Savchenko Problem 5.2.3
Physics · thermodynamics · optics
Problem
The number of molecules of a homogeneous ideal gas dN, whose velocity along an arbitrary x-axis lies in the interval (v_x, v_x + dv_x), out of the total number N of its molecules at a given temperature T is determined by the Maxwell distribution: where m is the mass of the molecule, and k is the Boltzmann constant. The function is called the distribution function. The figure shows the distribution function of nitrogen molecules at room temperature (T = 293 K). Using the graph, find: a) how many nitrogen molecules are contained in 1 cm³ of air that have a velocity in a certain direction in the range from 499 to 501 m/s, b) how many nitrogen molecules are contained in 1 m³ that have a velocity in a certain direction in the range from 498 to 502 m/s, if the number of nitrogen molecules in 1 cm³ is 2·10¹⁹Intuitive explanation
The Maxwell distribution tells you what fraction of gas molecules have velocities in a narrow range—you read the height of the curve, multiply by the width of your interval, and scale by the total count. The key is recognizing that the area under over your velocity interval gives the fraction of molecules in that range, so multiply by to get the actual count. For part (a), you'll estimate the curve's height from the graph at m/s and use the thin interval width; part (b) scales this idea to a larger volume and wider velocity window.
Intuitive explanation
The Maxwell distribution tells you what fraction of gas molecules have velocities in a narrow range—you read the height of the curve, multiply by the width of your interval, and scale by the total count. The key is recognizing that the area under over your velocity interval gives the fraction of molecules in that range, so multiply by to get the actual count. For part (a), you'll estimate the curve's height from the graph at m/s and use the thin interval width; part (b) scales this idea to a larger volume and wider velocity window.