# Pressure, temperature, and volume relation in liquids

Physics books and websites are full of Ideal Gas info and solutions when it comes to relating pressure, volume, and temperature. How about liquids? And solids?

For example let us say a liquid in a horizontal pipe is under double the static pressure than it was originally. By static pressure I do not mean hydrostatic - I mean static as in the other pressure factor of water (not sure what it is called.. static pressure ignoring hydrostatic pressure). Your water company decides, whelp, let's increase the city water pressure! Say originally we had a horizontal copper pipe full of water. At the one end of the pipe is a tap that is closed. At the other end of the pipe is the incoming city water company (city pressure). Water temperature is 20 degrees celsius (room temp) in the pipe originally, and a pressure of "1". Then the city increases water pressure to "2". The units do not matter, just say the pressure doubled. The water did not move, because the tap was shut. Yet the static pressure doubled. Would the temperature rise, twice as much to 40 degrees celsius due to the pressure increase? For simplicity, let's say the copper pipe never expanded and none of the taps can leak.

Ideal liquid laws?

Liquids transmit pressure - but, when they are not moving, AFAIK they hold a static pressure, almost like a "charge". I wonder if since liquids are virtually incompressible, an increase in static pressure causes the liquid to heat up ideally. We could think of a liquid under pressure like a gas under pressure which cannot change volume, due to its special container that doesn't let it change volume (the liquid itself!). i.e. if an ideal gas does not change volume, does not move, but increases in pressure - will it change in temperature proportional to change in pressure? (Yes, but ideal liquids?)

As for solids - well supposedly the fairly solid "earth" is under pressure causing tremendous heat in the center - kind of empirical evidence that pressure causes heat in solids. So this might answer my question - but still! I have to ask about liquids specifically too.

A practical aside: how does a water company actually create more pressure if the town demands it - larger pump? Or more pumps are added in series? I am also having trouble seeing how static water pressure can be increased without the pump turbine blades snapping in half due to the virtuallly incompressible water. I can visualize the dynamic pressure increasing once someone turns a tap on - but static pressure of liquids is a bit more tricky to visualize. Can a pump simply "shudder" and basically stay static, but still waste energy performing molecular work - increasing static pressure but not moving the water as a whole. Doesn't a pump motor have to move somewhat in order to perform work - but if the water cannot move..??? Well I suppose it is like turning on an electric lawn mower, and having a man grab the blade and hold it in one static position. Motor still performs work on the muscles of the man somehow, even though motor is not moving?

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Mapes
Homework Helper
Gold Member
The term I think you're looking for is

$$\left(\frac{\partial T}{\partial P}\right)_S$$

which is the isentropic increase in temperature per unit pressure, or the temperature increase when a material is pressurized quickly. (If it's pressurized slowly, its temperature stays the same as the surrounding environment.) Through a couple of partial derivative identities, we can show that this term is equivalent to

$$\frac{T\alpha}{\rho c_P}$$

where T is the temperature, $\alpha$ the volumetric thermal expansion coefficient, $\rho$ the density, and $c_P$ the constant-pressure specific heat capacity. The thermal expansion coefficient is much lower, and the density much higher, in liquids and solids compared to gases, so the temperature increase upon pressurization is relatively small. Try calculating it for different materials.

For the town water pressurization question: I'm just guessing, but couldn't one just fill the water tower to a higher level?

The term I think you're looking for is
....
the isentropic increase in temperature per unit pressure, or the temperature increase when a material is pressurized quickly.

Yes this helps, thank you.

For the town water pressurization question: I'm just guessing, but couldn't one just fill the water tower to a higher level?

That would be a simple practical solution. In that case gravity potential is used to store the energy..

I wonder if another potential could be used if they did not have that option. I am thinking that somehow the hidden elasticity of the molecules in a liquid can be charged up. Intra-molecular potential creating static charge. I mean, hydrostatic pressure from gravity is not the only charge water can hold? Or is it? Example: how people on farms far away increase water pressure using pumps? I'll look into it. That again in my mind right now is more easy to see as dynamic pressure in the moment when you need it.

I am visualizing a windshield washer pump and again I see the dynamic pressure easily, but the static pressure is very temporary and not stored for long period of time. Hmm. Can the static liquid pressure be stored without gravity? There is vapor pressure of water, which is kind of related.. almost like evidence that the hidden elastic in water exists. It is interesting that we have different pressure models - vapor pressure, hydrostatic pressure, and they are not all in the same pressure, we model them almost as separate pressures with different behaviors. Yet pressure as a whole is tied together in the end. The analogy I see is when water pressure increases, the vapor pressure elastic contracts. When water pressure decreases, the vapor pressure elastic expands. Once the elastic expands enough, the elastic breaks and snaps and water boils (intra molecular bonds are broken).

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Mapes