Molecular speeds: A cubic box with sides of 20.0 cm contains 2.00 × 10²³ molecules of helium with a root-mean-square speed (thermal speed) of 200 m/s. The mass of a helium molecule is 3.40 × 10^-27 kg. What is the average pressure exerted by the molecules on the walls of the container? The Boltzmann constant is 1.38 × 10^-23 J/K and the ideal gas constant is R = 8.314 J/mol ∙ K = 0.0821 L ∙ atm/mol ∙ K. A) 3.39 kPa B) 1.13 kPa C) 570 Pa D) 2.26 kPa E) 9.10 Pa

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Quizplus · Accepted Answer

The average pressure exerted by the gas molecules can be calculated using the ideal gas law formula $P = \frac{NkT}{V}$, where $P$ is the pressure, $N$ is the number of molecules, $k$ is the Boltzmann constant, $T$ is the temperature, and $V$ is the volume of the container. However, since temperature is not directly given, we use the root-mean-square speed ($v_{rms}$) to relate to the kinetic energy and thus to the temperature. The formula that connects $v_{rms}$ to the pressure in a container of volume $V$ is $P = \frac{1}{3} \frac{Nm\overline{v^2}}{V}$, where $m$ is the mass of a molecule and $\overline{v^2}$ is the square of the root-mean-square speed. Given the mass of a helium molecule, the number of molecules, and the root-mean-square speed, we can calculate the pressure without explicitly finding the temperature. The calculation involves substituting the given values into the derived formula, taking into account the conversion of units where necessary, such as converting the volume from cubic centimeters to cubic meters for consistency with the SI units of the other given quantities.

Molecular Speeds: a Cubic Box with Sides of 20