Collection of Solved Problems in Physics

Nitrogen in a Vessel

Task number: 2173

• Hint 1

  Using the formula for average molecular kinetic energy of a gas \(\bar{E}_k\), express the temperature T₁ of nitrogen before heating.

• Formula for the Average Kinetic Energy

  For the average kinetic energy \(\bar{E}_k\) of any gas, the following relation applies

  \[\bar{E}_k=\frac{3}{2}nRT,\]

  where n is the amount of the gas, T is its temperature and R is the molar gas constant.

• Hint 2: The Most Probable Speed

  To calculate the most probable speed v_p of the heated nitrogen molecules, use the following relation:

  \[v_p=\sqrt{\frac{2RT_2}{M_m}},\]

  where M_m is the molar mass of nitrogen, T₂ is the temperature of the heated nitrogen and R is the molar gas constant.

• Notation

  \(\bar{E}_k=3.5\,\mathrm{kJ}=3500\,\mathrm{J}\)	average translational kinetic energy of nitrogen molecules
  n = 1 mol	amount of nitrogen
  ΔT = 200 K	increase in temperature
  vp = ?	the most probable speed

  ---

  From the Tables:

  Mm = 28 g mol−1 = 0.028 kg mol−1	molar mass of nitrogen
  R = 8.31 J mol−1K−1	molar gas constant

• Analysis

  First we express the temperature of nitrogen before heating from the formula for the average kinetic energy of a gas, according to which the energy is directly proportional to the thermodynamic temperature. After that we add the temperature change to the calculated initial temperature and obtain the temperature of the heated nitrogen, which we subsequently substitute in the formula for the most probable molecular speed.

• Solution

  The average kinetic energy \(\bar{E}_k\) of any gas is directly proportional to its thermodynamic temperature T. The following relation applies

  \[\bar{E}_k=\frac{3}{2}nRT,\]

  where n is the amount of the gas and R is the molar gas constant.

  From this formula we can easily determine temperature T₁ of the nitrogen before heating:

  \[T_1=\frac{2\bar{E}_k}{3nR}.\]

  For temperature T₂ of the heated nitrogen it holds true

  \[T_2=T_1+\mathrm{\Delta}T=\frac{2\bar{E}_k}{3nR}+\mathrm{\Delta}T.\]

  Finally, to calculate the most probable speed v_p of the nitrogen molecules, we use the known formula

  \[v_p=\sqrt{\frac{2RT_2}{M_m}},\]

  where M_m is the molar mass of the nitrogen, and substitute for the obtained expression of temperature T₂:

  \[v_p=\sqrt{\frac{2R}{M_m}\left(\frac{2\bar{E}_k}{3nR}+\mathrm{\Delta} T\right)}.\]

• Numerical Substitution

  \[v_p=\sqrt{\frac{2R}{M_m}\left(\frac{2\bar{E}_k}{3nR}+\mathrm{\Delta} T\right)}=\sqrt{\frac{2\cdot{8.31}}{0.028}\cdot\left(\frac{2\cdot{3500}}{3\cdot{1}\cdot{8.31}}+200\right)}\,\mathrm{m\,s}^{-1}\] \[v_p\dot{=}530\,\mathrm{m\,s}^{-1}\]

   

  Just out of interest let's also try to determine the most probable speed of the nitrogen molecules before heating:

  \[v_p=\sqrt{\frac{4\bar{E}_k}{3nM_m}}=\sqrt{\frac{4\cdot{3500}}{3\cdot{1}\cdot{0.028}}}\,\mathrm{m\,s}^{-1}\] \[v_p\dot{=}410\,\mathrm{m\,s}^{-1}\]

• Answer

  The most probable speed of the nitrogen molecules after its temperature been increased is approximately 530 ms⁻¹.

• Comment

  Try to compare the determined speed of the nitrogen molecules with the speeds we encounter in everyday life.

  By comparing the calculated value with the speed of the air under normal conditions, we obtain the approximate value of the Mach number 1.6. So the speed is supersonic.