# Question d90c6

Jul 5, 2016

See explanation.

#### Explanation:

Let's start with the atomic mass and use carbon as an example.

Interestingly enough, what you're describing is actually the relative atomic mass, ${A}_{r}$, not the atomic mass, ${m}_{a}$, of carbon.

The atomic mass tells you the actual mass of a given atom and is measured in unified atomic mass units, $\text{u}$.

A unified atomic mass unit is defined as the mass of one proton or of one neutron present inside an atom's nucleus. More specifically, one unified atomic mass unit is equal to the mass of $\frac{1}{12} \text{th}$ of a single unbound carbon-12 isotope in its ground state

$\textcolor{p u r p \le}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\text{1 u" ~~ 1.660539 * 10^(-24)"g}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

In this example, the atomic mass of carbon is actually $\text{12.01 u}$, meaning that single carbon atom has a mass of

12.01 * overbrace(1.660539 * 10^(-24)"g")^(color(blue)("= 1 u")) ~~ 1.99431 * 10^(-23)"g"

The relative atomic mass of carbon is the atomic mass divided by the unified atomic mass unit

${A}_{r} = {m}_{a} / \text{1 u}$

Since atomic mass is expressed in unified atomic mass units, the relative atomic mass of a given atom will be unitless. In this example,

${A}_{r} = \left(12.01 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{u"))))/(1color(red)(cancel(color(black)("u}}}}\right) = 12.01$

Now, the molar mass of an element is simply the mass of one mole of atoms of said element. As you know, the conversion factor that takes you from moles to number of atoms looks like this

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \text{1 mole" = 6.022 * 10^(23)"atoms} \textcolor{w h i t e}{\frac{a}{a}} |}}} \to$ Avogadro's number

So, in the case of carbon, you know that a single atom has a mass of

$\text{12.01 u" = 12.01 * 1.660539 * 10^(-24)"g" = 1.99431 * 10^(-23)"g}$

which means that the mass of one mole of carbon atoms will be

6.022 * 10^(23)color(red)(cancel(color(black)("atoms of C"))) * (1.99431 * 10^(-23)"g")/(1color(red)(cancel(color(black)("atom of C")))) = "12.0097 g" ~~ "12.01 g"#

Since the molar mass of carbon tells you the mass of one mole of carbon atoms, you can say that the molar mass will be

${M}_{M} = {\text{12.01 g mol}}^{- 1}$

It's worth noting that the definition of the unified atomic mass unit includes the conversion to grams per mole, since

$\textcolor{p u r p \le}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\text{1 u" = "1 g mol}}^{- 1}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

So, you can say that

• the atomic mass of an element tells you the actual mass of an atom of said element and is measured in unified atomic mass units, $\text{u}$
• the relative atomic mass tells you the ratio that exists between the atomic mass of the element and the unified atomic mass unit, and is always a dimensionelss quantity
• the molar mass tells you the mass of one mole of an element and is measured in grams, $\text{g}$