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. M

    Decrease in pressure due to adiabatic expansion derivation?

    Homework Statement Show that \frac{dp}{p} =\frac{\gamma}{\gamma-1}\frac{dT}{T} if the decrease in pressure is due to an adiabatic expansion.Homework Equations Poisson equations: Pv^{\gamma} Tv^{\gamma - 1} Ideal Gas Law: Pv=R_{d}T, where R_{d} is the dry air gas constant. Hydrostatic...
  2. M

    Calculating compressed air expansion when heated

    Can anyone help me with this calculation? I'm trying to figure out how much compressed air (generated from screw compressor) will expand by a ratio of what when fed into a wood furnace. 1) Wood Furnace Fuel Input: 30kg dry wood fuel/hr 2) Compressed Air Pressure: 8 bar 3) Compressed Air Flow...
  3. D

    MHB How to Solve Inhomogeneous Differential Equations via Expansion?

    Solve inhomogeneous differential equation \[ y'' + k^2y = \phi(x) \] with homogeneous boundary conditions \(y(\ell) = 0\) and \(y(0) = 0\) by expanding \(y(x)\) and \(\phi(x)\) \begin{align*} y(x) &= \sum_na_nu_n(x)\\ \phi(x) &= \sum_nb_nu_n(x) \end{align*}...
  4. C

    Expansion of Van der waals for small pressure

    Homework Statement In the VDW eqn of state, ##(P + a/v^2)(v-b)=RT## write Pv as a function of {P,T} and by expanding the result in powers of P or otherwise show that the first terms of a Virial expansion in powers of P for a VDW gas are given by $$Pv = RT + \left(b - \frac{a}{RT}\right)P +...
  5. D

    Joule Expansion: Cooling Effects & Kinetic Energy

    I read on 27 chapter from Blundell's Thermal Physics 2nd edition. For ideal gases, Joule expansion doesn't cool gases. But for real gases, Joule expansion makes cooling effects. And this book(page 314) says that when gases expand, their potential energy of molecular interactions...
  6. K

    Determinant's solving without expansion

    Solve without expanding the determinant having elements which are listed as follows row-wise = {1^2,2^2,3^2,2^2,3^2,4^2,3^2,4^2,5^2} where , the determinant is of the order 3 I have tried some combinations of operations performed row or column wise that could produce 2 zeros either in one of...
  7. M

    Expanding Inhomogeneous Poisson Processes Using Taylor Series

    I'm at the end of a very long Poisson Processes question, involving inhomogeneous Poisson Processes. I just need to be able to expand the following expression to be able to complete the question. exp[{(sin ∏h)/∏} -h] Would anyone please be able to provide some help, with steps please!
  8. Y

    Complex analysis - partial fraction expansion

    Homework Statement Show that: Ʃ(-1)n/(n^2+a^2) (from n=0 to ∞) = pi/[asinh(pi*a)], a\neq in, n\in Z. Homework Equations f(z) = f(0) + Ʃbn(1/(z-an)+1/an) (from n=1 to ∞) , where bn is the residue of f(z) at an. The Attempt at a Solution The main problem is I don't how to pick the...
  9. J

    Please identify a series expansion

    Hi, Could someone please identify the following series expansion for me, if possible: $$f(x) = 1/h\int_x^{x+h}f(t)dt + A_{1}Δf(x) + A_{2}hΔf'(x) +...+ A_{m-1}h^{m-2}Δf^{(m-2)}(x) +r$$ where $$A_{m}$$ are, as far as I know, plain constants and $$Δ = [x, x+h]$$. I think this result was...
  10. R

    Taylor Expansion of f+df About x?

    How would one expand f+df about x? I'm messing something up in the process and can't seem to resolve it lol
  11. T

    Series Expansion: Proving ln(1+x)/x

    Homework Statement Show that 1- x/2 + x^2/3 - x^3/4 + x^4/5... (-1)^n (x^n)/(n+1) = ln(1+x)/x with |x| < 1 Homework Equations The Attempt at a Solution finding derivative of the function multiplied by x d/dx(xS(x)) = 1 -x + x^2 - x^3 + x^4 - x^5 +... absolute value of this...
  12. rakeru

    Finding a Coefficient in an Expansion (n-j is negative?)

