Atomic mass unit and Molar mass

[SpartanG345]

Hi

My textbook states that say 50 amu/atom = 50 g/mol.
Which does not really make any sense. I am not sure how to get the atom unit.

Since on wikipedia it says u = Molar-mass/Na

Molar-mass(g/mol) = u*Na

I also know that 1gram/Na = 1u

 

[morrobay]

Vanadium, atomic mass 50, 50g/mole
How many atoms in a mole ?

 

[SpartanG345]

The number of atoms in a mole is Na - Avogadro's number

 

[Borek]

What do you get when you divide molar mass by Avogadro's number? Mass of what?

 

[SpartanG345]

oh i see

you should get the mass of one atom/molecule

g/mole / Atoms/mole = g/atom which is u atomic mass unit

so g/mole = u*atom/mole
g/atom * atom/mole = g/mole

therfore i find g/mole = U*Na

but i am not sure what amu/atom is

 

[Borek]

amu/atom is a mass of atom express in amu units. Just like you can express mass in kg or pounds or stones, you can express it in amu.

 

[SpartanG345]

so does amu represent the number of grams per atom?
I think this is wrong... on wiki it says the unit of amu if kg - where 1 u is 1/12 the mass of C12

but if you divide molar mass M by Avagadros number Na - i think you get mass in amu

(g/mole) / (atoms/mole) = g/atom

since amu unit is measured in g this is equivalent to amu/atom - which i thaught would be multiplied by some conversion factor when
you change g/atom into amu/atom, which in this case is 1.

But i am still surprised that dividing molar mass by Na will get u/atom i am not sure why that happens..

 

[Borek]

so does amu represent the number of grams per atom?

No. It represent mass. Period.

I think this is wrong... on wiki it says the unit of amu if kg - where 1 u is 1/12 the mass of C12

Exactly. Mass. Period.

but if you divide molar mass M by Avagadros number Na - i think you get mass in amu

(g/mole) / (atoms/mole) = g/atom

you get mass of atom in g, hence per atom. This is subtle - usually when we say "mass of car" we think in kg, not kg/car. But it is exactly the same, no matter what is the object you weigh.

But i am still surprised that dividing molar mass by Na will get u/atom i am not sure why that happens..

amu was selected in such a way that mass of atom expressed in amu is identical to mass of mole expressed in grams. That's all.

Imagine new mass unit, barble. Barble is a mass of a single marble. If you have a bag if marbles, its mass - expressed in barbles - equals number of marbles in the bag. Why? Because we defined barble so that is happens.

 

[SpartanG345]

Ok i see

i guess its do with defining Na and defining the mass of amu with respect to C12.

Thanks for your time :)

 

[morrobay]

Make this analogy:
Substitute Avogadros number with one dozen, 12
Give 1 amu value of 1/12 g
Denote Hydrogen 1 amu. So the amu of H expressed in grams contains: 1g/1/12g = 1 dozen
Denote Carbon 12 amu , 12 (1/12g) = 1g So 12 grams Carbon = 1 dozen
Denote Magnesium 24 amu , 24 (1/12g) = 2g And 24 grams Magnesium = 1 dozen

 

[Dickfore]

Today, the "molar" quantities are all defined through the isotope ¹²C. The choice was made because it is useful in mass spectroscopy. It used to be defined through hydrogen (or oxygen). Since the mass of the hydrogen atom is approximately twelve times smaller than the mass of a carbon atom, we use the factor of 1/12 everywhere.

Ok, here come the definitions:

1. An atomic mass unit (a.m.u.) or unified atomic mass (u) is a unit of mass equal to $\frac{1}{12}$ of the mass of the atom of the isotope ¹²C. This means that the mass of one atom of ¹²C is exactly 12 u;
   
2. A mole is a quantity of substance that contains as many units as there are atoms in 12 grams of ¹²C.
   
3. Avogadro's number $N_{A}$ is the number of atoms in one mole.

From 1, 2 and 3 it follows that:

$$12 \, u \times N_{A} = 12 \, \mathrm{g} \Rightarrow u = \frac{1 \, \mathrm{g}}{N_{A}}$$

Until we find the exact value of Avogadro's number, we don't know what the numerical value of the unified atomic mass (or atomic mass unit) in grams is. Nevertheless, the above equality always holds. Also, everything would have worked if we simply substituted the 12 everywhere with 1 (for hydrogen) or 16 (for oxygen), since these factors would always cancel exactly.

We can measure the masses of atoms or molecules in units of u. Then, we say that they are the relative atomic or molecular masses:

$$A_{r} = \frac{m(\mathrm{atom})}{u}, \; M_{r} = \frac{m(\mathrm{molecule})}{u}$$

Now comes the business of the unit of mole. It is very inconvenient to always work with this abstract quantity of $N_{A}$ atoms. Also, since the advent of the atomic structure of matter, people saw that the law of conservation of mass in atomic reaction is a consequence of a more stringent law: The law of conservation of the number of atoms of each element in a chemical reaction (since chemical reactions have to do with the interchange of electrons, but do not alter the nuclei). This law is not valid in nuclear reactions. So, instead of quantifying the amount of substance through its mass, we can quantify it through its number of units and use the Avogadro's number as a unit. Then we say:

A mole (unit symbol mol, dimension N) is the quantity of substance (usually dentoted by the symbol n) that contains Avogadro's number of units.

If there are N units of something, then the quantity of substance of that something is found by:

$$n = \frac{N}{N_{A}}$$

However, n is a quantity with dimension of quantity of substance ($[n] = \mathrm{N}$, N is unitalicized, because it is a symbol for dimension) and N (italicized because it is a symbol for a physical quantity) is a number and, therefore, a dimensionless quantity. To preserve dimensional consistency, we must have $[N_{A}] = \mathrm{N}^{-1}$. Because this constant is with dimensions, it is called Avogadro's constant, in contrast to Avogadro's number. This ambiguity in symbols often leads to confusion:

$$N_{A}(\mathrm{constant}) = N_{A}(\mathrm{number}) \, \mathrm{mol}^{-1}$$

Now, you may define molar masses:

$$M = \frac{m}{n}, \; [M] = \mathrm{M} \, \mathrm{N}^{-1}$$

There is a connection between molar masses and relative masses. We can derive it from the above definitions, by simply expressing the quantity of substance through the total number and the total mass through the mass of one unit (atom, molecule) and the total number:

$$M = \frac{m_{a} \, N}{\frac{N}{N_{A}(\mathrm{constant)}}} = m_{a} \, N_{A}(\mathrm{constant}) = A_{r} (M_{r}) \, u \, N_{A}(\mathrm{constant})$$

But:

$$u \, N_{A}(\mathrm{constant}) = \frac{1 \, \mathrm{g}}{N_{A}(\mathrm{number})} \, N_{A}(\mathrm{number}) \, \mathrm{mol}^{-1} = 1 \frac{\mathrm{g}}{\mathrm{mol}}$$

Therefore:

$$M = A_{r}(M_{r}) \, \frac{\mathrm{g}}{\mathrm{mol}}$$

Notice that we don't need to know the exact value of Avogadro's number, since it cancels exactly in the above relation.