What is Equation of state: Definition and 97 Discussions

In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars.

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  1. grav-universe

    I Equation of state for Einstein field equations

    Years ago I posted a thread where I solved for the exterior vacuum metric for a static spherical mass using only a single one of the unknown functions, A, B, or D, where D = C r^2, since they are inter-related. A moderator here graciously supplied the EFE's as A, B, and D relate to the energy...
  2. C

    Find Isobaric Expansion & Pressure-Volume Coefficient for Solid

    The answer to this problem is However, I am confused how this relates to the question. My working is, ##V = cT^2 - bpT## ##\frac{dV}{dT} = 2cT - bp## (I take the partial derivative of volume with respect to temperature to get the isobaric expansion coefficient) ##\frac{dV}{dP} = 0## (I take...
  3. F

    A Quintessence field equation of state

    Why the equation of state parameter w for a quintessence field with potential m²Phi² is not in between 0 and 1/3 depending on the energy of its particles relative to the mass of the field instead of the w resulting from formula 9 of https://arxiv.org/pdf/1504.04037.pdf which seems to be...
  4. C

    Engineering Find the Canonical Equation for the Dieterici Equation of State

    In Elements of Gasdynamics the author describes what he calls The Canonical Equation of State where (∂E∂S)v=T and (∂E∂V)s=−P He does a simple one for a perfect gas and uses the enthalpy(T,V) for the Canonical Equation of state. Now he asks to find the Canonical Equation of State for E(V,S)...
  5. P

    Heat engine that uses a rubber band (Thermodynamics)

    Nota that the path a to b is isothermal since J is proportional to L, then we can find the value of T_a and T_b using the equation of state and the figure. We have, \begin{equation} J_0=\alpha L_0T_b \end{equation} or \begin{equation} T_b=T_a=\frac{J_0}{\alpha L_0}=T_0 \end{equation} Also, by...
  6. V

    A Equation of state of gravitational energy in open de Sitter?

    Hi, Gravitational energy in de Sitter has equation of state w=-1 (cosmological constant) and w=-1/3 (curvature energy in open or closed de Sitter). Is this just gravitational radiation, and how does this accord with the equation of state of photon radiation w=1/3? Does this mean that densities...
  7. Polyamorph

    I Inverting the Birch-Murnaghan Equation of State

    Hello, https://www.physicsforums.com/lib/Eqn026.pngI have the pressure (P)-volume (V) Birch-Murnaghan equation of state coeffcients (V_{0},K_{0}, K^{'}_{0}, K^{''}_{0}) for a number of different compositions. I'm interested in the volume at very specific pressures only and ideally I would like...
  8. Ranku

    I Equation of state and density parameter

    Does ##w## = -1 by itself imply that spacetime is flat, or is it only due to ##\Omega##Matter + ##\Omega##Λ = 1 ?
  9. WhiteWolf98

    Fluid Dynamics - Mass Conservation, State Equation for an Ideal Gas

    I understand that ##\dot m=\rho Q## and ##{\dot m}_{in}= {\dot m}_{out}## . So one can say that ##\rho Q_1 = \rho Q_2##. But I'm not sure if that equation is correct. I don't know if the density remains constant, or the volume flow rate. And then how I'm also supposed to tie a state equation in...
  10. W

    Equation of state for a real gas

    My work was as follows: The first law states ##dU = TdS - PdV##, and thus $$p =- (\partial U/\partial V)| _S$$ $$U = -RT \ln(V-b) + f(S)$$ To determine ##f(S)##, I reasoned that in the ideal gas limit of ##b = 0##, ##U## should take the form of the ideal gas' molar internal energy ##\frac{3}{2}...
  11. hnnhcmmngs

    How Does the Equation of State for a Solid Change with Temperature and Pressure?

