# What total volume of gases are produced at 33 degrees C and 0.968 atm by the decomposition of 2.00 g nitroglycerin (C_3H_5N_3O_9)? (R = 0.08206 atm/mol-K)

Jan 31, 2017

The volume of gas produced is 1.66 L.

There are four steps involved in this stoichiometry problem:

Step 1. Write the balanced chemical equation.

${M}_{r} : \textcolor{w h i t e}{m l l} 227.09$
color(white)(mm)"4C"_3"H"_5"N"_3"O"_9"(l)" → underbrace("12CO"_2"(g)" + "10H"_2"O(g)" + "6N"_2"(g)" + "O"_2"(g)")_color(red)("29 mol of gas")

Step 2. Convert grams of nitroglycerin to moles of nitroglycerin

$\text{moles of nitroglycerin" = 2.00 color(red)(cancel(color(black)("g nitroglycerin"))) × ("1 mol nitroglycerin")/(227.09 color(red)(cancel(color(black)("g nitroglycerin")))) = "0.008 807 mol nitroglycerin}$

Step 3. Convert moles of nitroglycerin to moles of gas

The molar ratio of gases to nitroglycerin is $\text{29 mol gas":"4 mol nitroglycerin}$.

$\text{0.008 807"color(red)(cancel(color(black)("mol nitroglycerin"))) × ("29 mol gas")/(4 color(red)(cancel(color(black)("mol nitroglycerin")))) = "0.063 85 mol gas}$

4. Use the Ideal Gas Law to calculate the volume of gas.

The Ideal Gas Law is

$\textcolor{b l u e}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} P V = n R T \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$

where

$P$ = the pressure of the gas,
$V$ = the volume of the gas,
$n$ = the number of moles of the gas,
$R$ = the universal gas constant
$T$ = the temperature of the gas

We can rearrange the Ideal Gas Law to get

$\textcolor{b l u e}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} V = \frac{n R T}{P} \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$

In this problem,

$n = \text{0.063 85 mol}$
$R = \text{0.082 06 L·atm·K"^"-1""mol"^"-1}$
$T = \text{(33 + 273.15) K" = "306.15 K}$
$P = \text{0.968 atm}$

V = ("0.063 85" color(red)(cancel(color(black)("mol"))) × "0.082 06 L"· color(red)(cancel(color(black)("atm"·"K"^"-1""mol"^"-1"))) × 306.15 color(red)(cancel(color(black)("K"))))/(0.968 color(red)(cancel(color(black)("atm")))) = "1.66 L"

The volume of gas is 1.66 L.

Here's a useful video on mass-volume conversions.