Root Mean Square Velocities

This study material notes on root mean square velocity defines root mean square velocity and the parameters on which it depends. The root mean square velocity of particles in a gas is calculated using the square root of the average velocity-squared of the individual molecules in the gas. The root-mean-square speed addresses both molecular weight and temperature, two parameters that have a direct influence on a material’s kinetic energy. The Maxwell-Boltzmann equation, which is the foundation of gas kinetic theory, defines the speed distribution for gas at a specific temperature. The most likely speed, the average speed, and the root-mean-square speed and velocity can all be calculated using this distribution function.

Before deriving the root square mean velocity, let’s first understand the Kinetic molecular theory of gases.

Kinetic Molecular Theory of Gas Molecules

According to Kinetic Molecular Theory, gas particles are always in motion and undergo elastic collisions. The following are the five basic tenets of the kinetic-molecular theory:

1. A gas is made up of molecules that are separated by average distances that are significantly greater than the molecules’ size. When compared to the volume of the gas, the volume occupied by the molecules is insignificant.
2. The molecules of an ideal gas exhibit no attraction to each other or the container’s walls.
3. The molecules are in continuous random motion and follow Newton’s laws of motion as material bodies. This means that the molecules move in a horizontal straight line until they clash with one other or the container’s walls.
4. Gas particle collisions are elastic. In other words, when particles collide, there is no overall loss or gain of kinetic energy.
5. The absolute temperature is exactly proportional to the average kinetic energy of the gas molecules.

Kinetic Interpretation of Absolute Temperature

The kinetic molecular theory states that the average kinetic energy of an ideal gas is linearly proportional to the absolute temperature. The energy a body has as a result of its motion is known as kinetic energy (K.E.):

K.E. = ½ mv2

Where The kinetic energy of a molecule is K.E., the mass of a molecule is m, and the magnitude of a molecule’s velocity is v.

The average velocity of molecules increases with rises in the temperature of a gas. Collisions with the container’s walls will cause the gas molecules to gain greater momentum, and hence more kinetic energy. If the walls of the container are cooler than the gas, they will return less kinetic energy to the gas, causing it to cool until thermal equilibrium is reached.

The Behaviour of Different Gases at the Same Temperature

The mass of molecules has an impact on the speed distribution. Gas molecules with a heavier mass move slower than lighter gas molecules at the same temperature. At the same temperature, lighter nitrogen molecules, for example, move quickly than heavier chlorine molecules. As a result, nitrogen molecules at any given temperature have a more probable speed than chlorine molecules.

The Behaviour of the Same Gas at Different Temperature

The average speed of a gas’s molecules increases as its temperature rises, moving the Maxwell-Boltzmann curve toward higher speeds. The most probable speed of hydrogen gas increases from 1414 to 1577 m s–1, and the root mean square velocity increases from 1926 to 2157 m s–1, both by 11 per cent, when the temperature is increased from 300 to 373 K, respectively.

Molecular Speed of Gas Molecules and Kinetic Energy

Individual molecules travel at different speeds, irrespective of the fact that the molecules in a sample of gas have average kinetic energy and thus an average speed. Some molecules are travelling quickly, while others are moving slowly. As a result, the speed and energy of all the gas molecules at any given time are not the same. As a consequence, we can only get an average value of molecule speed.

If there are n molecules in a sample, and their respective speeds are u1, u2, u3,……un, then the average speed of the molecules equals to uav :-

uav.=  (u1 +u2 +……….un)/n

Maxwell-Boltzmann and Root Mean Square Velocity

The speed directly beneath the peak of the Maxwell-Boltzmann graph may appear to be the average speed of a molecule in the gas, but this is not the case. The speed right behind the peak is the most probable speed vp because it is the speed at which a molecule in a gas is most likely to be found.

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Derivation of Root Mean Square Velocity

We overcome the “directional” component of velocity and simultaneously obtain the particles’ average velocity by squaring the velocities and taking the square root. We now refer to the root mean square velocity as the root mean square speed because it does not include the direction of the particles. The root-mean-square speed of particles in a gas is calculated by determining the square root of the square of average velocities of the molecules in the gas.

vrms=3RTM

Mm is the molar mass of the gas in kilograms per mole, R is the molar gas constant, and T is the temperature in Kelvin, where vrms is the root-mean-square of the velocity.

The root-mean-square speed considers both molecular weight and temperature, two factors that have a direct impact on kinetic energy.

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Conclusion

The distribution of velocities doesn’t vary indeed though the velocity of gaseous particles is continually changing. Because we can not determine the velocity of each molecule, we constantly reason in terms of the average behaviour of the molecules. The velocities of molecules travelling in opposite directions have opposite signs. There are exactly as numerous gas particles travelling right (+ velocity) as there are heading left (- velocity), the average gas-particle velocity is zero. That is why we start by squaring the velocities and making them all positive. This assures that calculating the mean (or average value) won’t operate on a value of zero. Hence lies the significance of root mean square velocity.

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