    Homework Statement [/B] Find the equation of state of a solid that has an isobaric expansion coefficient dV/dT = 2cT - bp and an isothermal pressure-volume coefficient dV/dp = -bT (Assume the solid has a volume Vo at zero temperature and pressure. Enter a mathematical equation. Use any variable...
  12. H

    Thermodynamics: using Peng-Robinson's equation of state

    MODERATOR NOTE: NO TEMPLATE BECAUSE THREAD WAS ORIGINALLY STARTED IN NON-HOMEWORK FORUM I'm struggling with a question on an assignment about thermodynamics: Nitrogen gas, initially at a temperature of 170 K and a pressure of 100 bar, escapes from a thermally isolated tank with a volume of...
  13. maistral

    Volume translated Peng-Robinson equation of state

    Hi. Please excuse my ignorance but this entire volume translation formulas for EOS confuses me to no end. Could someone tell me how the volume-translated Peng-Robinson exactly works? How do I calculate the fugacity expression of VTPR? Do I integrate the V + c terms against dV or do i integrate...
  14. AwesomeTrains

    Determing equation of state from thermodynamic coefficients

    Homework Statement The isobaric expansion coefficient and the isothermal compressibility are given by: $$\alpha_p = (1/V)(\partial V/\partial T)_p \quad \kappa_T = -(1/V)(\partial V / \partial p)_T$$ Suppose they have experimentally been determined to be: $$ \alpha_p = \frac{1}{T} +...
  15. maistral

    Cubic equation of state near critical point (~0.95 < Tr < 1)

    Hi. I have read somewhere that cubic equations of state seem to have convergence issues when vapor pressure calculations are done near the critical point. Sadly, I have forgotten where I have have read it :( I would like to ask some knowledgeable people regarding this, or can people point me I...
  16. Demystifier

    A Equation of State w=1: Why Not Considered in Cosmology?

    A free scalar field with Lagrangian density $${\cal L}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$$ has the energy momentum tensor with the equation of state ##p=\rho##, i.e. ##w=1##. The Lagrangian density above is a very natural Lagrangian, yet the equation of state ##w=1##...
  17. G

    Change in Temperature for Stretched Surface

    Homework Statement The surface tension of a layer of water obeys ## \sigma = a- bT##, where ##T## is the temperature. Find the change in temperaure, ##\Delta T## when the area is increased isentropically. Homework Equations ## dU = dQ -dW## , ##dW = \sigma dA##, ##dU = C_A dT + [\sigma...
  18. G

    ##\alpha_P =\frac{V-b}{TV}## Find equation of State

    Homework Statement The coefficient of thermal expansion and isothermal compressibility of a gas are given by ##\alpha_P =\frac{V-b}{TV}## and ##\kappa_T = \frac{V-b}{PV}## find: a) The equation of state b) If the heat capacity at constant volume ##C_V## is constant, what is ##\delta U##? c)...
  19. H

    Exact thermodynamics equation of state

    Dear all, I am using molecular dynamics to calculate energy for ionic liquid. In the article that is proposed equation of state that I want to use, the author would call the energy in the following formula(u) conformational(potential energy): (du/dv)T + T(dp/dT)V = ptot Firt T and V are...
  20. H

    Integral constant for internal energy of an ionic liquid

    Integral constant for internal energy of ionic liquid I have a question, and I will be really grateful if someone helps me. I have a polynomial equation for internal energy which I calculated by integration an equation of state formula, which is based on density. But, because I calculated this...
  21. D

    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)...
  22. D

    Free energy Helmholtz using only the equation of state

    Homework Statement This is a state ecuation of a gas: PV=AT+B/V, where A and B there are constants. First: Demonstrate that ##c_V## depends only of T Second: Find U(T,V) and S(T,V) Homework Equations ##\left(\frac{\partial U}{\partial S}\right)_V=T\text{ (1)}## ##\left(\frac{\partial...
  23. S

    A How Does the Stiffness of Supernova EOS Change with Various Parameters?

