Question

A 1.0 m ✕ 1.0 m ✕ 1.0 m cube of nitrogen gas is at 20℃ and 1.0 atm. Estimate the number of molecules in the cube with a speed between 700 m/s and 1000 m/s.

Accepted Answer

Step 1: Begin by understanding the problem. We are tasked with estimating the number of nitrogen gas molecules in a cube with a speed between 700 m/s and 1000 m/s. This involves using the Maxwell-Boltzmann distribution to determine the fraction of molecules within the specified speed range. Step 2: Calculate the total number of molecules in the cube. Use the ideal gas law, $$ PV = nRT $$, where $$ P  is $$pressure (1.0 atm), $$ V  is $$volume (1.0 m³), $$ n  is $$the number of moles, $$ R  is $$the gas constant (8.314 J/(mol·K)), and $$ T  is $$temperature in Kelvin (convert 20℃ to 293 K). Rearrange the equation to solve for $$ n $$: $$ n = \frac{PV}{RT} . $$Then, multiply $$ n  by $$Avogadro's number ($$ 6.022 	imes 10^{23} ) to $$find the total number of molecules. Step 3: Use the Maxwell-Boltzmann speed distribution formula to find the fraction of molecules with speeds between 700 m/s and 1000 m/s. The probability density function is given by $$ f(v) = 4 \pi \left( \frac{m}{2 \pi k_B T} ight)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}} $$, where $$ m  is $$the mass of a nitrogen molecule, $$ k_B  is $$Boltzmann's constant ($$ 1.38 	imes 10^{-23}  J/K$$), $$ T  is $$temperature, and $$ v  is $$speed. Integrate this function over the range $$ v = 700  m/s to$$ $$ v = 1000  m/s to $$find the fraction of molecules in this speed range. Step 4: Determine the mass of a nitrogen molecule. Nitrogen gas (N₂) has a molar mass of approximately 28 g/mol. Convert this to kilograms per molecule by dividing by Avogadro's number: $$ m = \frac{28 	imes 10^{-3}}{6.022 	imes 10^{23}}  kg. $$Step 5: Multiply the fraction of molecules (from Step 3) by the total number of molecules (from Step 2) to estimate the number of molecules in the cube with speeds between 700 m/s and 1000 m/s.

A 1.0 m ✕ 1.0 m ✕ 1.0 m cube of nitrogen gas is at 20℃ and 1.0 atm. Estimate the number of molecules in the cube with a speed between 700 m/s and 1000 m/s.

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Concetti chiave

Ideal Gas Law

Kinetic Molecular Theory

Maxwell-Boltzmann Distribution