The work done on the liquid is 2 aΓ 1 ∆ x, where Γ 1 is the surface tension at temperature T 1.I love teaching systems of equations, like seriously LOVE it! I always teach it right after the linear equations unit, and it’s the perfect transition from graphing one line, to graphing two. Let us start by moving the piston to the right, isothermally at a temperature T1, through a distance ∆ x, being the portion AB of figure XII.2. Notice that, as we move the “piston” to the right, provided that the temperature remains constant the surface tension force between the “piston” and the liquid does not change thus the isotherms are horizontal lines, with the warmer isotherms lying lower than the cooler isotherms. Let us now take the liquid around a Carnot cycle, as shown in figure XII.2. If we do it isothermally (slowly), the liquid has to absorb some heat from its surroundings. If we do this adiabatically (quickly), the liquid cools. If we pull the piston to the right through a distance dx, the work we do on the liquid is 2 aΓ dx. The factor 2 arises because there are two surfaces, above and below. If the width of the “cylinder” is a, the surface tension force with which the liquid is pulling on the “piston” is 2 aΓ, where Γ is the surface tension. If we allow the film to contract and to pull the “piston” to the left, the temperature will rise. Nevertheless, as explained above, the temperature of the liquid then drops. A difference between this picture and that of a gas inside a real cylinder is that when we pull the “piston” out, we are doing work on the liquid. ![]() We could even refer to these two parts as the “cylinder” A and the “piston” B. Another way, which might lend itself more easily to the sort of thermodynamical analysis we are accustomed to in discussing gases, is to imagine a film of soapy water held in a wire frame, constructed of a fixed U-shaped portion A (see figure XII.1), and a bridge B which we can move in and out, allowing us to do work on the liquid by pulling it to the right, or the liquid to do work by pulling the bridge to the left. One way in which we can imagine work being done on a liquid to increase its surface area is simply to imagine distorting a spherical drop into a nonspherical shape. It may at first seem surprising that doing work on a liquid, in order to create new surface, results in a fall of temperature, but the work is being used not to increase the kinetic energy of the molecules, but rather to increase their potential energy by pulling them to the surface. ![]() Increasing the area will result in a fall in temperature, so, if the temperature is kept constant, some heat must be absorbed from the surroundings, and hence the increase in the internal energy is a little more than the surface tension. We have already pointed out that the surface tension can be regarded as the work required to create new area. Conversely, if a spherical drop is distorted from its spherical shape, it becomes cooler. As many of them fall beneath the surface as the surface area is decreased, this potential energy is converted to kinetic energy. Molecules near the surface have a high potential energy. It should not come as a surprise to learn that, at least in principle, as the blob adjusts (in an adiabatic process) to its spherical shape of least surface area, it becomes warmer. a blob of least surface area for a given volume. How much work is needed to increase the surface area? And how is this related to what we have described as “surface tension”? It may be noted in passing that energy per unit area (J m −2) is dimensionally similar to force per unit length (N m −1).Ī non-spherical blob of liquid will, under the action of surface tension, contract into a spherical blob – i.e. However, from the point of view of thermodynamics, it is easier to think of surface energy. In this section I shall use the following symbols: ![]() It is expressed in dynes per cm or newtons per metre. We describe the tendency of a surface to contract by drawing an imaginary line in the surface, and we say that the surface to one side of the line pulls the surface of the other, and we call the force per unit length perpendicular to the line the surface tension. The effect is often conveniently described in terms of “surface tension”. In the absence of other forces, this means that it will become spherical. It is well known that a liquid tends to contract to a shape that minimizes its surface area. \)įor a second example of non- PdV work we shall consider the phenomenon of “surface tension”.
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