Computation of 2nd and 3rd Virial Coefficients

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Discussion Overview

The discussion revolves around the computation of the 2nd and 3rd Virial coefficients for Argon (Ar) at 300K, focusing on the methodology of fitting data to determine these coefficients. The scope includes aspects of statistical fitting and the implications of using different numbers of terms in polynomial fits.

Discussion Character

  • Homework-related
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents their approach using Maple to fit data for the Virial coefficients, resulting in a polynomial with seven terms.
  • Another participant questions the necessity of fitting with seven terms when only the 2nd and 3rd coefficients are required, suggesting that fitting too many terms may lead to loss of information.
  • A participant defends their choice of fitting seven terms, stating that it provides a good fit and clarifies that reducing the fit to only include the 2nd and 3rd coefficients would still allow for the calculation of Z.
  • There is a discussion about the nature of polynomial fitting, where one participant notes that using more coefficients can lead to overfitting, making the polynomial less useful between data points.
  • Another participant agrees with the concern about overfitting and emphasizes that fitting more coefficients than necessary can lead to misleading results.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of fitting with seven terms versus a more limited approach. There is no consensus on the optimal number of terms to use in the polynomial fit.

Contextual Notes

Participants highlight the potential pitfalls of overfitting and the implications of using a polynomial with a number of coefficients equal to the number of data points, which could lead to a curve that fits the data exactly but lacks predictive power.

amattiol
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Homework Statement


Compute the 2nd and 3rd Virial coefficients at 300K of Ar. Experimental parameters are mentioned in xlist and ylist below :)


Homework Equations


My professor has given a hit, which is as follows: "Use the data to determine the (z) compressability (sp?) and plot Z as a function of the reiprocal of molar volume."


The Attempt at a Solution



In Maple this is where I am at:

xlist := [.4, .5, .6, .8, 1, 1.5, 2, 2.5, 3, 4];
ylist := [6.2208, 4.9736, 4.1423, 3.1031, 2.4795, 1.6483, 1.2328, .98357, .81746, .60998];
xylist := zip(proc (x, y) options operator, arrow; [x, y] end proc, xlist, ylist);
xylist:= [[0.4, 6.2208], [0.5, 4.9736], [0.6, 4.1423], [0.8, 3.1031], [1, 2.4795], [1.5, 1.6483], [2, 1.2328], [2.5, 0.98357], [3, 0.81746], [4, 0.60998]]

Then I have done non linear fitting for 7 Virial coefficients with the resulting equation of fit:
f(x) =0.733475918379429/x^6-5.86019434244588/x^5+18.1883313118305/x^4-27.6759879335724/x^3+21.3651589091868/x^2-5.25691168982818/x+1
 
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Why fit with 7 terms when you are asked for only 3?

Not strong on the statistical theory but I think you positively lose information by fitting to too many terms.
 
epenguin, I fit 7 terms because it was such a good fit I couldn't resist keeping it fit to the 8th coefficient. If I reduce the fit eqn to z = 1 + B/Vbar + C/(Vbar^2) then B and C would be the 2nd and 3rd coefficients correct? Then I use those to calculate Z?
Thanks for the response...I realize I didn't format the question very well.
 
When fitting polynomials, the more coefficients you use, the better fit you will get. When number of coefficients equal number of data points the curve will go exactly through all the points (think why), but it will be useless between points.
 
Borek said:
When fitting polynomials, the more coefficients you use, the better fit you will get. When number of coefficients equal number of data points the curve will go exactly through all the points (think why), but it will be useless between points.
:approve:
Exactly what I was about to say. and fitting 10 points with 6 coefficients you are getting that way.
 

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