# Bug in linear thermal expansion, $L_0$ must be $0$

Assume we change the temperature of an object with negligible size in $2^{nd}$ and $3^{rd}$ dimensions from $T_0$ to $T_1$ to $T_2$, with all of them pairwise different. We choose a substance with coefficient $\alpha \ne 0$.

$\alpha L_0(T_2 - T_0) = L_2 - L_0 = (L_2 - L_1) + (L_1 - L_0) = \alpha L_1(T_2 - T_1) + \alpha L_0(T_1 - T_0)$
$\Leftrightarrow L_1T_2 - L_1T_1 + L_0T_1 - L_0T_2 = 0$
$\Leftrightarrow (T_2 - T_1)(L_1 - L_0) = 0$
$\Leftrightarrow \alpha L_0(T_2 - T_1)(T_1 - T_0) = 0$
$\Rightarrow L_0 = 0$

Where is my mistake?

• The second line, 4th term should be -L0T0 instead of -L0T2. – Kevin Kostlan Sep 21 '13 at 13:45
• no, that just cancels out with the term on the other side of the equation. – Bojan Serafimov Sep 21 '13 at 13:55

In principle, you need to integrate the relevant equation for linear expansion (http://en.wikipedia.org/wiki/Thermal_expansion#Linear_expansion ). You, however, use just a linear approximation and make some conclusions based on the term quadratic in $\triangle T$. So you use the linear approximation beyond the limits of its applicability.

Of course you are wrong. The eqs in first line are ok. In the 2nd line the Kostlan comment is pertinent, imo.

An simple explanation of the whys of the formula of the linear elongation in function of the temperature change is presented PSE-here
In short: the first relation in the post reads $\alpha L_0 \Delta T=\Delta L$, for any $L_0$, i.e. $L$.
$\alpha L =\frac{\Delta L}{\Delta T}$ has the solution $$L=L_0\ e^{\alpha T}$$

This is a self-similar system;
Choosing the temperature units (ut) such that $alpha=1$, for a unit length bar ($L_0=1$), we have a simplified relation $l(x) = e^{x}$ where x is dimensionless, depending on Temperature and the material.

the elongation due to two x units variation can be calculated in several ways:
one step of 2x, i.e. a single elongation :
$l=e^2=7.3890560989$
two steps of 1x each, i.e. two successive elongations :
$l=l1+l2; l1=e^1; l2=l1*(e^1)=(e^1)*(e^1)=(e^2)=7.3890560989$
etc ,...

Note: the first relation used in the question is not an approximation, but it is the very same exponential relation I wrote. Thus, the accepted answer is completely off target, imo.

No object can have length=0 in one direction, or else the universe is impossible.