    Homework Statement Find the coefficient of x8 in the expansion of (x2-3)7 Homework Equations This one: (n n-j) an-jxj The Attempt at a Solution Hi! Well, I know that x= x2 a= -3 n= 7 and apparently, j=8. This is what confuses me. n-j is a negative number.. how would...
  13. B

    What is the Temperature at which the Gap Between Two Bars will be Closed?

    Homework Statement A brass bar and an aluminium bar are each attached opposite each other to immovable walls. (There is a diagram, but I think the idea is clear enough). There is a gap between the two bars of 1.3x10-3m at 28°C. At what temperature will the gap be closed? coefficient of...
  14. J

    MHB 57^(th) digit in decimal expansion

    Calculation of $57^{th}$ digit in the decimal expansion of $\displaystyle \frac{1}{57}$
  15. mnb96

    Question on Taylor expansion of first order

    Hello, according to my textbook, the Taylor expansion of first order of a scalar function f(t) having continuous 2nd order derivative is supposed be: f(t) = f(0) + f'(0)t + \frac{1}{2}f''(t^*)t^2 for some t^* such that 0\leq t^* \leq 1 Quite frankly, I have never seen such a formulation...
  16. U

    Laurent Expansion for 1/(z^2-1) in the Annulus 1 < |z-2| < 3

    Homework Statement Find the Laurent expansion for \frac{1}{z^2-1} in the annulus 1 < |z-2| < 3 The Attempt at a Solution I've gotten to the last parts but getting stuck there. First I expanded the denominator and did a partial fraction decomposition and arrived at...
  17. M

    Universe expansion speed vs gravitational waves speed

    From my own(simplistic)perspective,dark energy is expanding the universe by creating further spacetime at a velocity faster than the speed of light,if gravitational waves propagate at the speed of light how is it possible that M31 and the milky way are still bound by gravity when the fabric of...
  18. D

    Binomial expansion of a function with x raised to a power

    Hey guys. So I need to know how to Binomial expand the following function \frac{1}{(1-x^{2})}. I need this because I have to work out \prod^{∞}_{i=1}\frac{1}{(1-x^{i})} for i up to 6. But I have to do it with Binomial expansion. If i can learn how to do \frac{1}{(1-x^{2})} then the rest...
  19. I

    Series expansion of an integral at infinity

    Hello, I'm fiddling with Wolfram Alpha and I can't find a definition of what do they mean by the "Series expansion of the integral at x -> inf". In particular, I have two divergent integrals and I am wondering whether their ratio is some finite number. Here it is: \left[\int_0^{\infty}...
  20. M

    Laurent Series Expansion of Electrostatic Potential

    Homework Statement Consider a series of three charges arranged in a line along the z-axis, charges +Q at z = D and charge -2Q at z = 0. (a) Find the electrostatic potential at a point P in the x, y-plane at a distance r from the center of the quadrupole. (b) Assume r >> D. Find the...
  21. C

    How Does Gas Expansion Relate to Changes in Molar Internal Energy and Enthalpy?

    Homework Statement A rigid thick walled insulating chamber containing a gas at a high pressure ##P_i## is connected to a large insulating empty gas holder where the pressure is held constant at ##P_A## with a piston. A small valve between the two chambers is opened and the gas flows...
  22. P

    Trigonometry Expansion: Expanding cos(8x) Using sin(2⋅x) or cos(2x)

    Homework Statement We know that sin(2⋅x)=2⋅sin(x)⋅cos(x)and cos(2x)=cos^2(x)−sin^2(x). Use the appropriate formula to expand cos(8x) in terms of one of these two formulas. Homework Equations sin(2⋅x)=2⋅sin(x)⋅cos(x) cos(2x)=cos^2(x)−sin^2(x) The Attempt at a Solution I am not...
  23. H

    Adiabatic Expansion: Constant pressure and temperature

    I'm new to the forum, so please be kind. I was reading through my pchem textbook, and I noticed something. We're given the equation: ΔU = q + w For an adiabatic expansion, we're told that q = 0. Fair enough, no heat transfer. But when there is a constant T and change in V, my book...
  24. marcus

    Can Observations Prove the Existence of Quantum Gravity in the Early Universe?

