Collection of Solved Problems in Physics

Unknown Nitrogen Oxide

Task number: 1798

• Hint

  Using the ideal gas law, determine the molar mass of the unknown gas first.

  Then find such a combination of the number of oxygen and nitrogen atoms in the molecule, so that th sum of molar masses of these atoms corresponds to the found molar mass of the gas.

• Notation

  V = 1 l = 10-3 m3	volume of the gas
  m = 1 g = 10-3 kg	mass of the gas
  t = 17 °C ⇒ T = 290 K	temperature of the gas
  p = 31.7 kPa = 31.7·103 Pa	pressure of the gas
  NxOy  x = ?  y = ?	unknown formula of the gas

  From the Handbook of Chemistry and Physics:

  R = 8.31 J K-1 mol-1	molar gas constant
  Ar(N) = 14	relative atomic mass of nitrogen N
  Ar(O) = 16	relative atomic mass of oxygen O

• Analysis

  From the task assignment we know all state quantities of the unknown gas that we need for the ideal gas law. Let us assume that properties of the unknown gas are not so different from the properties of an ideal gas. We can determine the unknown gas’s molar mass from the equation of state.

  Molar mass in g mol^-1 is numerically equal to the relative molecular mass, which could by determined as the sum of the relative atomic masses of the atoms that the molecule is composed of. Hence we determine the molecule structure of the unknown gas.

• Solution

  We use the ideal gas law

  \[ pV\,=\,nRT\,\mathrm{,} \]

  into which we substitute the unknown amount of substance n expressed by the total mass of the gas m and its molar mass M_m:

  \[ n\,=\,\frac{m}{M_m}\,. \]

  From the ideal gas law, we express the unknown molar mass M_m and substitute given values into obtained equation:

  \[ pV\,=\,\frac{m}{M_m}RT\,\mathrm{,} \] \[ M_m\,=\,\frac{mRT}{pV}\,=\,\frac{10^{-3}\cdot 8{.}31\cdot{290}}{31{.}7\cdot{10^3}\cdot{10^{-3}}}\,\mathrm{kg\,mol^{-1}} \] \[ M_m\,\dot=\,0{.}076\,\mathrm{kg\,mol^{-1}}\,=\,76\,\mathrm{g\,mol^{-1}}. \]

  Hence it follows that the relative molar mass of the compound is 76 and it holds that:

  \[ A_r\left(\mathrm{N}_x\mathrm{O}_y\right)\,=\,xA_R\left(\mathrm{N}\right)\,+\,yA_R\left(\mathrm{O}\right) \] \[ 76\,=\,14x\,+\,16y. \]

  where x and y have to be small natural numbers.

  Thus let us try to compute the value of y for values of x:

  \[x\,=\,1\hspace{30px}y\,=\,3{.}88\] \[x\,=\,2\hspace{30px}y\,=\,3\] \[x\,=\,3\hspace{30px}y\,=\,2{.}13\] \[x\,=\,4\hspace{30px}y\,=\,1{.}25\]

  From this summary we see, that the compound of two nitrogen atoms and three oxygen atoms, i.e. dinitrogen trioxide N₂O₃, suits the relative molecular mass the best.

• Answer

  The unknown compound is dinitrogen trioxide N₂O₃.