Question

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`?

Answer 1

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

Explanation:

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

`PV=nRT`,

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^@"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="0.3060 g"`

`P="747 torr"`

`V="0.111 L"`

`R=62.364color(white)(.)"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="373 K"`

Unknown

`n=?`

`"molar mass = ? g/mol"`

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

`n=(PV)/(RT)`

`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.

`"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 `"C"_4"H"_6"O"_2"` and its known molar mass to three sig figs is `"86.09 g/mol"`. https://www.ncbi.nlm.nih.gov/pccompound?term=%22diacetyl%22

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

`"Percent error"=abs((86.09-85.83)/(86.09))xx100="0.3020%"`

Answer 2

`85.9 "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 = (dRT)/P`

where `M` is the molar mass of the substance,
`d` is its density, in `"g"/"L"`,
`R` is the universal gas constant, equal to `0.08206 ("L"-"atm")/("mol"-"K")`,
`T` is the absolute temperature (in `"K"`), and
`P` is the pressure of the gas (in `"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, `"mol"/"L"`, which is `n/V`:

`PV = nRT`

`P = (nRT)/V`

`P/(RT) = n/V`

Now, let's multiply both sides of the equation by `M`, the molar mass with units `"g"/"mol"`:

`(PM)/(RT) = (nM)/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 `(nM)/V` is the density of the gas, and if we plug this back into our equation:

`(PM)/(RT) = (nM)/V = d`

Thus, `d = (MP)/(RT)`, and rearranging to solve for the molar mass yields our original equation, `M = (dRT)/P`.

The density of the diacetyl is

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

The temperature, in Kelvin, is

`100^oC + 273 = 373 "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 `("CH"_3"CO")_2`, and from this the molar mass is found to be `86.1 "g"/"mol"`, which measures up well with our result.