614 | |
| 2,3 | 8,285 | 1,581 | 1,784 | 2,494 | 2,336 | 8,894 | 8,861 | 8,427 | 1,638 | |
| 2,4 | 8,534 | 1,601 | 1,818 | 2,616 | 2,440 | 9,181 | 9,144 | 8,669 | 1,661 | |
| 2,5 | 8,779 | 1,621 | 1,858 | 2,756 | 2,556 | 9,506 | 9,462 | 8,893 | 1,684 | |
| 2,6 | 9,022 | 1,641 | 1,893 | 2,889 | 2,669 | 9,809 | 9,759 | 9,144 | 1,706 | |
| 2,7 | 9,262 | 1,659 | 1,928 | 3,021 | 2,779 | 10,106 | 10,049 | 9,377 | 1,728 | |
| 2,8 | 9,501 | 1,678 | 1,959 | 3,151 | 2,888 | 10,389 | 10,327 | 9,607 | 1,749 | |
| 2,9 | 9,738 | 1,695 | 1,994 | 3,286 | 3,000 | 10,688 | 10,618 | 9,835 | 1,769 | |
| 3,0 | 9,972 | 1,713 | 2,029 | 3,430 | 3,118 | 10,992 | 10,915 | 10,029 | 1,789 | |
| 3,1 | 10,205 | 1,729 | 2,064 | 3,572 | 3,231 | 11,299 | 11,213 | 10,282 | 1,809 | |
| 3,2 | 10,436 | 1,745 | 2,099 | 3,718 | 3,347 | 11,605 | 11,507 | 10,502 | 1,828 | |
| 3,3 | 10,666 | 1,761 | 2,135 | 3,871 | 3,467 | 11,922 | 11,812 | 10,719 | 1,847 | |
| 3,4 | 10,894 | 1,777 | 2,170 | 4,025 | 3,588 | 12,236 | 12,142 | 10,935 | 1,866 |
Notes: a) \(V\)—true volume of the drop; b) \(S\)—true surface area of the drop; c) \(d\)—equatorial diameter of the drop; d) \(H\)—height of the drop, according to formula (2); e) according to formula (3); f) according to formula (4); g) according to formula (5); h) according to formula (6); k) according to formula (7); l) \(S_0\)—surface area of the spherical segment according to formula (10).
Table 2
Volume, surface area, equatorial diameter, and height of a hanging mercury drop.
Contact diameter 0.02 cm
| a) \(V \times 10^{+3}\), cm\(^3\) | b) \(S \times 10^{+2}\), cm\(^2\) | c) \(d \times 10^{+1}\), cm | d) \(H \times 10^{+1}\), cm | e) \(V_1 \times 10^{+3}\), cm\(^3\) | f) \(V_2 \times 10^{+3}\), cm\(^3\) | g) \(S_1 \times 10^{+2}\), cm\(^2\) | h) \(S_2 \times 10^{+2}\), cm\(^2\) | i) \(S_3 \times 10^{+2}\), cm\(^2\) | k) \(d_k \times 10^{+1}\), cm | l) \(S_0 \times 10^{+2}\), cm\(^2\) |
|---|---|---|---|---|---|---|---|---|---|---|
| 0,1 | 1,011 | 0,574 | 0,565 | 0,097 | 0,097 | 1,019 | 1,019 | 1,042 | 0,580 | 1,011 |
| 0,2 | 1,623 | 0,721 | 0,726 | 0,198 | 0,198 | 1,644 | 1,644 | 1,654 | 0,726 | 1,623 |
| 0,3 | 2,137 | 0,823 | 0,843 | 0,299 | 0,303 | 2,180 | 2,180 | 2,167 | 0,831 | 2,136 |
| 0,4 | 2,596 | 0,905 | 0,938 | 0,402 | 0,409 | 2,668 | 2,667 | 2,625 | 0,914 | 2,594 |
| 0,5 | 3,018 | 0,972 | 1,021 | 0,505 | 0,518 | 3,120 | 3,118 | 3,047 | 0,985 | 3,015 |
| 0,6 | 3,413 | 1,032 | 1,095 | 0,610 | 0,630 | 3,553 | 3,550 | 3,440 | 1,046 | 3,410 |
| 0,7 | 3,787 | 1,084 | 1,163 | 0,716 | 0,743 | 3,965 | 3,961 | 3,813 | 1,102 | 3,782 |
| 0,8 | 4,145 | 1,132 | 1,226 | 0,823 | 0,858 | 4,367 | 4,360 | 4,168 | 1,152 | 4,136 |
| 0,9 | 4,488 | 1,175 | 1,286 | 0,931 | 0,977 | 4,757 | 4,747 | 4,508 | 1,198 | 4,477 |
| 1,0 | 4,819 | 1,216 | 1,344 | 1,040 | 1,097 | 5,147 | 5,134 | 4,836 | 1,241 | 4,805 |
| 1,1 | 5,140 | 1,253 | 1,399 | 1,150 | 1,220 | 5,524 | 5,507 | 5,153 | 1,281 | 5,122 |
| 1,2 | 5,452 | 1,288 | 1,452 | 1,262 | 1,347 | 5,896 | 5,875 | 5,461 | 1,318 | 5,430 |
| 1,3 | 5,757 | 1,321 | 1,506 | 1,375 | 1,478 | 6,277 | 6,250 | 5,760 | 1,354 | 5,729 |
| 1,4 | 6,055 | 1,352 | 1,558 | 1,491 | 1,613 | 6,651 | 6,617 | 6,052 | 1,388 | 6,021 |
| 1,5 | 6,348 | 1,382 | 1,611 | 1,610 | 1,754 | 7,036 | 6,994 | 6,337 | 1,420 | 6,306 |
| 1,6 | 6,635 | 1,410 | 1,664 | 1,787 | 1,900 | 7,422 | 7,371 | 6,616 | 1,451 | 6,584 |
| 1,7 | 6,918 | 1,437 | 1,717 | 1,856 | 2,053 | 7,813 | 7,751 | 6,888 | 1,481 | 6,857 |
Note: The meaning of the symbols in the column headings and the formulas by which they were calculated are explained in the note to Table 1.