    On what does the stiffnes/softness of the SN EOS depends? How does it change with temperature, progenitor mass, or other parameters like compresibility, symmetry energy etc? Would softening in NS EOS lead to softening in SN EOS? Tnx, Cheers
  24. S

    A Neutron star mass-radius relation plot

    Hi there... I have a question about the tipical mass-radius relation plot for a neutron star, something that looks like this for different equations of state: My question is, what is exactly happening at the ends of the EOS lines? I know that after passing the maximum mass the curve starts...
  25. Leonardo Machado

    A EoS BPS-BPP + Pure Neutron Matter/ Baryonic Matter

    Hello everyone, Today I'm asking you for help to put together these two equations of state for high densities BPS-BBP with a eos of a pure neutron gas or a non interacting n + p + e gas. I have already the data of BPS-BBP equation of state, with includes neutron drip and let a open space to...
  26. arpon

    I Can internal energy be calculated from equation of state?

    We know, $$dU=TdS-PdV$$ ##\int PdV## can be calculated if the equation of state is given. I tried to express ##S## as a function of ##P ,V## or ##T## (any two of those). $$dS=\left(\frac{\partial S}{\partial V}\right)_T dV+\left(\frac{\partial S}{\partial T}\right)_V dT$$ $$=\left(\frac{\partial...
  27. J

    A The equation of state of non-relativistic energy

    I just have a quick question about a concept I think I haven't fully grasped from my cosmology course. Why does non-relativistic energy have an equation of state with w=0? Also, is the concept of pressure different in general relativity than in thermodynamics or statistical mechanics?
  28. F

    A Hadronic era equation of state

    Hello, I was wondering : going back in time up to the time when the universe is hot enough to create nucleons particle antiparticle pairs, when we are not much above this temperature threshold, the nucleons can not be considered ultrarelativistic and if they carry a fraction of the energy...
  29. Useful nucleus

    A Nonhydrostatic equation of state for a solid

    I could find many equation of states for solids under hydrostatic compression such as Birch-Murnaghan and Vinet but I could not find any derived equation of state for non-hydrostatic case. Are there any texts or papers that discuss such equations of states in some detail ?
  30. Y

    Heat Capacities Given Equation of State

    Homework Statement Given the equation of state ##V(P,T)=V_1\cdot exp(\frac{T}{T_1}-\frac{P}{P_1})## where ##V_1\;,T_1\;,P_1## are constants: a. derive an equivalent equation ##P(V,T)##; b. given ##C_V=DT^3## where D is a const, calculate the entropy of the system ##s(V,T)## up to a const; c...
  31. S.Daedalus

    Einstein Equation of State vs Gravity from Entanglement

    There's a somehow related set of issues I find myself pondering time and again: In 1995, Ted Jacobson derived Einstein's equations from thermodynamics across a horizon. Roughly, he showed that if the horizon's entropy is given by the Bekenstein-Hawking formula, then the second law of...
  32. VoteSaxon

    Van der Waals and boiling point

    Homework Statement The bulb of a constant volume gas thermometer is immersed in an ice/water/water vapour mixture at equilibrium and the recorded pressure is 0.400 atm. It is then immersed in a boiling liquid and the pressure is 0.844 atm. Sufficient gas is then removed from the bulb such that...
  33. S

    Enthelpy's Equation of state vs Energy's equation of state

    I'm really confused about Energy and Entropy's equations of state, and how their differentials work. So I find everywhere that H = U + PV. And so dH = d(U +PV) = dU + PdV + VdP Ok, that makes sense. But then I look at energy and try to get the equation of state dU = δqrev + δwrev. That also...
  34. K

    Equation of State of PVT system

    Homework Statement Find the Equation of State of a fluid with Volume expansivity ##\alpha_P## and an isothermal compresibility ##\kappa_T## are given by $$\alpha_P=\alpha_0 \Big(1-\frac{P}{P_0}\Big) \\ \: \: \: \: \kappa_T=\kappa_0[1+\beta_0(T-T_0)]$$ ¿ What conditions should the constants...
  35. T

    How is the virial equation of state derived?

