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Introduction
The square root of the aggregate velocity-squared of the molecules in a gas is the root mean square speed of gas. Towards the end of the 19th century, scientists put forth a theory on the properties of gases based on a few assumptions.
These basic ideas and assumptions formed the basis for the Kinetic molecular theory. They are:
- The gas particles are negligible in comparison to the entire volume of gas.
- In the case of an ideal gas, there are no forces of attraction exerted by the particles on one another or with their surroundings.
- Gas particles are never at rest and often collide as they move along straight lines.
- The collisions being elastic by nature, the total K.E remains constant.
- The K.E of the molecules is directly related to the absolute temperature of the gas. The gas particles lose all the kinetic energy if the absolute temperature is Zero.
Kinetic Theory of Gases
The combination of the Kinetic theory of molecules and the gas laws like Charles’s law and Boyle’s law is known as the Kinetic theory of gases.
With the volume and temperature of a gas, Charles’s law states that V1/t1 =V2/t2. According to Kinetic theory, the rise in the temperature of the gas increases the kinetic energy of the gas particles. The faster-moving particles stay apart from each other to compensate for the increase in the number of collisions with container walls at constant pressure, which leads to an increase in the volume of gas.
Boyle’s law says that at a fixed temperature, pressure and volume become inversely related or p1v1= p2v2. In kinetic theory, at a particular temperature pressure of a gas is determined by the number of collisions the gas molecules have with the walls of the cabin. When compressed in a smaller space, the gas molecules become packed, thus increasing the number of collisions or increasing the pressure.
Molecular Velocity of Gas Particles
The kinetic theory states that all gas particles are in constant linear motion, leading to collisions, which can change the speed or direction based on the nature of the collision. The velocity of the gas can be calculated by considering the average of the squares of individual velocities and finding the square root of the average. This is known as the RMS velocity of the gas.
Velocity = Vrms = √3RT/Mm; R is the ideal gas constant, T is absolute temperature or temperature in Kelvin and Mm refers to the molar mass of that particular gas.
KE = ½ * m* v2.
With an increase in temperature, the average kinetic energy of the gas molecules also increases, leading to a larger range of velocities of the gas molecules. The larger molecular weights of molecules decrease the bandwidth of velocity distribution as all molecules have similar energy at the same temperature.
Root Mean Square Speed
In a gas sample, each gas molecule moves at random speed and directions. Selecting a particular molecule to understand the properties of a gas can be useless and misleading as some molecules may have the velocity at 0 owing to the constant collisions and conserved kinetic energy of the gas as a whole.
Due to this, it is inconvenient to study each molecule of the gas whose values of kinetic energy or velocity keep changing. Thus, researchers consider the average velocity of all the molecules of gas present in the gas sample. However, the more exact calculation of the velocity of gas molecules used in scientific calculations is the RMS velocity or the Root Mean Square method.
If there are m molecules in a sample of gas with each molecule, Mk, having the velocity Vk where k = 1, 2, 3…..k. Then,
C = (C-2) 1/2 = √[m1v12 + m2v22+……..)/m]
The average velocity is given by C = [(m1v12 + m2v22 + …….)/n].
Where m = total number of molecules.
We calculate the Root Mean Square velocity of the gas sample by taking the average of the squares of the individual velocities of the gas molecules and finding the square root of that sum which gives us an ideal representation of the collective velocity of all the molecules of the gas sample. Since the direction of each of the molecules becomes insignificant, thus it all comes down to the average speed of the molecules.
Vrms = √3kT/M where k is Boltzmann constant, T is the absolute temperature in Kelvin and M represents the molecular mass or molar mass of the gas sample.
Diverse Applications of Root Mean Square Method
Apart from chemistry, the root mean square method is used in physics to determine the root mean square speed of an ideal gas.
The root mean square method is used in electricity to find out the average electrical power concerning a fluctuating voltage. Electrical engineers use this method to calculate average power in a circuit where timely fluctuations occur. With alternating currents, the waveform imitates the sine curve, which helps analyse the average power at any given load.
Due to the efficiency of the root mean square method in deducing the power output. The voltages of power output in many countries are always mentioned in the root mean square values but not in peak values as they can be deduced from the root mean square values.
The applications of RMS extend even into error calculation whereby the RMS of the two sets of data is checked to find out the average deviation of error from zero. RMS is used when data obtained from theory as well as the practical is deduced in comparison to each other to find the errors in the pair. The RMS value shows the deviation of the data from Zero errors.
Conclusion
The Root Mean Square method is used to calculate the absolute velocity of the gas sample in a closed chamber or container. The square root of the mean velocity of the molecules in a gas is used to calculate the root-mean-square speed of particles in a gas. The root-mean-square speed considers combined molecular temperature and weight, two elements that have a direct impact on a material’s kinetic energy. The RMS velocity of a gas can be deduced based on the kinetic theory of gases, whereby it is said that the velocity of the molecules of the sample is dependent on the absolute temperature and molar mass of the gas.
