Rms speed

Solve the rms-speed equation (6.20) for temperature by first squaring both sides. 

The average kinetic energy, e, is related to the root-mean-square (rms) speed u through the equation e = /i mi/ Because the MM of CH4 (16) is slightly less than that of NH3 (17), the root-mean-square speed of CH4 is slightly higher than that of NH3. Root-mean-square speed is inversely proportional to the square root of the molar mass of the gas. 1 point for correct answer and explanation. 

Kinetic Theory of Gases Velocity Distribution Speed Distribution Collision Frequency Meen Free Path Determine average and RMS speeds from data Plot radial distribution functions... 

If we solve this equation for velocity, we arrive at an expression representing the root mean square (rms) speed of a gas. The rms speed is symbolized u. 

Sample Calculate the rms speed of nitrogen gas (N2) particles in a tank that has a temperature of 27°C. 

In this section we shall be concerned with a molecular theory of the transport properties of gases. The molecules of a gas collide with each other frequently, and the velocity of a given molecule is usually changed by each collision that the molecule undergoes. However, when a one-component gas is in thermal and statistical equilibrium, there is a definite distribution of molecular velocities—the well-known Maxwellian distribution. Figure 1 shows how the molecular velocities are distributed in such a gas. This distribution is isotropic (the same in all directions) and can be characterized by a root-mean-square (rm speed u, which is given by... 

Schematic Maxwellian velocity distribution Up) du is the fraction of the molecules with velocities between v and v + dv. Values of the rms speed u and the mean speed c are shown.

For each gas (including air), calculate the maximum value of the Reynolds number from Eq. (11) and verily that it is below 1000. Use Eq. (12) to verily that U, the length of the transition region, is small compared with the length L of the capillary. Estimate the largest value of V from Eq. (5) and compare it with the molecular rms speed u and mean speed c. 

By taking square roots wc find that the rms speed is given by =... 

At the same temperature, the He, O2, and Xe molecules all have the same average kinetic energy lighter molecules move faster to compensate for their smaller masses. These rms speeds convert to 3050, 1080, and 532 mph, respectively. The average molecule moves along quite rapidly at room temperature ... 

At 22°C, CI2 molecules have some rms speed (which we need not calculate). At what temperature would the rms speed of F2 molecules be the same ... 

How fast does a molecule move, on the average, at any temperature 77 One way to estimate molecular speed is to calculate the root-mean-square (rms) speed (u, which is an average molecular speed. One of the results of the kinetic theory of gases is that the total kinetic energy of a mole of any gas equals 7 T Earlier we saw that the average kinetic energy of one molecule is mu and so we can write... 

For gas molecules, the heat capacity is a constant equal to C = (n/2)pkB where n is the number of degrees of freedom for molecule motion, p is the number density, and kB is the Boltzmann constant. The rms speed of molecules is given as v = V3kBTlm, whereas the mean free path depends on collision cross section and number density as = (pa)-1. When they are put together, one finds that the thermal conductivity of a gas is independent of p and therefore independent of the gas pressure. This is a classic result of kinetic theory. Note that this is valid only under the assumption that the mean free path is limited by inter-molecular collision. 

At every temperature, hydrogen has an rms speed four times as great as that of oxygen, while their average kinetic energies are the same. 

At room temperature, the usual range of molecular speeds is 300 to 500 m/s. Hydrogen is unusual because of its low mass its rms speed is about 1900 m/s. 

The average speed and the rms speed occur most frequently in physico-chemical calculations. 

Note that the speed corresponding to the maximum number of molecules is called the most probable speed. This is always smaller than the average speed, which is in turn smaller than the root-mean-square speed (rms speed). 

If you calculate the rms speeds as we will in Section 10.8, you will find that the rms speed is almost 6 X 10 m/s for the 100 °C sample but slightly less than 5 X 10 m/s for the 0 °C sample. Notice that the distribution curve broadens as we go to a higher temperature, which tells us that the range of molecular speeds Increases with temperature. 

The rms speed is important because the average kinetic energy of the gas molecules in a sample is equal to — (Section 5.1) Because mass does not change with... 

For a molecule traveling at the rms speed, the impulse imparted by a collision with a wall depends on the momentum of the molecule that is, it depends on the product of the molecule s mass and speed /nUrms- The coUision rate is proportional to the number of molecules per unit volume, nj V, and to their speed, which is be-... 

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