Expansion of Solids due to increases in Temperature

[jnbfive]

I was given a question about a certain material. The idea was that when it was heated from zero degrees C to 200 degrees C, it would be 10.06 mm from its original length of 10. Doing it in reverse order, length of 10.06 mm from 200 C to 0 C gave me an answer of 9.99964, which is less than the initial 10 mm. The question is, why does this happen. It also lists a description of this paradox given by H. Fakhruddin. I was wondering if anyone had a link or a pdf file that I'd be able to read in order to draw my own conclusions on this.

Another question I was given was as follows:

The volume expansion of a solid or a liquid can be written as V = V₀(1+Beta*Change in Temperature), where beta is the coefficient of volume expansion. Starting with this equation and with the definition of mass density, use the binomial expansion to show that the mass density, rho, of a substance can be written as rho = rho₀(1-Beta*Change in Temperature).

Now, I looked this up and couldn't really understand what I found. I was wondering if someone could break this down for me so that I can figure it out. I'm assuming that rho = mass/volume in this question.

 

[marcusl]

1) Your calculations are correct. The "paradox" comes from using a simple linear model

$$L=L_0*(1+T*CTE)$$,

where CTE is the linear coefficient of thermal expansion, to describe a more complex behavior.

You can verify the same "paradox" by many everyday examples. If you buy a $100 item at 10% off, you pay $90. Someone who bought it at full price didn't pay 10% more than you did, however, they paid 1(1-0.10)=11.1% more than you. The difference arises from "linearizing" a multiplicative process.

The linear model is accurate for most purposes because the CTE's are so small (the error in your case is just 36 ppm). More complex models are available when greater accuracy is needed.

2) The binomial expansion is

$$(1+\epsilon)^{-1}\approx 1-\epsilon$$.

This holds true when $$\epsilon<<1$$, so the coefficient times the temperature must be a small number.

To get the result you asked about, substitute your expression for V into rho = mass/volume and apply the binomial expansion.

 

[marcusl]

A typo in the fifth line was pointed out: it should read ".. 1/(1-0.1) gives 11.1% more.."