# Compressibility Definition and 59 Threads

In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. In its simple form, the compressibility

κ

{\displaystyle \kappa }
(denoted β in some fields) may be expressed as

κ
=

1
V

V

p

{\displaystyle \kappa =-{\frac {1}{V}}{\frac {\partial V}{\partial p}}}
,where V is volume and p is pressure. The choice to define compressibility as the negative of the fraction makes compressibility positive in the (usual) case that an increase in pressure induces a reduction in volume. The reciprocal of compressibility at fixed temperature is called the isothermal bulk modulus.

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1. ### Engineering Exploring the Confusing Impact of Compressibility on CL Measurements

This situation is really confusing me. First of all, does accounting compressibility affect the measurements for CL? If yes, how does CL change as a function of the increasing mach number? Back to the problem, above is the "in-flight" lift curve for a plane traveling at high speeds, so the CLs...
2. ### B NIST database and compressibility factor Z

There's a huge volume of data on NIST database: https://webbook.nist.gov/cgi/fluid.cgi?T=293.15&PLow=10&PHigh=1000&PInc=10&Applet=on&Digits=5&ID=C7727379&Action=Load&Type=IsoTherm&TUnit=K&PUnit=bar&DUnit=mol%2Fl&HUnit=kJ%2Fmol&WUnit=m%2Fs&VisUnit=uPa*s&STUnit=N%2Fm&RefState=DEF I'm interested...
3. ### A substance has an isothermal compressibility kappa = (aT^3)/(P^2)...

Starting from v(P,T), dv=(dv/dp)_T dp + (dv/dT)_P dt i implemented conditions when T and P are constant and ended up with ln V = aT^3/P + constant and ln V = bT^3 /3P + constant If i assume that the constant is 0, i can say that a/b = 1/3 but how do i justify this assumption?
4. ### B Exploring Bulk Modulus: Examining Caesium and Beyond

Today, while studying about bulk Modulus, I encountered a doubt. Please consider this thought experiment. I'm considering Caesium as an example as it seems to have a quite low Bulk Modulus (comparatively) of 1.6 GPa. Let's say I apply a pressure of X GPa. Volume change ratio can be given by...
5. ### Accoustic waves notions (propagation in a tube)

Please help me with this problem I am facing, I am lacking notions of acoustics and I would be very grateful if someone could clarify them: A tube has a revolution symmetry arounf the ##x## axis and has a section dependent of the value of the abscissa (x), so the profile ##S(x)## is known. The...
6. ### How to find the thermal compressibility of a gas

Homework Statement : [/B] A gas obeying the equation of state PV=RT undergoes a hypothetical reversible process PV^\frac{5}{3} e^\frac{-PV}{E_0} = c_1 Can we prove that the thermal compressibility of the gas undergoing this process tends to a constant value at very high temperature? Here, E_0...
7. ### Pressure trace of a tank fed by a compressor

G'Day All, This is my first post so please let me know if I have completed this form incorrectly, or missed a point of etiquette etc... 1. Homework Statement The problem is to determine the pressurisation rate of a tank being filled by a pipe connected to a compressor. Assumptions: Pipe...
8. ### Specified equation of state from heat capacity

Homework Statement The constant-volume heat capacity of a particular simple system is c_v = AT^3 where A is a constant. In addition the equation of state is known to be of the form (v-v_0)p = B(T) where B(T) is an unspecified function of T. Evaluate the permissible functional form of B(T)...
9. ### Why Does the Compressibility of Helium Gas Diverge in This System?

Homework Statement A cylinder is fitted with a piston, and the cylinder contains helium gas. The sides of the cylinder are adiabatic, impermeable, and rigid, but the bottom of the cylinder is thermally conductive, permeable to helium, and rigid. Through this permeable wall the system is in...

18. ### How Much Water is Needed to Increase Pipeline Pressure for a Hydro Test?

I am trying to figure if there is a calculation for working out how much water I would have to pump into a system to increase the pressure. For example, I have a pipeline which is 2" and 6.5 km long. I worked the volume out using πr² × height so π×0.0254m²×6500m which is 13.17m³. How do I now...
19. ### Heating of liquid when pressure 1bar-->5000bars

Hello! I have some troubles finding a way to estimate how much temperature of liquid would rise when it is suddenly adiabatically pressurized to thousands of bars. In normal conditions liduids such as water are considered to be incompressible, but certainly not in 5000 bars. Because water (or...
20. ### Chorin Artificial Compressibility Equations

Hi! I have the following problem: pt + (c2u)x + (c2v)y = 0 ut + (u2+p)x + (uv)y = α(uxx+uyy) vt + (uv)x + (v2+p)y = α(vxx+vyy) It is a formulation of the incompressible Navier-Stokes equations. I would like to know an exact solution. Can anyone help me? Thanks
21. ### What does "compressibility is abridged" translate to?

