Expansion Definition and 1000 Threads

Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.Temperature is a monotonic function of the average molecular kinetic energy of a substance. When a substance is heated, molecules begin to vibrate and move more, usually creating more distance between themselves. Substances which contract with increasing temperature are unusual, and only occur within limited temperature ranges (see examples below). The relative expansion (also called strain) divided by the change in temperature is called the material's coefficient of linear thermal expansion and generally varies with temperature. As energy in particles increases, they start moving faster and faster weakening the intermolecular forces between them, therefore expanding the substance.

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  1. I

    Thermal expansion of a liquid in a cylindrical container

    Homework Statement There is liquid in a cylindrical container at some level. Now when the container is heated, the level of the liquid remains the same in the container. What is the relationship between the coefficient of linear thermal expansion of liquid and the container ? Homework...
  2. F

    How to Expand Noncommuting Variables in a Formal Power Series?

    Homework Statement Need to show that [a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger} Homework Equations [a,a^\dagger]=1 The Attempt at a Solution Need to expand f(a,a^\dagger) in a formal power series. However I don´t know how to do it if the variables don´t commute.
  3. T

    B Is the Expansion Rate Truly Accelerating?

    I'm studying about accelerating expansion rate. I am reading that the rate of expansion is accelerating. Now the farther objects are from us the faster they are moving away from us due to expansion. Farther and farther, faster and faster. But that would occur even if the expansion rate is...
  4. E

    Thermal expansion over a large temperature range

    So I've been working at a steel mill where we deal with billets cooling from temperatures around 1200 C to between 10-25 C. I have access to average thermal expansion coefficients over this temperature range. First question: Over a large temperature range as aforementioned am I correct in...
  5. T

    B What is the nature of spacetime?

    I've been trying to learn about cosmological expansion (some weeks ago), I think I understand as much as any lay-person could, regarding why everything is moving away from our galaxy. However I still don't understand what spacetime is. The fact that space can deform indicates to me that...
  6. R

    B CO2 Expansion: Will Temperature Fall Below 0C?

    Hi .I'm a beginner at thermodynamics,very limited in my understanding of math.I have a handbook on CO2 properties.According to it the properties for CO2 at 47.3 bar are T=12C,H= -3792,S= -42.6,and for 34.85 bar, T= 0 C,H= -3339.8,S=-39.33.If the gas expands from 47.3 to 34.85 bar,the gas will...
  7. DaTario

    I Non unicity of decimal expansion and extremes of intervals

    Hi All, The famous proof of the theorem: ## 1 = 0.9999999...## seems to point to a statement more or less like this: "There is no uniqueness in decimal expansions of real numbers, specially if one wishes to compare numbers (and their decimal expansions) extremely close of one another." Is this...
  8. T

    Exploring Expansion of Space/Time Framework

    I know expansion explains the size of the space/time framework. Is the "expansion" we see between objects relative to each other (objects seem to be getting farther away in all directions) due to continued expansion of space/time or is there also an inter-framework explosive (for lack of a...
  9. U

    Find the first 3 terms of the asymptotic expansion of Jn(x)

    Homework Statement The bessel function Jn(x) is defined by the integral Jn(x)=1/(inπ)∫0πeixcosφcos(nφ)dφ From this formula, find the first 3 terms of the asymptotic expansion of Jn(x) when x=n and n is a large positive integer. Homework EquationsThe Attempt at a Solution I tried combining the...
  10. Z

    Stat. States, Orthonorm. expansion coefficients etc qn?

    I'm reading through some lecture notes for QM in a subsection about stationary states where the definition of orthonormality involving a kronecker delta \int_{-\infty}^{\infty}\psi_n(x) \ \psi_m^*(x)dx=\delta_{m,n}and the formula for some wavefunction that is a superposition of energy...
  11. T

    Gravitational waves, distance and space-time

    So the discovery of gravitational waves observed a contraction and stretching of space-time, and I've been trying to understand how the expansion of the universe means that space itself is growing. I want to understand how this actually works in relation to 'things' like a photon and an atom...
  12. H

    I Does the Universe's Expansion lead to Critical Mass < 1?

