At 20^∘ C , the surface tension of water is 72.8 dynes/cm and that of carbon tetrachloride (CCl4

At 20^∘ C , the surface tension of water is 72.8 dynes/cm...

Key Concepts

Surface Tension

Surface tension is the property of a liquid’s surface that makes it behave like a stretched elastic membrane. It causes the liquid to minimize its surface area and plays a crucial role in forming spherical droplets, as the cohesive forces among the liquid molecules strive to reduce the total surface energy.

Laplace’s Equation

Laplace's equation relates the pressure difference across a curved interface to the surface tension and the radius of curvature. For a spherical drop, it states that the excess pressure is proportional to the surface tension divided by the radius, explaining how liquids under the same pressure will have different drop sizes depending on their surface tension.

Gauge Pressure

Gauge pressure refers to the pressure measured relative to the ambient atmospheric pressure. In the context of droplets, it represents the excess pressure inside the drop compared to the outside, and this excess pressure is responsible for maintaining the curved surface against the inward pull of surface tension.

Scaling Relations in Droplets

The volume of a spherical droplet scales with the cube of its radius. Since the radius itself is determined by the balance between the gauge pressure and the surface tension (via Laplace’s law), differences in surface tension lead to different droplet sizes and consequently a cubic impact on the volume. This concept is key in understanding how variations in surface tension result in significant differences in drop volumes under the same gauge pressure.

Transcript

00:02 In this problem, we're told we have two drops of liquid, a drop of water, h2o, with a surface tension of 72 .8 dines per centimeter, and a drop of carbon tetrachloride, which is ccl4, with a provided surface tension, which is 26 .8 dines per centimeter.

00:18 We're told that the pressure inside both of these drops is the same, and what we are looking for is we are looking for the ratio of the volume of the water drop to the volume of the carbon tetrachloride drop.

00:33 Now to make this a little bit easier, i'm going to convert these surface tensions into si units, into standard units.

00:40 They're in dines per centimeter.

00:41 I'm going to put them into newtons per meter.

00:43 To do that, i need to know that one dine per centimeter is equal to 10 to the negative 3 newtons per meter.

00:52 So what that means is that for our water droplet, the surface tension is equal to 72 .8 times 10 to the negative 3 newtons per meter.

01:06 And that for our carbon tetrachloride, our surface tension is equal to 26 .8 times 10 to the negative 3 newtons per meter.

01:17 It's going to make that a little bit easier to work with.

01:21 Now, what we also need to know is we need to know that for a droplet of a material, it's with one surface, the gauge pressure in that droplet is equal to 2 times the surface tension divided by the radius of that droplet.

01:35 And we know that the pressure in both of these droplets is the same.

01:38 So what i can do is i can set up this equation, our gauge pressure equation, with the variables for water and for carbon tetrachloride, and set them equal to each other.

01:47 So for example, for water, the gauge pressure would be equal to two times the surface tension of water divided by the radius of the water droplet.

01:59 And for the carbon tetrachloride, the pressure would be equal to two times the surface tension of water, divided by the radius, would be equal to two times the surface tension of carbon tetrachloride divided by the radius of the carbon tetrachloride bubble.

02:13 Because those pressures are the same, i can set them equal to each other.

02:16 In order to work our way toward a ratio, what i'm going to do is i'm going to cross -multiply these two values to get them all in one nice line.

02:24 So i'm going to do that on our next page...

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