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Simple question, in materials publications I often see the relative change of volume in a system reported as

$$ \Delta \left (V \right )/V $$

is the denominator volume supposed to be initial or the final volume? I would assume it is the final volume as it likely parallels the relative error calculation, but I'd like to make sure.

-- It occurs to me that this is actually very likely dependent on the situation, still input would be appreciated.

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up vote 3 down vote accepted

Usually, as for example in the formula that estimates volumetric thermal expansion

$$\frac{\Delta V}{V} = \beta \Delta T$$

$V$ represents initial volume.

Actually, the real definition of volumetric thermal expansion coefficient $\beta$ is stated in the differential form

$$\frac{\text{d} V}{V} = \beta \text{d} T,$$

which means that the first expression is only an integrated version of the second one under an assumption that $\beta$ is temperature independent (and that $V$ is not a variable but initial volume). Since such coefficients are constant only for very small temperature ranges, obviously $\Delta V \ll V$, so it is almost irrelevant whether $V$ represents initial or final volume.

If however, if $\Delta V \approx V$, it would be IMHO appropriate from the author to explicitly specify which volume is represented by $V$.

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Thanks for your post. With the temperature ranges I'm working with, not to mention phase transitions, delV/V can be large (>10%). Usually delV/V is presented to give a general idea of how much the volume changes not necessarily to present the thermal expansion coefficient, again because of phase transitions. If anything, in this instance, it is more relevant to Clausius-Clapeyron equation than thermal expansion. – acadien Apr 25 '12 at 23:40
@acadien My think it would be fair to say that there is an unwritten convention that $V$ is an initial volume. If the author should think otherwise, one should explicitly tell so. – Pygmalion Apr 26 '12 at 6:50

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