Competition - post a photo you've taken of nature >>

Mass, Volume and Density

Hi, please could someone help answer the following question.

Question: The ratio of the masses of the ATOMS for copper and aluminium is equal to the ratio of densities of their METALS.

What can you deduce from this relationship?

The answer is that the volume of a copper atom must be roughly equal to the volume of an aluminium atom.

Please could someone explain why this must be the case and how to approach such questions as I had no idea how to approach this question and what they were looking for.

Reply 1

lerjj

Original post by Bibloski

You can work this out using the definition of density as mass per unit volume.

The difference between atoms and the whole metal is really just there as a confusion- all the mass of the metal comes from the mass of the atoms and it would be sensible to approximate the volume of the metal as simply the combined volume of the atoms.

Thus, we have two materials, with masses in the same ratio to their densities. If the statement that their volumes are equal doesn't seem to follow naturally, we can use maths:

m_1 : m_2 = \rho_1 : \rho_2

\dfrac{m_1}{m_2}=\dfrac{\rho_1}{\rho_2}

\dfrac{m_1}{\rho_1} = \dfrac{m_2}{\rho_2}

then you just note that the last line is the formula for volume on both sides.

Reply 2

Bibloski

Original post by lerjj

m_1 : m_2 = \rho_1 : \rho_2

\dfrac{m_1}{m_2}=\dfrac{\rho_1}{\rho_2}

\dfrac{m_1}{\rho_1} = \dfrac{m_2}{\rho_2}

then you just note that the last line is the formula for volume on both sides.

Thanks for your help. So basically you re-arrange the ratio so that you have the mass of the copper atom over the density of the copper metal equal to the mass of the aluminium atom over the density of the aluminium metal. Since density over mass is volume you can see that the volumes are equal. Is that right?

Reply 3

lerjj

Original post by Bibloski

Yes, although that result should be pretty intuitive. The trick in the question is to first realise that because it's talking about mass and density, you're probably going to need to rope in volume at some point. Then to get past the confusion about atoms/metal.

Reply 4

Bibloski

Original post by lerjj

The only thing I'm not quite 100% on is how do you know that the mass of the atom of a particular metal over the density of the metal means that the volumes of the atom are the same. What I mean is that we have divided the mass of the atom by the density of the metal, how is this equal to the volume of an ATOM?

Reply 5

lerjj

Original post by Bibloski

What you're essentially doing is saying that a block of metal with say 1,000,000 atoms in it has a total volume equal to 1,000,000 the volume of an atom and a mass equal to 1,000,000 times the mass of each atom. You can just ignore the difference, because the mass and volume of a block of metal is to a good approximation the same as the total volume.

It would be good if someone else could check me on this though, because obviously the approximation's not as good for volume because of small gaps. I'll have a think about whether those gaps 'cancel out' anywhere.

Reply 6

Bibloski

Original post by lerjj

Ok thanks. So are we basically saying that the density of a metal is the same as the density of the atom of that metal?

Reply 7

lerjj

Original post by Bibloski

Ok thanks. So are we basically saying that the density of a metal is the same as the density of the atom of that metal?

On the average, yes.

Or at least that it's off by a constant factor at a fixed temperature. That feel like the minimum requirement for the answer your book gives to be correct. If the density of the metals was independent of the density of the metals, this question would be unanswerable I think.