The speed of propagation of a sound wave in air at 27^∘ C is about 350 m / s . Calculate, for co

step 1 Hello, so in this problem, to find the VRMS of the nitrogen molecules which is the formula of VR

The speed of propagation of a sound wave in air at 27^∘ C is...

The speed of propagation of a sound wave in air at $27^{\circ} \mathrm{C}$ is about 350 $\mathrm{m} / \mathrm{s} .$ Calculate, for comparison, (a) $v_{\mathrm{rms}}$ for nitrogen molecules and (b) the rms value of $v_{x}$ at this temperature. The molar mass of nitrogen $\left(\mathrm{N}_{2}\right)$ is 28.0 $\mathrm{g} / \mathrm{mol} .$

Key Concepts

Root Mean Square Speed

The root mean square (rms) speed is a statistical measure of the speed of particles in a gas, defined as the square root of the average of the square of the velocities. It is derived from the kinetic theory of gases and is fundamental in linking macroscopic thermodynamic properties with microscopic molecular behavior.

Maxwell–Boltzmann Distribution

The Maxwell–Boltzmann distribution describes the statistical distribution of speeds for particles in a gas at thermal equilibrium. This distribution forms the basis for calculating various average speeds, such as the rms speed and the one-dimensional (x-direction) rms velocity, and illustrates the spread of molecular velocities in different directions.

Equipartition Theorem

The equipartition theorem is a principle in statistical mechanics which states that energy is equally divided among all available degrees of freedom in a system. For a gas, each translational degree of freedom contributes an equal amount to the total kinetic energy, which is essential in deriving the expressions for rms speeds in three dimensions.

Molar Mass to Molecular Mass Conversion

Converting molar mass to the mass of a single molecule is crucial for microscopic calculations. The molar mass given in grams per mole must be converted to kilograms and then divided by Avogadro's number to obtain the mass of one molecule, which is used in equations relating temperature, kinetic energy, and molecular speed.

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