I'm doing an independent research in aeronautics (realistically I'm just learning the material on my own and testing out of it due to unforeseen circumstances last semester). One area of study that my professor advised me to research in depth was just this simple phrase "compressibility is...
22. ### Compressibility factor and van der Waals equation for temp

The pressure exerted on the walls of the container by a real gas is less compared to an ideal gas. This is due to the attractive forces of the gas pulling the molecules back towards the rest of the gas molecules. However, there is also a relationship whereby at lower temperatures, the z is even...
23. ### Question about compressibility factor

At low temperatures, z falls below 1 and the reason for that is because the intermolecular interactions cause the pressure exerted to be lesser than expected. PVm/RT=z and since P is less than expected z drops below 1. However, as the pressure increases z increases to be above 1 because as P...
24. ### Work and isothermal compressibility

Homework Statement 1 kg of water is at room temperature and the pressure is isothermally increased on the system from 1 atmosphere to 1000 atmospheres. What is the work done? What is the change in heat? What would be the temperature change if this was done adiabatically? The volumetric...
25. ### Equations of state -- Partial derivatives & Expansivity

Homework Statement Show that the coefficient of volume expansion can be expressed as β= -1÷ρ (∂ρ÷∂T) keeping P (pressure) constant Where rho is the density T is Temperature Homework Equations 1/v =ρ β= 1/v (∂v÷∂T) keeping P (pressure ) constant The Attempt at a Solution I started with...
26. ### Finding vapour pressure using compressibility chart

Homework Statement I was asked to find the vapour pressure and saturated liquid molar density of propane at 263.15K using a generalised compressibility chart. (not allowed to use NIST or steam tables either, the chart i was given does not have reduced volume lines) Homework Equations Tr=T/Tc...
27. ### How to Calculate Isothermal Compressibility for Question 6b?

Homework Statement Homework Equations No idea... The Attempt at a Solution Does anyone know how to do question 6b)? Its a past paper question for an exam in 2 days and i have literally no idea where to start.
28. ### Solving an Equation of State Using Compressibility & Expansivity

Homework Statement Obtaining an equation of state from compressibility and expansivity. States of superheated steam are observed to have an isothermal compressibility k=(rNT) / (VP^2) and a volume expansivity B=(N/V)((r/P)+(am / T^m+1)). r,m and a are constants. a) Find dv in terms of dP...
29. ### Linear compressibility Lennard-Jones potential

Homework Statement Please see attachment. The correct answer is supposed to be ##\alpha = .15 k_B/U_0##. The Attempt at a Solution The minimum of the potential is at ##x = a##. Expanded about this minimum we get ##U = -U_0 + 36\frac{U_0}{a^2}(x - a)^2 - 252\frac{U_0}{a^3}(x - a)^3 + ...##...
30. ### Isothermal compressibility and spring constant solids

Homework Statement Consider a solid of compressibility ##\kappa##. Assume that the atoms in this solid are arranged on a regular cubic lattice, the distance between their nearest neighbors being ##a##. Assume further that a restoring force ##-k_0 \Delta a## acts on a given atom when it is...
31. ### Thought experiment about fluid compressibility

I'm a physics noob but this is something I was thinking about when I read in my textbook that fluids are generally incompressible. Let's say I had 1000 L of water enclosed in a sphere several meters thick (say, 2 meters thick) made of an extremely hard, dense metal with 0 outlets or holes of...
32. ### Compressibility Factor Pressure For Calculating Fan Power

What is the correct type of pressure (static or total) used in the compressibility factor, KP, when calculating fan power? Howden's Fan Engineering book seems to indicate total pressures should be used, but I also have a PDF from Howden that indicates static pressures should be used. Online...
33. ### Finding out compressibility of a fluid when bubbles are present?

I want to know approximately how compressible a fluid can become when H2O bubbles are present. Does anyone know how to do this? Thanks.
34. ### Volume using Compressibility Factor

Homework Statement Find the volume of 2 kg of ethylene at 270 K, 2500 kPa using Z Homework Equations Method to Solve for Z, Using Tr and Pr PV = ZnRT The Attempt at a Solution Tr = 0.9561 Pr = 0.496 Z found to be approx. 0.75 R given on a table at 0.2964 kJ/kg K From...
35. ### Prove expansibility / isothermal compressibility

β is expansibility , (1/V)(∂V/∂T) at constant temperature κ is isothermal compressibility , (-1/V)(∂V/∂P) at constant pressure How to prove (∂P/∂T) at constant V = (β/κ) Thank you
36. ### Finding the compressibility factor (Z)

