Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 170 C . Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of N2 = 28.0 u).
Mean free path =1.11×10−7m Collision frequency =4.58×109s−1 Successive collision time ≃500× (Collision time) Pressure inside the cylinder containing nitrogen, P=2.0atm=2.026×105Pa Temperature inside the cylinder, T=17oC=290K Radius of a nitrogen molecule, r=1.0A˚=1×1010m Diameter, d=2×1×1010=2×1010m Molecular mass of nitrogen, M=28.0g=28×10−3kg The root mean square speed of nitrogen is given by the relation: υrms=M3RT where, R is the universal gas constant =8.314Jmol−1K−1 ∴υrms=28×10−33×8.314×290=508.26m/s The mean free path (l) is given by relation: l=2×d2×PkT Where, k is the boltzmann constant =1.38×10−23kgm2s−2K−1 ∴l=2×3.14×(2×10−10)2×2.026×1051.38×10−23×290 =1.11×10−7m Collision frequency =lυrms =1.11×10−7508.26=4.58×109s−1 Collision time is given as: T=υrmsd =508.262×10−10=3.93×10−13s Time taken between successive collisions: T′=υrmsl =508.261.11×10−7=2.18×10−10 ∴TT′=3.93×10−132.18×10−10=500 Hence, the time taken between successive collisions is 500 times the time taken for a collision.