    Homework Statement It's just that in my textbook, for section titled "Second virial coefficients can be used to determine intermolecular potentials," I have an equation that I do NOT understand how it was derived---I tried to do it over and over, but couldn't quite figure how. If anyone could...
  36. Robsta

    Equation of state of a rubber band

    Homework Statement For a stretched rubber band, it is observed experimentally that the tension f is proportional to the temperature T if the length L is held constant. Show that: a) The internal energy U is a function of temperature only b) Adiabatic stretching of the band results in an...
  37. L

    Adiabatic condition and equation of state

    Homework Statement 8.02 × 10−1 moles of nitrogen gas ( γ= 1.40) is contained in a volume of 2.00 × 10−2 m3 at a pressure of 1.00 × 105 Pa and temperature of 300 K. The sample is adiabatically compressed to half its original volume. IT behaves as an ideal gas. (i) What is the change in...
  38. M

    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...
  39. N

    What is the Equation of State for an Ideal Gas?

    Homework Statement Hi Often I bump into the relation p=\rho c_s^2, where p is the pressure, rho the density and cs the speed of sound in the ideal gas. It is apparently called the equation of state for an ideal isothermal gas. I know about pV=NkT, which becomes p=\rho kT, but does this mean...
  40. electricspit

    Gibbs Free Energy and Equation of State

    I'm wondering why the Gibbs function is related to the equation of state as follows (supposedly): V=(\frac{∂G}{∂P})_T I found a thread on here that mentions this relationship, but doesn't explain it at all. Any help understanding this would be appreciated, this is my first introduction to...
  41. G

    Container with two gases and equation of state

    Homework Statement We have a cylindrical container filled with a gas: the container has an upper compartment with 2 moles of this gas, and another compartment below with 1 mole of the same gas, separated by a diathermic boundary, and also has an adiabatic mobile plunger over the upper...
  42. G

    Thermodynamics: find internal energy given the equation of state

    Given the following equation of state and C_p = 3R, find the equation of the internal energy in terms of T and V p(V-b) = RT Any clues? I can use also the specific heat at constant pressure or volume. Thanks!
  43. 4

    Equation of state of photon gas

    Dear all, I am using stress-energy tensor to derive equation of state of photon gas (assuming it as a perfect fluid). I completed all the steps except one: average value of [cos(θ)]^2 over unit sphere = 1/3. I have no idea how this is so. (θ is polar angle). I tried integrating over...
  44. M

    Equation of State: Internal Interventions

    It is known that the combination of (p, v, t) gives Equation if State. Okay, what if we have the intervention of the internal as well, i.e, U=cte*PV for example. Can this be also considered as an equation of state?
  45. F

    Why is it not possible to derive Cv from equation of state?

    Assume you have no knowledge about the microscopic information, all you know about the system of pure material is it's equation of state and number of particles. Why can't we get Cv=(∂U/∂T)V from here(if this is wrong how do we solve it?)? Why can we obtained some other quantities just from...
  46. J

    Equation of state in gravity vs microphysics

    Hi, I have a very simple question. Consider a free scalar field in the realm of GR. Then its stress energy tensor, in a Roberton-Walker Universe, is the one of a perfect fluid with pressure=denity, hence an equation of state : w=p/rho=1 However this scalar model is an archetype of maslesss...
  47. N

    Inversion Curve for a gas obeying Dieterici's equation of state

    Homework Statement For a gas obeying Dieterici's equation of state: P(V-b) = RTexp(-a/RTV) for one mole, prove that the equation of the inversion curve is P = ((2a/b^2) - (RT/b)) * exp((1/2) - (a/(RTb))) and hence find the maximum inversion temperature.Homework Equations N/A The Attempt...
  48. O

    Equation of State: Understanding f(P,v,T)=0

    I know Pv=RT, but I don't understand why my book said f(P,v,T)=0 mean? Why the function of P,v,T equals to zero? Why not equal to a constant? Thank you.
  49. 2

    Fugacity, from the virial equation of state

    Homework Statement Find the fugacity from the virial equation of state, where B is a constant. Homework Equations Z=\frac{PV}{RT}=1+\frac{B}{V} Don't know how to do underbars in TeX, but the V terms are on a per-mol basis. B is a constant and no further expansions of the EOS...
  50. J

    Help with Ideal gas equation of state

    I am reading through my thermodynamics book. Going over a air-standard otto cycle example problem. For reference the example problem is as follows: The temperature at the beginning of the compression process of an air-standard Otto cycle with a compression ratio of 8 is 5408R, the pressure...