Homework Statement Determine the volume, in m^3, occupied by 20 kg of hydrogen (H2) at 1170 kPa, 2220°C. Homework Equations Z=pv/rt, Pr=P/Pc, Tr=T/Tc, and for hydrogen M = 2.016 (kg/kmol) Tc = 33.2 (K) Pc = 13.0 bar Zc=pc*vc/(RTc) The Attempt at a Solution I know if I find Z then...
37. ### Isothermal compressibility of bose-Einstein condensate

Homework Statement Variables: N (number of particles), μ (chemical potential), P (pression), V (volume). k is Boltzmann's constant. I often use β=1/kT. The (isothermal) compressibility is given by \kappa_{T} = -\frac{1}{V}\left (\frac{\partial V}{\partial P}\right )_{N,T} The...
38. ### Help with Compressibility of liquid and relation to pressure

Homework Statement Assume that the distance across a microscopic cell is larger than the correlation length of the liquid, so whatever is happening in one cell is statistically uncorrelated with what is happening in an adjacent cell. Further, assume that each cell has two distinct possible...
39. ### Is Infinite Compressibility Possible in a System?

Correct me if I am wrong, but I think that compressibility, generally tells how difficult it is to compress an object. Meaning, the higher the compressibility, the easier it will be to compress an object with a given pressure. I was asked in my University to describe an system of infinite...
40. ### Compressibility of Ice vs Water: Answers & Explanations

Dear PF, Is ice compressible? What its compressibility... or compressibility relatively to water compressibility Thanks much
41. ### Water density and compressibility

Water is compressible, but it takes a lot of pressure to increase the density by 10 or 20% Deep Ocean pressure measurements http://nctr.pmel.noaa.gov/Dart/Pdf/Eble_J_atmo_91.pdf http://seagrant.gso.uri.edu/factsheets/abyss.html http://www.windows2universe.org/earth/Water/density.html...
42. ### Finding coefficient of thermal expansion and isothermal compressibility

Homework Statement To a very good approximation, ammonia obeys the Bertholet equation of state, which readsPV=nRT+\frac{9}{128}(\frac{nRTc}{Pc})(1-6\frac{Tc^2}{T^2})Pa)Suppose we have 500 grams of ammonia under a pressure of P=3.04 atm and at T=323K. Calculate the volume of ammonia according...
43. ### Increase in compressibility when pressure is applied

I was running some structural relaxations under constant pressure, and found out that my system has compressibility which increases when pressure is applied. Of course, this is only up to a point, since at pressure high enough, phase transition will occur and compressibility will drop sharply...
44. ### An example of a fluid with low compressibility

Hi, Does anyone know a fluid with low compressibility? Thanks, H
45. ### Real gas compressibility isotherms(graphs)

Homework Statement Sketch the following compressibility isotherms(fancy word for graphs) (compressibility factor, PV/nRT versus pressure :) for oxygen at a temperature just above the critical temperature ::) for oxygen at a temperature well above the critical temperature eg. 200 degrees...
46. ### Isothermal Compressibility always positive proof

I have a question on the quantity -(dV/dP)T,N where V = volume, P = pressure, T = temp, N = number of moles and T, N are held constant. I see in textbooks that this quantity is always positive at equilibrium. It makes intuitive sense, as if it were negative, it would be unphysical. I've been...
47. ### Compressibility of a Refrigerant

I am going to be running some experiments with refrigerants. They will be at various temperatures and pressures. I will then be doing some modeling of certain phenomena. I would like to be able to quantify whether the ideal gas model is reasonable during certain stages of the experiment. My...
48. ### How Do You Calculate Gas Volume Using the Ideal Gas Law?

Homework Statement Calcualte the volumes in cubic feet of the gases in the attached chart, assuming the gases are ideal Homework Equations Ideal gas equation PV=nRT The Attempt at a Solution Please see the attached sheet also, this has my attempt, if i use my method and then use a...
49. ### [Thermo.] thermal expansion and isothermal compressibility coefficients.

Homework Statement A certain metal whose thermal expansion coefficient \beta is 5,0 × 10^-5 °C^-1 and whose isothermal compressibility \kappa_T is 1,2 × 10^-6 atm^-1 is at an initial pressure of 1 atm and an initial temperature of 20°C. A thick layer of Invar is thermally insulating the...
50. ### How Does Adiabatic Compressibility of an Ideal Gas Compare to Isothermal?

I have been asked to show that if an ideal gas is compressed isothermally its compressibility is 1/P whereas if the same gas is compressed adiabatically its compressibility is 1/yP Where y is gamma I have managed to do the first bit about isothermal compressibility, but cannot work out...