    Frank Wilczek and Larry Krauss point out (what is perhaps obvious but nevertheless important) that the universe acts as a giant magnifying glass for observing Quantum Geometry effects. Because of its expansion the cosmos, they point out, can serve as a classical detector of microscopic QG...
  25. G

    Exploring Universe's Expansion | Georgia's Question

    Hello. I am very curious about astronomy, and nothingness. I have a question though: Many people say the the universe is expanding, and say that beyond the universe, there is "nothing". What I think though, is that "something must exist beyond all that is already here, and this something I think...
  26. D

    Determinants and taylor expansion

    I'm doing a proof, and near the last step I want to write the expression, \frac{d}{dt} \det{A(t)} = \lim_{\epsilon \to 0} \frac{\det{(A+\epsilon \frac{dA}{dt})} - \det{A}}{\epsilon} which produces the right answer, so I believe that it may be correct. This looks very much like a Taylor...
  27. E

    Space expansion, gravity waves, warp

    If gravity waves are supposed to propagate at c, how could the theoretical warp drive wave propagate at greater than c values? I have heard and red and understand that space itself can expand (and warp? ) faster than c and, in fact, that will happen in the future when space in between...
  28. C

    Decimal Expansion Homework: Terminating & Non-Terminating 9's

    Homework Statement How many decimal expansions terminate in an infinite string of 9's? How many dont? The Attempt at a Solution If we have a number terminate with an infinite amount of 9's then it will be a rational number. So there would be countably many of these. And since...
  29. D

    Expansion of a gas against a piston

    So let's say a gas is doing some work against a piston and decreasing its internal energy. Does the gas also do work against gravity if it is expanding in the positive y direction? If so would it even be measurable? Thanks for the help
  30. A

    Isothermal Expansion and the 2nd Law

    The book that I use (Concepts in Thermal Physics by S. and K. Blundell) states the second law in two ways. The way they state the Kelvin version is "no process is possible whose sole result is the complete conversion of heat into work." How does that fit in with the isothermal expansion of an...
  31. M

    Help Solving a Volume Expansion Problem

    I have no idea how to solve this problem. I feel like I need more information. For one the new height of the liquid should depends on the volume expansion of the glass tube. I am not given any dimensions of the glass tube other than it's height. Therefore if I am to believe that only the...
  32. zhangwfjh

    On representation of expansion in fluid flow

    In Helmholtz original thesis On integrals of the hydrodynamical equations, which express vortex-motion, he mentioned in the first section that the change undergone by an arbitrary infinitesimal volume of water under the time dt is composed of three different motions. One of them is an expansion...
  33. Fernando Revilla

    MHB Verification of Limit Using Taylor Expansion: x-ln(1+x)/x^2 = 1/2

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  34. R

    An infinite universe is necessarily in expansion

    What do you think of this argument? Lets suppose an infinite and eternal universe that is homogenous on a large scale. Since there is no privileged inertial reference frame, regardless of the chosen inertial reference frame, the observed statistical velocity distribution of matter is nearly...
  35. C

    Expansion of gas through valve

    Homework Statement A thick walled insulating chamber contains ##n_1## moles of helium gas at a high pressure ##P_1## and temperature ##T_1##. It is allowed to leak out slowly to the atmosphere at a pressure ##P_o## through a small valve. Show that the final temperature of the ##n_2## moles...
  36. T

    Metric expansion in terms of planck pixels

    what if the evolution of the universe over time and the expansion of space a series of binary events such that the same quantity of energy is divided geometrically from one into two, four, eight, sixteen pieces, such that each Planck time is equivalent to one such split and each Planck length is...
  37. D

    Could the Universe Be Shrinking Instead of Expanding?

    greetings gentlemen, i found this interpretation of red shift observations interesting: http://arxiv.org/abs/1303.6878/ my question is whether any new understanding can be gained by equating this increasing-mass explanation to the old expansion theory, if they both provide accurate...
  38. T

    A question relating to expansion rate of the Universe

    1. Forgive me for this novice question, but can variation in the expansion rate of the universe, be partially explained by an increasing number of stars turning mass into energy, hence reducing gravitational attraction between galaxies, and at the same time applying radiation pressure (a small...
  39. J

    Is the expansion of the universe relative to our clocks and rulers?