    We understand that for the Universe to be flat, the critical mass of everything that has mass must total an omega of 1.0. At first, with our lack of knowledge regarding dark energy, we thought that the universe that we detected (ordinary matter and dark matter) did not even add up to a third of...
  13. TheSodesa

    A difficult series expansion (finding a limit)

    Homework Statement Find \lim_{x \to 0}\frac{ln(1+x^2)}{1-cos(x)} by using series representations. Check using L'Hospitals rule. Homework Equations Taylor polynomial at x=0: \sum_{k=0}^{\infty}\frac{f^{k}(0)}{k!}(x)^{k} = f(0) + f'(0)(x) + f''(0)x^{2} +... The Attempt at a Solution Using...
  14. Korbid

    Virial expansion: Potential that depends on velocities

    I'm studying this potential that depends on positions and velocities. http://motion.cs.umn.edu/PowerLaw/ E(\tau)=\frac{k}{\tau^2}e^{-\tau/\tau_0} where τ is the time to collision. By extrapolation \tau=\frac{b-\sqrt{d}}{a} a=||\vec{v}_{ij}||^2 b=-\vec{r}_{ij}\cdot\vec{v}_{ij}...
  15. Titan97

    Thermal expansion of a rod with variable alpha

    Homework Statement The coefficient of linear expansion of a rod of length 1 meter (at 300K) varies with temperature as ##\alpha=\frac{1}{T}##, where T is the temperature. Find the increment in length when the rod is heated from 300K to 600K Homework Equations $$\Delta L=L\alpha \Delta T$$ The...
  16. dakota224

    Problem: using thermal expansion to calculate sea level rise

    Homework Statement Since the 1980's a total of 20 x 10^22 J of additional energy has been absorbed by the worlds oceans. This is 90% of the additional heat our planet is storing as a consequence of global warming. The surface area of the worlds oceans is 361,900,000 km^2 and we will assume this...
  17. Q

    DiffEq, Binomial Expansion and limits

    Homework Statement Use algebra to show that U(x) = −√x − 1 and L(x) = −√x satisfy the ’funnel condition’ U(x) − L(x) → 0 as x → ∞ Homework Equations Funnel condition: The two fences come together asymptotically, i.e. U(x) − L(x) is small for large x. The Attempt at a Solution I think that...
  18. T

    Observational evidence for expansion of universe....

    Is there any observation evidence for the expansion of the universe besides redshift?
  19. empty_set

    Change in density due to thermal expansion

    Homework Statement I want to find the mathematical proof to show that the density of an object changes with thermal expansion. My professor showed this in class and it was horribly wrong because he let a few things out. The book I'm reading showed it in 4 steps and left out a lot of the crucial...
  20. ognik

    Wave Packet Expansion Homework - Understand Abbreviations

    Homework Statement The text states: "Let us consider a wave packet whose Fourier inverse ##\phi (\vec{k})## is appreciably different from zero only in a limited range ##\Delta \vec{k}## near the mean wave vector ##\hbar \vec{\bar{k}} ##. In coordinate space, the wave packet ##\psi(\vec{r}, t)##...
  21. ognik

    MHB Wave Packet Expansion: Understanding $\omega (\vec{k})$

    The text states: "Let us consider a wave packet whose Fourier inverse $\phi (\vec{k})$ is appreciably different from zero only in a limited range $\Delta \vec{k}$ near the mean wave vector $\hbar \vec{\bar{k}} $. In coordinate space, the wave packet $\psi(\vec{r}, t)$ must move approximately...
  22. Hepth

    Puiseux/Taylor Expansion of an Integrand pre-Integration

    My problem : I have a function that I want to integrate, in the limit that a parameter goes to zero. I have a function ##f[x,r]## I want to compute ##F[r] = \int dx f[x,r]## and then series expand as ##r \rightarrow 0## This is impossible algebraically for me, but may be possible if I can...
  23. Daniel Lobo

