# Question 75905

Nov 1, 2015

Sodium.

#### Explanation:

The idea here is that you need to use the ideal gas law equation to find a relationship between the density of the gas and its molar mass.

As you know, the ideal gas law equation establishes a relationship between pressure and volume, on one side, and number of moles and temperature on the other.

$P V = n R T$

Here $R$ represents the universal gas constant and is usually given as

$R = 0.082 \left(\text{atm" * "L")/("mol" * "K}\right)$

Now, the number of moles of can be written as the ration between the mass of the sample and the gas' molar mass

$n = \frac{m}{M} _ M$

Plug this into the idea lgas law equation to g et

$P V = \frac{m}{M} _ M \cdot R T$

Multiply both sides of the equation by ${M}_{M}$ to get

$P V \cdot {M}_{M} = \frac{m}{\textcolor{red}{\cancel{\textcolor{b l a c k}{{M}_{M}}}}} \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{{M}_{M}}}} \cdot R T$

$P V \cdot {M}_{M} = m \cdot R T$

Now look what happens when you divide both sides by the volume of the gas

$\frac{P \textcolor{red}{\cancel{\textcolor{b l a c k}{V}}} {M}_{M}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{V}}}} = \frac{m}{V} \cdot R T$

$P \cdot {M}_{M} = \frac{m}{V} \cdot R T$

But since density is defined as mass per unit of volume, you will have

$P \cdot {M}_{M} = \rho \cdot R T$

Now plug in your values and solve this equation for ${M}_{M}$, the molar mass of the gas - do not forget to convert the temperature from degrees Celsius to Kelvin and the pressure from torr to atm!

${M}_{M} = \frac{\rho \cdot R T}{P}$

M_M = (2.9 * 10^(-3)"g"/color(red)(cancel(color(black)("L"))) * 0.082(color(red)(cancel(color(black)("atm"))) * color(red)(cancel(color(black)("L"))))/("mol" * color(red)(cancel(color(black)("K")))) * (1000 + 273.15)color(red)(cancel(color(black)("K"))))/((10.)/760color(red)(cancel(color(black)("atm"))))#

${M}_{M} = \text{23.009 g/mol" ~~ "23 g/mol}$

Now, the important thing to realize here is that you're dealing with a gaseous element, not a compound.

The high temperature at which the element is kept is a clue - in this case, sodium has a molar mass of approximately $\text{23 g/mol}$, and a boiling point of about ${883}^{\circ} \text{C}$.

This means that at ${1000}^{\circ} \text{C}$, sodium atoms will exist in the gaseous state. Therefore, the element you're looking for is sodium.