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I am reading Modern Theory of Polymer Solutions by Hiromi Yamakawa and in the context of the virial expansion the following puzzles me a bit:

"...expanded in terms of solute concentration $c$ (in gramms per unit volume)...

"...Thus, by use of the relation $ρ = N_A c/M$ with $N_A$ the Avogadro number, $c$ the solute concentration (g/cc), and $M$ the solute molecular weight..."

What exactly does unit volume mean in this context? And how can I interpret the unit g/cc?

Asking Questions
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1 Answers1

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Unit volume is a volume equal to 1 standard volume unit in the system of units you are using.

In most contexts (like this one) it shows up as "per unit volume". In this case it is because we are discussing density. Density is mass "per unit volume".

In your question they give the units $\frac{g}{cc}$. This is a bit confusing if you don't know much about unit labeling. "$cc$" is one way to say "cubic centimetres" (probably more common some places than others, in Canada I rarely see it that way besides in the medical community and engine sizing). I find it far less confusing to keep it as $cm^3$, as then it's obviously cubic centimeters.

You can also have "unit mass" and "unit time" for example, and things are often measured in "per unit mass" and "per unit time".

I can think of a couple very common and good reasons to use "unit values". Often times materials and situations can be expressed in a way where they scale linearly with some variable. Mass of an object scales linearly for an object with uniform density (mass per unit volume). This means that if you know the density of a material, and it's volume, you can calculate the weight.

It also makes it easy to compare quantities measured relative to the same unit value. An example of that is speed, measured in $\frac {m}{s}$ (metres per second). In this case the unit value is time, and you can compare the speed of two objects and determine which would travel further in the same amount of time.

They are very simple and helpful once you get used to them, and this probably explains a lot of stuff you already understood. (I'm guessing part of what confused you was the unit "$cc$")

JMac
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  • Thank you, the comment on "cc" being cubic centimeters cleared everything up. It is very confusing to call the particle density rho and the mass density c. So that is what got me off track. – Asking Questions Mar 02 '17 at 15:00
  • @JMac, Let's say I'm using SI units in my calculations. If a quantity of matter has a total volume $0.01 m^3$ then how will I be able to calculate the mass of $1 m^3$ of it? as $1 m^3 > 0.01 m^3$? – Harshit Rajput Jul 11 '22 at 09:04
  • @HarshitRajput Huh? Is it 0.01 m^3 or 1 m^3 of material, you mention both? It works the same as finding mass from any density though, multiply the volume by the density, making sure the units of volume match the units in the value of density you have (or convert accordingly). – JMac Jul 11 '22 at 21:13
  • @JMac I mean, we know that density is the mass of a unit volume, so let's say we're working with SI units and the quantity of matter that we have has a volume 0.01m^3, since density is mass per unit volume and since a unit volume in this case (since I'm using SI units) will be 1m^3, how will I be able to calculate the mass of 1m^3? I just have 0.01m^3 of material. Does it mean that 'unit volume' can even be 1cm^3 or 1mm^3 even if I'm working with SI units? It's just that at last I can convert the units? – Harshit Rajput Jul 12 '22 at 08:34
  • This is the best explanation I have read! Most people do not understand this expression. Thank you! – Fossil Doctor Apr 16 '23 at 18:23