How Much Does A Drop of Water Have?
By Zimo Wang, 4th Mar 2022
Translated by Xinyi Cheng, 6th Jun 2023
The main character we are talking about today is “water”. Water, which plays a vital role in the human body, functions not only in maintaining homeostasis but also as a solvent for various substances. Today, we are looking at “water” from a microscopic point of view, so the topic is actually “a drop of water”.
1. Using Tate’s Law to Describe “A Drop of Water”
Before introducing Tate’s Law, let’s see the most basic principle of how a droplet forms --- surface tension. It is surface tension that enables water to form small droplets. Imagine a water molecule: If it exists in the center of a droplet, which means that it is surrounded by other molecules, the intermolecular forces among those molecules will be equal (as Figure 1a shows). Then imagine another water molecule: This molecule exists on the fringe of a droplet, so there are no other molecules exerting forces from outside; it is only affected by inward forces exerted by molecules aside and inside (as Figure 1b shows) to form the surface tension.
Therefore, water will be spontaneously “wrapped” to form a droplet, and this is the basic principle of how surface tension forms. A simple formula to calculate the surface tension is below:
In this formula, F represents the surface tension; σ is a constant; l is length. This is a comprehensible formula, and we can substitute it into Newton’s First Law of Motion to do calculation.
As shown in Figure 2, there are two forces exerted on a droplet in a capillary tube: a gravitational force W, and an upward surface tension F. We suppose that the cross-section of the droplet is in the shape of an ideal circle; then “l” in the formula can be converted into the perimeter of the circle. In Figure 2, assume that the radius of the circle is r, and l = 2πr. Finally, when the magnitude of gravitational force equals to surface tension, the droplet reaches equilibrium:
The formula above is the basic form of Tate’s Law—— quite juicy, isn’t it? Solve this equation to get:
Since the mass of droplet is related to its radius, we can take into the formula:
Take in specific number and we can get the approximate radius of a droplet:
0.0728 is the factor of surface tension for water. Therefore, according to Tate’s Law, droplets that are naturally formed in capillary tubes should have the same radius under the same conditions, which is 3.3 mm.
2. Is the magnitude of σ accurate?
Here is a question about the calculations above:
What is the exact value of σ?
σ is the prefactor of surface tension for water, the value we use is 0.0728 N/m (the unit can be attained by using dimensional analysis). 0.0728 is the answer given by the Internet, but can it really work for this question? Let’s see how Ines M. Hauner and her colleagues carry out their experiment and publish the paper.
This formula is from Ines M. Hauner’s paper, so the inference process is unavailable; let’s treat it as an axiom. In this formula, is the minimum diameter of a droplet during its process of dripping, as shown in figure 3; represents “numerical prefactor”; σ is the factor of surface tension that we have mentioned before; ρ represents the fluid density.
, where t is the time when water starts to drip, and t0 is the time when water is about to drop, so τ represents the episode between the water dripping and reaching the critical point.
We’ve said that we are going to determine the exact value of σ, and the formula can be used to conduct a series of experiments and calculations. After analysing every character in this formula, we can find that except A and σ are unknown, and other numbers are available by measuring. Therefore, here are what we doing: the first step is to solve the value of A according to different kinds of liquids, and then substitute the value and get the factor of surface tension, σ. We need to measure that corresponds with τ. As shown in following figure.
It can be seen from the figure that there are different liquids that are not constrained to water of different pH and some organic solutions (like octane). Also, the y- axis is , which simplifies the sequent calculations. What does the slope represent? You can take out your pen and paper and solve it by yourself—— it's not hard!
According to the formula mentioned before, raising both side of the equation to the two-thirds power, we can get:
Corresponding to Figure A, it becomes an understandable proportional function, in which the slope is . We can know from Figure A that the slopes of water are greater compared to those of organic solutions. Then, our purpose is to solve the value of A, so we let A to be the y-axis, and the x-axis is . The figure we get is:
Since the slopes of water are comparatively greater in Figure A, the corresponding value of y in Figure B is higher, which indicates that our comprehension is surefooted. Now, what is the slope in Figure B? Using the formula of calculating slope, we can get the slope k:
After getting the slope, we can calculate the value of A. The value of A in this paper is 0.90±0.01. Finally, we take this number into the original formula, and we can get σ~0.090N/m. There are still differences between our result and 0.0728 that is mentioned before. Why? Is 0.0728 the wrong value? Absolutely not. Here are two explanations: First, also the most probable one, is that the experimental error in the paper is too large. The second explanation is the one provided by the paper: the surface tension of a newly formed droplet might be greater than that of a droplet in a balanced state. Not completely impossible, but forget it. Finally, let’s take factor 0.09 into the formula, and the radius of a droplet is around 3.3 mm-3.7 mm.
That’s all the content of today’s passage! Thanks for your reading!
Ref:
Hauner, I., Deblais, A., Beattie, J., Kellay, H., & Bonn, D. (2017). The Dynamic Surface Tension of Water. The Journal Of Physical Chemistry Letters, 8(7), 1599-1603. https://doi.org/10.1021/acs.jpclett.7b00267