Estimate the collision frequency of an argon atom in liquid argon at its normal boiling temperature, 85 K. Assume that the atom is moving at the mean speed for this temperature. Estimate the time between collisions as the time required to travel twice the distance from the minimum in the potential

Question

Estimate the collision frequency of an argon atom in liquid argon at its normal boiling temperature, 85 K. Assume that the atom is moving at the mean speed for this temperature. Estimate the time between collisions as the time required to travel twice the distance from the minimum in the potential function of Figure 9.15 to a value of r such that the potential energy is equal to the kinetic energy of a particle moving at the mean relative speed.

Accepted Answer

[latex]〈v〉=\sqrt{\frac{8RT}{πM} }=\sqrt{\frac{8(8.3145 J K^{−1} mol^{−1})(85K)}{π(0.039948 kg mol^{−1})} }=212ms^{−1} [/latex]

The kinetic energy of an argon atom moving at this speed is [latex]1.49×10^{−21} J[/latex]. To find the value of r that corresponds to this value of the potential energy function, we set

[latex]1.49 × 10^{−21} J=4ε\left[(σ/r^{′})^{12}-(σ/r^{′})^{6}\right] [/latex]

where [latex]r^{′}[/latex] is the value of r that we seek. This equation must be solved by numerical approximation, using a spreadsheet or a simple computer program. With values of ε and σ from Table A.14, we find that [latex]r^{′}=3.3×10^{−10} m[/latex]. The minimum of the potential energy is at [latex]3.8×10^{−10} m[/latex]. Twice the distance from [latex]r^{′}[/latex] to this value of r is [latex]1.0×10^{−10} m[/latex]. The time required to traverse this distance at [latex]212ms^{−1}[/latex] is equal to [latex]4.7×10^{−13} s[/latex], giving an estimate of the collision rate equal to [latex]2.1×10^{12} s^{−1}.[/latex]

Table A.14 Parameters for the Lennard–Jones Potential
Substance	[latex]σ × 10^{10}/m [/latex]	[latex](ε/k_{B})/K[/latex]
Argon	3.40	120
Carbon dioxide	4.5	189
Helium	2.56	10.22
Krypton	3.60	58.8
Neon	2.75	35
Nitrogen	3.7	95
Oxygen	3.5	118
Xenon	4.1	87
From J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids,Wiley, New York, 1954, p. 1110ff.

Question 9.21: Estimate the collision frequency of an argon atom in liquid ...

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