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We can only speculate about what happens at the Planck length, we are a long way from any experiment that might probe into those length scales. This has virtually no effect on any calculations we ever do however. As such it is not something worth worrying about when trying to solve a problem which is at scales we encounter in day to day life. In your ...

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As stated above, the mass of the whole system (sugar + water) doesn't change. In addition, with "ideal" mixing, the total volume of the water plus the total volume of the sugar equals the total volume of the mixture. However, this is not a sure bet, and there are many cases of a volume of one material mixed with a different volume of water, and the total ...

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The mass doesn't change at all, it will be just the sum of the water mass and the mass Added, what happens is the change of density because the mixture, in general the molecules get closer to each other ( through the intermolecular forces) and, this way, the volume become lower to the same mass quantity, what increase the density by the equation $$\rho = ... 0 You have to take the log of a dimensionless quantity. I assume you have a model for the relationhship between D and V that looks like D=m \log V + b, but since you can only take the log of a dimensionless quantity, this should really be D=m \log \left(V/V_0\right) + b. Typically V_0 is just 1 in some choice of units, so for example, V_0 = 1 ... 0 You cannot meaningfully evaluate the logarithm of a dimensionful number. The reason for this should become clear if you attempt to series expand a logarithm with a dimensionful argument. The usual way to get around this is to write something like:$$\frac{\frac{D}{1\,{\rm Gray}}}{\log_{10}(\frac{V}{{1\,\rm cm}^3})} = 1.23 Some people prefer to omit the ...

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