Mean square velocity of gas molecule

Problem 7 From kinetic equation of gases, how will you calculate the root mean square velocity of gas molecules under different conditions ... 

Root mean square velocity of gas molecules is always greater than the average velocity under similar conditions. 

Problem 49-1. Derive Equation 49-5 with the use of the results of Section 14. By equating W to the kinetic energy lAmvi (v being the velocity), derive the Maxwell distribution law for velocities, and from it calculate expressions for the mean velocity and root-mean-square velocity of gas molecules. 

Mean square velocity of gas molecules. The mean velocity of a gas molecule is given by the following equation ... 

The root mean square velocity of a collection of gas particles is proportional to the square root of the temperature in kelvins and inversely proportional to the square root of the molar mass of the particles (which because of the units of R must be in kilograms per mole). The root mean square velocity of nitrogen molecules at 25 °C, for example, is 515 m/s (1152 mi hr). The root mean square velocity of hydrogen molecules at room temperature is 1920 m/s (4295 mi/hr). Notice that the lighter molecules move much faster at a given temperature. 

The square root of the average of the square of the velocity, vrms, is not the average velocity, but a quantity called the "root mean square velocity. (b) The pressure of the gas does not matter, (c) The identity of the gas is important, because the mass of the molecule is included in the calculation. (Contrast this conclusion with that of the previous problem.)... 

Table 1.4 The average speeds of gas molecules at 273.15 K, given in order of increasing molecular mass. The speeds c are in fact root-mean-square speeds, obtained by squaring each velocity, taking their mean and then taking the square root of the sum...

EXAM PLE 9.6 Rate of Atomic Collisions as a Function of Pressure. Assuming 1019 atoms per square meter as a reasonable estimate of the density of atoms at a solid surface, estimate the time that elapses between collisions of gas molecules at 10 6 torr and 25°C with surface atoms. Use the kinetic molecular theory result that relates collision frequency to gas pressure through the relationship Z = 1/4 vNIV, for which the mean velocity of the molecules v = (BRTI-kM) 12 and NIV is the number density of molecules in the gas phase and equals pNJRT. Repeat the calculation at 10 8 and 10 10 torr. 

This is obtained from the well-known expression for the pressure of a gas pv — 1/3 mnffi where p = pressure, v = volume, m = mass of a molecule, n = number of molecules in the volume v, and ii = root mean square velocity. 

C is the mean square velocrty for all the molecules in a gas. Equation 2.13 tells us that the velocity of gaseous molecules is inversely proportional to the square root of the density. Thus, it follows that the rate of diffusion is inversely proportional to the square root of the molecular weight (M) of a gas. 

The root mean square velocity is defined as the square root of the mean ofthe squaresofdjfferentvelodtiespossessedby molecules of a gas at a given temperature. Evidently, the root mean square velocity would be given by... 

Each average value of velocity can be used to best describe some particular property of the ensemble of molecular velocities. For example, in a gas all molecules have the same average kinetic energy. Hence, the root-mean-square velocity is the best estimate of velocity to use for computing parameters that are a function of kinetic energy... 

Consider a 1.0-L container of neon gas at STR Will the average kinetic energy, root mean square velocity, frequency of collisions of gas molecules with each other, frequency of collisions of gas molecules with the walls of the container, and energy of impact of gas molecules with the container increase, decrease, or remain the same under each of the following conditions ... 

The kinetic theory of gases was first propounded by Daniel Bernoulli in 1738. It was rediscovered and worked out in detail about the middle of the nineteenth century by Kr5nig, Waterston, Maxwell, and above all by Clausius. According to the kinetic theory the molecules of a gas move in straight lines unless they are deflected from their path by impacts with other molecules or with the walls of the containing vessel. They, therefore, exert on every solid wall a pressure which is measured by the momentum which the molecules impart to the wall in unit time. As this momentum is proportional to the number of collisions per unit of timclosed vessel is proportional to the square of the velocity of its molecules. It is assumed in this that all the molecules have the same velocity. When this is not the case, the pressure can be shown to be proportional to the mean square of the velocity. In any case we may write... 

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