    Finding a limit using power series expansion

    Homework Statement The problem wants me to find the limit below using series expansion. ##\lim_{x \to 0}(\frac{1}{x^2}\cdot \frac{\cos x}{(\sin x)^2})## Homework EquationsThe Attempt at a Solution (1) For startes I'll group the two fractions inside the limit together ##\lim_{x...
  24. T

    Expansion of the universe and quantum perturbations

    So I'm just learning about the higgs, electro-magnetic, strong and weak nuclear fields/forces. Is dark energy one of those? I understand there is as difference in the universe expanding from expansion and everything 'receding' from dark energy. I was watching a UC video on expansion, the Hubble...
  25. S

    Adiabatic Expansion Pressure Temperature Relation

    Homework Statement The problem is in the context of convection in the troposphere Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation \frac{dT}{dP} = \frac{2}{f+2} \frac{T}{P} Homework Equations Ideal gas law PV = nRT...
  26. Q

    Question Concerning the Rate of Expansion

    I admit that I do not have much knowledge pertaining to relativity and space expansion, however I do know that all speeds are relative excluding light. This makes sense to me, however a new article published says that "Space itself is pulling apart at the seams, expanding at a rate of 74.3 plus...
  27. D

    Series expansion for 2D dipole displaced from the origin

    I learn that we can expand the electric potential in an infinite series of rho and cos(n*phi) when solving the Laplace equation in polar coordinates. The problem I want to consider is the expansion for the potential due to a 2D line dipole (two infinitely-long line charge separated by a small...
  28. Karan Punjabi

    What causes the anomalous expansion of water?

    Guys, I was studying thermal expansion and i saw an unusual example of expansion in water when i asked my teacher what's the reason behind it, he says its advanced chemistry. Can someone explain this unusual behavior of water?
  29. Ontophobe

    Accelerating or Decelerating Universe Expansion?

    The farther out we look, the faster the galaxies are moving away from us. Closer galaxies are moving away from us, but not as quickly. We reason that when they're as far away as the more distant ones, they'll be going that fast, too. So we infer that the universe's expansion is accelerating...
  30. Cora

    Ideal Gas Expansion State Properties & Exergy Balance

    Homework Statement Two well-insulated rigid tanks of equal volume, tank A and tank B, are connected via a valve. Tank A is initially empty. Tank B has 2 kg of Argon at 350 K and 5000 kPa. The valve is opened and the Argon fills both tanks. State 2 is the final equilibrium state. The temperature...
  31. A

    I Expanding Universe: Big Bang & Beyond

    Hi all, After the 'big bang' it is said that space is expanding. Atom is almost 99% space then why are they not expanding, same can go with all the stars and galaxies and all stuffs. Hope this is not sounding crazy. If so pls forgive me. Cheers.
  32. T

    Expansion of the universe, acceleration limit

    So if the universe expansion is accelerating due to dark energy, does that mean that (assuming there is) one end of the universe relative to the other end of the universe will see it moving away at speeds greater than the speed of light? Or is the expansion capped by relativity? Or does the...
  33. C

    Is There Evidence for Redshift Drift in High-z Galaxies?

    I am interested in locating information of the evidence of z evolution that has been collected of the spectra of relatively high z sources over the previous 75(35) years. Before I get too far into the question, I thought I would try to find what research might have been done on the subject...
  34. S

    Taylor expansion of a scalar potential field

    Consider the potential ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##, where ##\phi## is a scalar field and the mass dimensions of the couplings are: ##[\lambda]=0##, ##[a]=1##, and ##[\epsilon]=4##. Expanding the field ##\phi## about the point...
  35. sinkersub

    Inverse Binomial Expansion within Laurent Series?

    Homework Statement Find the Laurent Series of f(z) = \frac{1}{z(z-2)^3} about the singularities z=0 and z=2 (separately). Verify z=0 is a pole of order 1, and z=2 is a pole of order 3. Find residue of f(z) at each pole. Homework Equations The solution starts by parentheses in the form (1 -...
  36. A

    Does mass slow the expansion of space?

