# Calculate the molar mass in "g"/"mol"of diacetyl (butanedione) given that in the gas phase 100 degrees Celsius and 747 torr, a 0.3060 g sample of diacetyl occupies a volume of 0.111L?

May 26, 2017

The molar mass of diacetyl, $\text{C"_4"H"_6"O"_2}$, as calculated based on the question parameters is $\text{85.83 g/mol}$. Its actual molar mass is $\text{86.09 g/mol}$.

#### Explanation:

You can use the ideal gas law to answer this question. The equation is:

$P V = n R T$,

where $P$ is pressure, $V$ is volume, $n$ is moles, $R$ is the gas constant, and $T$ is the temperature in Kelvins. Add $273.15$ to the Celsius temperature to get the temperature in Kelvins: ${100}^{\circ} \text{C" + 273.15="373 K}$.

We will use the ideal gas law to determine moles of diacetyl gas, then divide the given mass by the moles of diacetyl gas to determine its molar mass.

Known

$m = \text{0.3060 g}$

$P = \text{747 torr}$

$V = \text{0.111 L}$

$R = 62.364 \textcolor{w h i t e}{.} {\text{L torr K"^(-1) "mol}}^{- 1}$
https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Physical_Properties_of_Matter/States_of_Matter/Properties_of_Gases/Gas_Laws/The_Ideal_Gas_Law

$T = \text{373 K}$

Unknown

n=?

$\text{molar mass = ? g/mol}$

Solve for $n$.
Rearrange the equation to isolate $n$. Insert the known data and solve.

$n = \frac{P V}{R T}$

n=(747color(red)cancel(color(black)("torr"))xx0.111color(red)cancel(color(black)("L")))/((62.364color(white)(.)color(red)cancel(color(black)("L"))color(red)cancel(color(black)("torr")) color(red)cancel(color(black)("K")))^(-1) "mol"^(-1)xx373color(red)cancel(color(black)("K")))="0.003565 mol"

Now that you have moles, divide the mass of diacetyl given in the question by the moles.

$\text{Molar mass diacetyl" = (0.3060"g diacetyl")/(0.003565"mol diacetyl")="85.83 g/mol diacetyl}$ rounded to four significant figures

The molecular formula for diacetyl is ("CH"_3"CO)"_2 or $\text{C"_4"H"_6"O"_2}$ and its known molar mass to three sig figs is $\text{86.09 g/mol}$. https://www.ncbi.nlm.nih.gov/pccompound?term=%22diacetyl%22

"Percent error" = abs(("known value"-"experimental value")/("accepted value"))xx100

$\text{Percent error"=abs((86.09-85.83)/(86.09))xx100="0.3020%}$

May 26, 2017

$85.9 \text{g"/"mol}$

#### Explanation:

What we can do here is calculate the density of the diacetyl, and use that to directly calculate the molar mass. We will use the equation

$M = \frac{\mathrm{dR} T}{P}$

where $M$ is the molar mass of the substance,
$d$ is its density, in $\text{g"/"L}$,
$R$ is the universal gas constant, equal to $0.08206 \left(\text{L"-"atm")/("mol"-"K}\right)$,
$T$ is the absolute temperature (in $\text{K}$), and
$P$ is the pressure of the gas (in $\text{atm}$)

Where is this equation derived from? Read the steps below if you would like to know, otherwise, skip to the next step.

Well, let's recall our ideal-gas equation, and rearrange it to solve for units similar to that of density, $\text{mol"/"L}$, which is $\frac{n}{V}$:

$P V = n R T$

$P = \frac{n R T}{V}$

$\frac{P}{R T} = \frac{n}{V}$

Now, let's multiply both sides of the equation by $M$, the molar mass with units $\text{g"/"mol}$:

$\frac{P M}{R T} = \frac{n M}{V}$

If we list the right side of the equation in terms of units, we have

cancel("mol")/"L" xx "g"/cancel("mol") = "g"/"L"

Which is the units for density. Thus, the value $\frac{n M}{V}$ is the density of the gas, and if we plug this back into our equation:

$\frac{P M}{R T} = \frac{n M}{V} = d$

Thus, $d = \frac{M P}{R T}$, and rearranging to solve for the molar mass yields our original equation, $M = \frac{\mathrm{dR} T}{P}$.

The density of the diacetyl is

d = (0.3060"g")/(0.111 "L") = 2.76 "g"/"L"

The temperature, in Kelvin, is

${100}^{o} C + 273 = 373 \text{K}$

and the pressure, in atmospheres, is

747 cancel("torr")((1 "atm")/(760 cancel("torr"))) = 0.983 "atm"

Finally, plugging in all our known variables, we have

M = ((2.76 "g"/cancel("L"))(0.08206 (cancel("L")-cancel("atm"))/("mol"-cancel("K")))(373 cancel("K")))/(0.983 cancel("atm")) = color(red)(85.9 "g"/"mol"

To check this, the molecular formula of diacetyl is ${\left(\text{CH"_3"CO}\right)}_{2}$, and from this the molar mass is found to be $86.1 \text{g"/"mol}$, which measures up well with our result.