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I have a very basic doubt. I knew that $$ \rho = \frac{m}{V} $$ And since mass is constant so volume may decrease or increase depending upon density. But suppose I have water in a beaker and I read it somewhere while trying to understand some concept (Surface tension) that no. of molecules in upper layers of water surface is lesser than that of the layers present below.

Consider in the case of water, H2O molecules are attracted to each other by Hydrogen bonding. And since the vessel surface area is same, How can be the number of molecules different in one layer since It would imply the gaps left out in the layer?

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Are number of molecules per unit volume constant(say in water) ?

No. The density of water changes with temperature - between $0^o$C and $100^o$C it changes by about 4% (see https://en.wikipedia.org/wiki/Water_(data_page)). Since its density (which is mass per unit volume) changes, then the number of molecules per unit volume must change as well.

But I think you have misunderstood the explanation of surface tension. Surface tension is not due to any change in the density of water molecules at the surface of the liquid. Instead, surface tension arises because the distribution of surrounding water molecules is asymmetric for a molecule at the surface. As this Wikipedia article says:

Surface tension results from the greater attraction of liquid molecules to each other (due to cohesion) than to the molecules in the air (due to adhesion)

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  • $\begingroup$ Can you please also explain why there are no gaps since number of molecules are also same(say in 1 cm$^3$ of liquid). $\endgroup$ Dec 16, 2022 at 2:01
  • $\begingroup$ @ShekharDangi I don't understand your question. You seem to be thinking of a liquid as if it were a crystalline solid, when there might be gaps or dislocations in the layers of atoms/molecules. But molecules in a liquid are free to move around - why should they leave any gaps ? $\endgroup$
    – gandalf61
    Dec 16, 2022 at 8:19
  • $\begingroup$ But how can different number of molecules occupy same volume even though they are randomly moving. $\endgroup$ Dec 16, 2022 at 10:20
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    $\begingroup$ @ShekharDangi Number of molecules per unit volume is proportional to density. If density changes then the number of molecules per unit volume changes. The underlying cause of this change in the case of temperature dependence is that a higher temperature means that molecules have higher average kinetic energy, and so are moving faster on average. $\endgroup$
    – gandalf61
    Dec 16, 2022 at 10:29
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I read it somewhere while trying to understand some concept (Surface tension) that no. of molecules in upper layers of water surface is lesser than that of the layers present below.

Yes. This is due to the effects of gravity on water pressure. Water pressure increases with depth because of the weight of the each succesive layer of water. As pressure increases, the water molecules squeeze closer together so there are more molecules of water for a given volume at the bottom of the glass than the top:

`$$\frac{ΔV}{V_0}=\frac{−ΔP}{B}$$

Where B is the bulk modulus of water, $\Delta P$ is the pressure difference, $V_0$ is the original volume.

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    $\begingroup$ But, to a first approximation, water is incompressible. So this might be true, but it isn't that significant. $\endgroup$
    – matt_black
    Dec 15, 2022 at 20:55
  • $\begingroup$ @matt_black Are there more molecules of water at room temperature in the bottom of a glass of water then at the top? If this is true, I stand by my answer. All that is needed is one molecule more at the bottom for a given volume then at the top $\endgroup$ Dec 16, 2022 at 14:48
  • $\begingroup$ At the extreme pressures of the ocean bottom, water is compresses <2%. So, while in principle, this is also true at the bottom of a glass of water, the effect is so small it is probably impossible to even measure it. $\endgroup$
    – matt_black
    Dec 16, 2022 at 17:59
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Yes, this is evident by water supply in household or electricity made from reservoir or dam. In fluids, density of lower level is high and that creates pressure and cause of viscosity. This change in density with depth of fluid causes pressure gradient, and this may be cause of gravity. Yes, gravity is similar to buyoant force.

Now you say that I am using gravity to explain gravity as pressure created by weight. No this is not, here density causing gradient and everywhere mass is measure of matter contain. Other way is that mass is same, but volume measuring from a point causes change in density.

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    $\begingroup$ This is a very unclear and confusing answer. $\endgroup$
    – gandalf61
    Dec 15, 2022 at 8:47
  • $\begingroup$ @gandalf61 What confusing. It is clear the fluids have tendency to have pressure at bottom of given height. This is why dam are constructed, so water from depth have more pressure to pump or produce electricity. This is why fluid have viscosity because bottom is denser. $\endgroup$ Dec 15, 2022 at 9:19
  • $\begingroup$ Yes, pressure increases with depth, but it is not clear how this is relevant to answering the original question, which is about density and surface tension rather than pressure or viscosity. Also, pressure and viscosity are not entirely due to gravity, since a volume of water still has both pressure and viscosity in the absence of gravity. $\endgroup$
    – gandalf61
    Dec 15, 2022 at 9:35
  • $\begingroup$ @gandalf61 Question was about change in density, surface tension was an illustration in support. Now if pressure increase then density may increases, how pressure increases in fluid, by decreasing volume. $\endgroup$ Dec 15, 2022 at 15:40
  • $\begingroup$ Higher pressure at greater depth would be true for an ideal incompressible fluid. Density is not the issue. In fact, most common liquids are well approximated by incompressibility. $\endgroup$
    – matt_black
    Dec 15, 2022 at 20:53

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