Velocity most probable

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

From the physical point of view the condition a < 3/2 and b < 3 can be interpreted as a criteria for a relativistic ideal gas to be degenerate. In other words, this means that if these conditions are satisfied, there exists a non-zero most probable velocity. 

A simpler first-order evaluation of the experiments may be obtained by using the most probable velocities and intersection angles as representative... 

In method B, which seems to be devoid of theoretical support, the two functions are fitted in magnitude at the most probable velocity and also at r0, the point of zero potential on the Lennard-Jones model. A comparison with experiment is given in Table 1. Z is the reciprocal probability or number of gas kinetic collisions required for deactivation. The calculated values are derived from equations (17) and (18). 

Problem 6 Define the terms average velocity, root mean square velocity and most probable velocity of molecules and how are they related to each Other (Meerut 2006)... 

The relationship of most probable velocity to the average velocity and root mean square velocity can be developed from the following expressions ... 

Most probable velocity, a = 0.8164 x R.M.S. velocity, u From the above equations, it can be seen that the ratio of the three velocities are given by ... 

The problem of passing from the dimensionless parameter x = p/lT, to values of 7s in s-1 has been approached in different ways. The simplest idea is to use extrapolation in the x 1(N) dependence to N = 0, leading to 7coi = 0 and 7s = 70 if one assumes that (3.2) holds. This allows us to obtain Tp = 7ox(0) by evaluating the fly-through relaxation rate 70 as a reciprocal transit time of molecules with the most probable velocity through the effective diameter of the laser beam [102]. 

There is a certain velocity for which the fraction of molecules is maximum. This is called the most probable velocity. 

The most probable velocity of a gas is the velocity possessed by maximum number of molecules of the gas at a given temperature. It corresponds to the peak of the curve. Its value, at a given temperature, depends upon the volume of the gas. 

The most probable velocity increases with rise in temperature, as shown in the figure given above. The entire distribution curve, in fact, shifts to the right with rise in temperature, as shown. The rise in temperature, therefore, increases the fraction of the molecules having high velocities considerably. This can readily be understood from the presence of the factor, exp -mc2/2kT), in Eq. 3.7. The exponent has a negative sign and the temperature T is in the denominator. The factor, therefore, increases markedly with increase in temperature. This factor is known as the Boltzmann factor. 

Types of Molecular Velocities Three types of molecular velocities are reckoned with in the study of gases. These are (i) the most probable velocity, cp (ii) the average velocity, , and (iii) the root mean square velocity l/2. 

The most probable velocity is defined as the velocity possessed by maximum number of molecules of a gas at a given temperature. 

With the help of the Maxwell equation Eq. 3.8, it is possible to derive mathematical expressions for the three types of velocities, viz., the most probable velocity, cp the average velocity, and the root-mean-square velocity, /2. These expressions are as follows ... 

Fig. Molecular distribution of the three types of velocities. Example For hydrogen gas, calculate (a) the root mean square velocity /2 (b) the average velocity and (c) the most probable velocity cp at 0 C.

Example. Calculation the temperature at which the root mean square velocity, the average velocity and the most probable velocity of oxygen gas are all equal to 1,500 m s-1. 

Example Calculate the value of p(c) for oxygen molecules at 27 C when c = cp, where c is the most probable velocity. 

The most probable velocity in the x direction is zero with positive and negative velocities having equal probabilities. 

The most probable velocity is obtained by taking the derivative of f[v) dv with respect to v and setting it equal to zero. 

Example 3.5 Compute the most probable velocity of an air molecule at standard pressure and 20°C. Remember that m - MW/NA. 

We see that for the individual components the most probable velocity is the same as the mean velocity, namely, zero (v, = Vj, = v = 0). That means that the most frequently observed component in a trial sampling of the gas will be zero. Similarly, by drawing on the properties of the Gaussian distribution (Sec. VI.8), the rms components are a/2 ... 

The velocity distribution for nitrogen gas at 273 K, with the values of most probable velocity (ump, the velocity at the curve maximum), the average velocity (uavg), and the root mean square velocity (urms) indicated. 

Analysis of the expression for f(u) yields the following equation for the most probable velocity ump (the-velocity possessed by the greatest number of gas particles) ... 

Thus we have three ways to describe a typical velocity for the particles in an ideal gas the root mean square velocity, the most probable velocity, and the average velocity. As can be seen from the equations for wrms, wmp, and wavg, these velocities are not the same. In fact, they stand in the ratios... 

J 27. The most probable velocity, ztmp, is the velocity possessed by the greatest number of gas particles. At a certain temperature, the probability that a gas particle has the most probable velocity is equal to one-half the probability that the same gas particle has the most probable velocity at 300. K. Is the temperature higher or lower than 300. K Calculate the temperature. 

C 125. 16.03 g/mol 127. From Figure 5.16 of the text, as temperature increases, the probability that a gas particle has the most probable velocity decreases. Since the probability of the gas particle with the most probable velocity decreased by one-half, the temperature must be higher than 300. K. The temperature is 1.20 X 103 K. 129. 42% C6H14, 58% CjH8 131. Xco = 0.291 XCo,... 

---