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#?

Question

Olivia Anton · Accepted Answer

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"#.

This question can be answered using the ideal gas law, which has the following equation: #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"#. The molar mass of the given mass will be found by dividing it by the moles of diacetyl gas, which will be found using the ideal gas law. Recognized #m="0.3060 g"# #P="747 torr"# #V="0.111 L"# #R=62.364color(white)(.)"L torr K"^(-1) "mol"^(-1)# https://tutor.hix.ai #T="373 K"# Not sure #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"# Calculate the mass of diacetyl given in the question by dividing the number of moles you now possess. #"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://tutor.hix.ai #"Percent error" = abs(("known value"-"experimental value")/("accepted value"))xx100# #"Percent error"=abs((86.09-85.83)/(86.09))xx100="0.3020%"#

Henry Antrim · Answer

#85.9 "g"/"mol"#

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.

Kai Bowman · Answer

undefined To calculate the molar mass of diacetyl (butanedione), we first use the ideal gas law equation:

PV = nRT

where:
P = pressure (in atm)
V = volume (in L)
n = number of moles
R = ideal gas constant (0.0821 L atm / K mol)
T = temperature (in Kelvin)

First, we convert the temperature from Celsius to Kelvin:
100°C + 273.15 = 373.15 K

Now, we rearrange the ideal gas law equation to solve for the number of moles (n):
n = (PV) / (RT)

Then, we plug in the given values:
n = (747 torr * 0.111 L) / (0.0821 L atm / K mol * 373.15 K)

Next, we convert torr to atm:
1 atm = 760 torr
747 torr / 760 torr/atm = 0.98158 atm

Now, we plug in the converted values:
n = (0.98158 atm * 0.111 L) / (0.0821 L atm / K mol * 373.15 K)

Calculate the number of moles (n):
n = 0.001525 moles

Now, we use the definition of molar mass to find the molar mass (M):
M = mass / moles

We're given the mass as 0.3060 g, so we plug in the values:
M = 0.3060 g / 0.001525 moles

Calculate the molar mass:
M ≈ 200.66 g/mol

Therefore, the molar mass of diacetyl (butanedione) is approximately 200.66 g/mol.