Mean square molecular speed

The speed of sound is about two-thirds of the root-mean-square molecular speed. This is reasonable because sound travels by moving molecules by reorienting, but not changing, the magnitude of their velocity. Sound can therefore travel no faster than the molecules through which it is traveling. 

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... 

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... 

The root-mean-square (rms) speed (u d of gas molecules in a sample at a given temperature is inversely proportional to the molecular mass. 

At this stage, we relate (vj.) to the root mean square speed, frms = (v2)111, the square root of the average of the squares of the molecular speeds (this quantity is explained in more detail in the text following this derivation). First, we note that the speed, v, of a single molecule is related to the velocity parallel to the x, y, and 2 directions ... 

This important result is used to find the root mean square speeds of the gas-phase molecules at any temperature (Fig. 4.25). We can rewrite this equation to emphasize that, for a gas, the temperature is a measure of mean molecular speed. From... 

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...

Before we leave the Kinetic Molecular Theory (KMT) and start examining the gas law relationships, let s quantify a couple of the postulates of the KMT. Postulate 3 qualitatively describes the motion of the gas particles. The average velocity of the gas particles is called the root mean square speed and is given the symbol rms. This is a special type of average speed. 

FIG U R E 9.14 The Maxwell-Boltzmann distribution of molecular speeds in nitrogen at three temperatures. The peak in each curve gives the most probable speed, u p, which is slightly smaller than the root-mean-square speed, Urms The average speed Uav (obtained simply by adding the speeds and dividing by the number of molecules in the sample) lies in between. All three measures give comparable estimates of typical molecular speeds and show how these speeds increase with temperature. 

Use the Maxwell-Boltzmann distribution of molecular speeds to calculate root-mean-square, most probable, and average speeds of molecules in a gas (Section 9.5, Problems 41-44). 

Calculate the root-mean-square speed of molecular chlorine in m/s at 20°C. 

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