\[ S_1=\pi\left(\frac{d+H}{2}\right)^2, \tag{4} \]
\[ S_2=\pi dH, \tag{5} \]
\[ S_3=\pi d_*^2=6^{2/3}\pi^{2/3}V^{2/3}, \tag{6} \]
where
\[ d_*=\left(\frac{6}{\pi}\right)^{1/3}V^{1/3}. \tag{7} \]
The results of calculations by formulas (2)—(7) are given in columns 5–10 of the table. As can be seen, the results closest to the value of the true volume and the true surface area of a hanging mercury drop are given by calculations using formulas (3) and (5) (see columns 6 and 8). Noticeable errors are produced by calculating the surface area of a hanging drop by formula (6), under the assumption that a drop of known volume \(V\) is a complete sphere (see column 9).
If the volume of a hanging mercury drop is unknown, but experimental data are available on the equatorial diameter and the height of the hanging drop (obtained, for example, by measuring these quantities in a photograph of the drop with a measuring microscope), then Tables 1 and 2 can also be used to find the true volume and surface area of the drop.
To determine the true volume of a hanging mercury drop, one should calculate an approximate value of the volume by formula (3), and then, from Tables 1 and 2 (columns 6 and 1), use interpolation to find the true value of the volume.
To determine the true surface area of a hanging mercury drop, one should calculate an approximate value of the surface area by formula (5), and then, from the table (columns 8 and 2), use interpolation to find the true value of the surface area.
It is possible to find from Tables 1 and 2 the true value of the volume and surface area of a hanging mercury drop directly from the experimental data \(d\) or \(H\), using linear interpolation (columns 3, 4 and 1 or 2). But this is less accurate, since \(d\) and \(H\) are not proportional to \(V\) or \(S\).
If the contact diameter lies in the range of values \(0.02\)—\(0.5\ \mathrm{cm}\) (and is not exactly equal to the values 0.02 and 0.05 given in the tables), then the true value of \(V\) or \(S\) can be found by interpolating the data given in Tables 1 and 2.
Column (11) gives the results of calculations of the surface area of a hanging mercury drop under the assumption that, for a given volume, it has the form of an incomplete sphere (spherical segment). For this purpose, from the known volume \(V\) and the base (chord) \(a\) of this spherical segment, its diameter \(d_0\) and height \(H_0\) were first determined by successive approximations using the formulas
\[ V=\frac{\pi}{6}d_0^3-\frac{\pi}{6}(d_0-H_0)\left[\frac{4}{3}a^2+(d_0-H_0)^2\right], \tag{8} \]
\[ \left(\frac{d_0}{2}\right)^2=\left(H_0-\frac{d_0}{2}\right)^2+\left(\frac{a}{2}\right)^2. \tag{9} \]
And then, from the found values \(d_0\) and \(H_0\), the surface area of the spherical segment was calculated by the formula
\[ S_0=\pi d_0^2-\pi d_0(d_0-H_0). \tag{10} \]
The results of calculations by formula (10) are presented in column (11) of both tables.
In this case the calculated values of the surface agree best with the true surface (see columns 2 and 11).
Conclusions
-
On the basis of numerical material obtained previously on the MINSK-1 electronic digital computer in the numerical solution of the Laplace equation, tables have been compiled for calculating the exact value of the surface area of hanging mercury drops for the case of two contacts of different diameter (0.02 and 0.05 cm) in the interval of drop volumes 0.1–3.4 mm³ (in steps of 0.1 mm³).
-
It has been shown that the approximate empirical formulas usually used for mercury drops give results that differ from their true value.
-
It should be recommended that the tables obtained in this work be used in all physicochemical and electrochemical investigations involving the use of hanging mercury drops.
LITERATURE
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- A. G. Stromberg, E. A. Zakharova. Zavod. labor., 30, 261 (1964).
- V. I. Melik-Gaikazyan. Izv. TPI, 136, p. 110, 1965.
- H. K. Adam. Physics and Chemistry of Surfaces. Gostekhizdat (1947).
- F. Bashfort, J. Adams. Capillary Action, Cambridge, 1883.
- A. N. Krylov. Lectures on Approximate Calculations. Akademizdat, Moscow (1950).
- S. Fordham. Proc. Roy. Soc. London, 194, 1, 1948.