Question

Consider a monatomic gas of particles each with mass m. what is vx,rms=⟨v2x⟩−−−−√, the root mean square (rms) of the x component of velocity of the gas particles if the gas is at an absolute temperature t? express your an

Answer 1

$\rm v_{rms}=\sqrt{\dfrac{3k_BT}{m} }$

Further explanation

Kinetic Molecular Theory (KMT) states that a gas consists of molecules that move at a constant and random speed. The collisions between molecules are perfectly elastic so that no energy is wasted.

The molecules move in straight lines until they collide

Energy because this motion is expressed as Kinetic energy (KE) which can be formulated as:

 $\displaystyle KE = \frac {1} {2} mv ^ 2$

The average kinetic energy value is only affected by temperature changes. The higher the temperature, the average kinetic energy of the molecule increases

This molecule is very small when compared to the distance between molecules, so the volume of gas contains mostly empty space

Gas particles move randomly (both speed and direction, as vector)

Average velocities of gases can be expressed as root-mean-square averages. (V rms)

$\large {\boxed {\bold {v_ {rms} = \sqrt {\dfrac {3RT} {Mm}}}}$

R = gas constant, T = temperature, Mm = molar mass of the gas particles

From the question

R = 8,314 J / mol K

T = temperature

Mm = molar mass, kg / mol

for absolute temperature 273.15 K

, and For Boltzmann constant (k), vrms :

$\rm v_{rms}=\sqrt{\dfrac{3k_BT}{m}}\\\\k=1,38.10^{-23},m=mass\:of\:a\:particle\:of\:the\:gas$

where :

$\rm k=\dfrac{R}{6.022\times 10^{23}}$

Learn more  

inertia affect during a collision  

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Henry's Law  

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ideal gas  

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Answer 2

The root mean square velocity is directly proportional to the square root of temperature.

The root mean square velocity is inversely proportional to the square root of the mass of the gas molecule and directly proportional to the Boltzmann constant.

So:

The expression for the root mean square velocity along x-direction is:

ν(rms) = $\sqrt {\frac{KT}{M}}$ when n = 1 mole or it equals:

ν(rms) = $\sqrt {\frac{nKT}{M}}$ when n not 1 mole