    Is the expansion of the universe relative to our clocks and rulers? If so, it seems correct to include relativistic effects. And yet I am told that this is wrong. Regards, Jack Martinelli
  40. Z

    'Expansion' of fluid world lines

    hello In MTW excercise 22.6, given a fluid 4-velocity u, why the expression : ∇.u is called an expansion of the fluid world lines ? Is the following reasoning correct ? We know that the commutator : ∇BA - ∇AB is (see MTW box 9.2) is the failure of the quadrilateral formed by the vectors...
  41. A

    Understanding the Expansion of P(A)

    P(B/A) = P(A/B).P(B) / P(A) Later we expand P(A) as P(A/B).P(B) + P(A/B).P(B) ... B is complement of B I don't understand how we can expand P(A) like that. Doesn't that assume that A ℂ B?
  42. J

    Thermal expansion of aluminum disc

    I have an aluminum disc that is 15in in diameter and about 1in thick. I am going to put it in a 400°F oven and I need to know how much the diameter will expand. One of the engineers I work with(i'm an intern) is using what I found to be the linear expansion equation (ΔL/Li=αΔT)...so Length...
  43. C

    What is the complete expansion of |1+a|^2 for a complex number a?

    Hi, Could you help me to understand the following expansion I found in a book of qunatum mechanics. |1+a|^2=1+a+a*+... where a* is the complex conjugate of a
  44. R

    How can we prove the work done by gas expansion in any deformable container?

    In my textbook W=∫p.dV is only proved for a syringe with a piston. This is quite easily done but the book never explains how it extrapolates to the general situation for a gas expanding in any deformable container. It seems the point is to prove dV= S.h where S is the surface area of a given...
  45. L

    On the Expansion of Exponential Function by Integration

    I ask members here kindly for their assistance. I'm having some confusion over the process of integrating inequalities, in particular for obtaining the series expansion for the exponential function by integration. The text by Backhouse and Holdsworth (Pure Mathematics 2), shows the expansion of...
  46. S

    Simple exposition of universal expansion law in General Relativity

    Hello. For all those to whom the concepts of general relativity, big bang and universal expansion are not clear, I just updated my page of cosmology. There I presented a rather simple proof of the Friedmann equations, that are the particular case of the Einstein's equation of General Relativity...
  47. J

    How to Expand Finite Series in a Shorter Form?

    Hello :blushing: How to do expand this: (\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 - t)^2 where X(t_j)-X(t_{j-1}) = \Delta X_j to this: (\sum_{j=1}^{n}(\Delta X_j)^4 + 2*\sum_{i=1}^{n}\sum_{j<i}^{ }(\Delta X_i)^2(\Delta X_j)^2 -2*t*\sum_{j=1}^{n}(\Delta X_j)^2+t^2I get near the North Pole... but...
  48. E

    Cauchy expansion of determinant of a bordered matrix

    The Cauchy expansion says that \text{det} \begin{bmatrix} A & x \\[0.3em] y^T & a \end{bmatrix} = a \text{det}(A) - y^T \text{adj}(A) x , where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar. There is a proof in Matrix Analysis by Horn and...
  49. T

    How Do You Find the Power Series Expansion of \( e^z \) at \( \pi i \)?

    "[F]ind the power-series expansion about the given point for each of the functions; find the largest disc in which the series is valid. 10. ##e^{z}## about ##z_{o} = \pi i##" (Complex Variables, 2nd edition; Stephen D. Fisher, pg. 133)$$f(z) = e^{z} = e^{z-a} \cdot e^{a} = e^{a} \cdot \sum...
  50. C

    Matching Inner and Outer Expansions for Approximating ODE Solutions

    I'm trying to approximate f'(r) for the following equation using matched asymptotic expansions -\frac{1}{2}\epsilon ff''=\left[\left(\epsilon+2r\right)f''\right]' where \epsilon \ll 1 and with the boundary conditions f(0)=f'(0)=0, \quad f'(\infty)=1 The inner expansion which satisfies...
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