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How can I get from a photograph of a liquid surface to a value for the surface tension.

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There is a possibility that this is a good question, but you need to tell us more: do you have something in the image or not to tell you what the absolute scales are (something like a ruler or knowing the diameter of the tube), or at least a guarantee that all the images have the same scale? Do you understand the geometry of the liquid column? What part of this physics do you understand and where are you stuck? Give us something to go on here. –  dmckee Nov 15 '12 at 21:29
    
Have a look at this simple link : ehow.com/how_6019173_calculate-surface-tension.html . There is a formula for getting newtons/meter . You cannot do it just from a photograph if you do not know the temperature. Yes, the slit is covered with water because of surface tension. You could make wider and wider slits until this no longer happens and then use approximations to calculate the tension, given temperature. Complicated method. (this refers to your related slit question) –  anna v Nov 16 '12 at 5:07
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The word you are looking for is Contact Angle and your photograph would need to be good enough to measure this. –  Mark Rovetta Nov 19 '12 at 18:40
    
The Sessile Drop Technique is a method used for the characterization of solid surface energies, and in some cases, aspects of liquid surface energies. –  Mark Rovetta Nov 19 '12 at 20:47
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It is not completely clear what you consider a 'liquid surface', but a nice general method to determine the surface tension of a liquid is the pendant drop test. The basic thought is to balance gravity and surface tension by hanging a known droplet volume from a syringe and analyzing the shape of the droplet (image from this article).

pendant drop schematic

For a high surface tension the droplet will be more or less spherical whereas for a low surface tension the droplet will deform more and stretch in the direction of gravity. A pretty good explanation of the details of the method can be found here.

Also, if you have access to a standard apparatus for this type of measurement, the calculation will be available in the software of the machine.

The sessile drop technique that @MarkRovetta mentions is also an option to determine the surface tension but it is less commonly used because you introduce an extra `experimental difficulty'. Namely you need an (almost) perfectly smooth surface to make sure that the droplet is axisymmetric, which is almost guaranteed for the pendant drop technique. The sessile drop principle is almost the same: you put a droplet of known volume on a surface and determine contact angle and circumference from a sideview. This image you can then fit to a mathematical model which balances the Laplace pressure with the hydrostatic pressure: $$\frac{2}{R_0}+\rho g z = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right), \;R_1=\frac{\partial s}{\partial \theta},\; R_2=\frac{x}{\sin \theta} $$ where the parameters are indicated in the image below. With simple geometrics this system can be converted to a system of ODEs dependent on $s$. By integrating over $s$ to the desired contact angle $\theta$ and adjusting $R_0$ to get the correct volume $\left(\frac{\partial V}{\partial s}=\pi x^2 \sin \theta\right)$ the surface tension can be found by fitting the experimentally obtained droplet profile.

sessile drop

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