    I read that the mass of the universe slows the expansion of the universe. I also heard that if the mass of the universe was above a critical value the universe would contract. I believe this is called the big crunch scenario. And that it is currently not considered a probable scenario. Is this...
  37. U

    Gravity wave cancellation cause space expansion?

    Is it possible that the increased acceleration of the universe could be caused by the cancellation of gravity waves? If not, what would be the impact on space-time when gravity waves do cancel out?
  38. B

    Taylor expansion with multi variables

    I was reading a book on differential equations when this(taylor expansion of multi variables) happened. Why does it not include derivatives of f in any form? The page of that book is in the file below.
  39. L

    Conservation of energy and space expansion

    This had me thinking for a while. Imagine a photon emitted by a very distant object at a redshift of z = 2.0 for example. As the photon travels through space, due to space expansion the photon's wavelength will shift towards red. With an increase in the wavelength there must come a decrease in...
  40. DaveC426913

    Accelerating Expansion and Mediocrity Priniciple

    This other thread got me reading about metric expansion and this graph: https://en.wikipedia.org/wiki/Metric_expansion_of_space#/media/File:Universe.svg I am possibly misreading the graph, but it seems to suggest that, if we are in an expanding universe (large dotted line), and curvature is...
  41. EdColider

    How does the expansion of the universe work?

    How does universe expansion work? I thought that the universe was infinite and the celestial corps were getting further distance from each other. If the universe is infinite, how does someone calculate something when infinity is getting bigger? From a reference point? Is the rate of expansion...
  42. 8143902066

    Universe Expansion: Is it Possible Beyond Light Speed?

    I had seen somewhere that Universe is expanding with a speed greater than Light Speed... it really sounds absurd for me... As per my sense, light speed is the universal acceptable speed limit and we can't get back if we reach that speed until any other thing outside the object makes it to...
  43. J

    Relationship Between Spatial Expansion and Gravity's Force?

    Fellow Nerds, I'm looking for a quantitative relationship between the gravitational strength of a point on a field and the speed of expansion of space at that point. Given a cosmological constant and a metric, is it possible to pinpoint a certain point of space and ask how quickly that space is...
  44. G

    Expansion of the Universe and Thermodynamics

    Hello, I have a very basic question about the expansion of the Universe and it could be that I'm being very stupid here: if the universe is expanding into the empty surroundings as the red shift evidence seems to demonstrate, then surely the universe must be doing work on the surroundings like...
  45. N

    Measuring space expansion without redshift

    Hi, Apart from the redshift of electromagnetic waves (which we do observe), is there any other evidence of space expansion that is not based on redshift? If not, is there a theoretical experiment that could let us observe and measure space expansion? Here I am assuming that space is not just...
  46. T

    Taylor Expansion to Understanding the Chain Rule

    I don't understand this as isn't according to chain rule, . So where is the in the above derivative of F(t)? Source: http://www.math.ubc.ca/~feldman/m226/taylor2d.pdf
  47. ErikZorkin

    I Space expansion and Universe as computation

    Good day. I do not know much about cosmology, rather computer science, but the following theoretical question bothers me a little. Some scientists, like Tegmark, Wolfram, Zuse or Fredkin, support the idea that the Universe might be just computation. Computable means that something can be...
  48. D

    Question about cooling due to expansion of compressed air

    Hi! I'm a teacher in Mechanical Engineering, and I organize a (yearly) student's project where they design and create a cooling machine. The machine works by compressing air, cooling the compressed air, and then rapidly expanding it. This can lead to temperatures of minus 7 degrees Celsius. I...
  49. Searay330

    Thermal Expansion linear vs area expansion

    this is a picture of my notes for thermal expansion for linear vs area. my question is why does the area coefficient of expansion for the area = 2(liner coefficient of expansion). any insight would be appreciated.
  50. vetgirl1990

    Adiabatic Expansion of a Gas: Final Pressure-Volume Product Calculation

    Homework Statement A gas consisting of diatomic molecules that can rotate but not oscillate at a given range of temperatures expands adiabatically from pressure of 365Pa and volume of 70m3, doing 101J of work, while expanding to a final volume. What is its final PV (pressure volume) product...
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