Maxwell’s speed distribution

This expression is the so-called Maxwell s speed distribution of gas particles. A first impression can be gained in an experiment using glass beads as a model gas (Experiment 10.3). 

Fig. 10.9 Maxwell s speed distribution of N2 molecules at different temperatures. [Included is the corresponding root mean square speed dashed line)J ...

Fig. 10.10 Maxwell s speed distribution fra gases with varying molar masses at 298 K.

If a system is not uniform, it is not in thermodynamic equilibrium then can obey law of Maxwell s speeds distribution. However, if "equilibrium absence" is not big, can considered like good approximation, all little volume (microscopic scale) is in equilibrium (considered like subsystem). This is for two reasons. First little portions of gas contain a big number of molecules. Second the necessary time for established the equilibrium in a little volume is brief in comparison with necessary time for that transport processes get equilibrium in little volume with rest of system (it is true when concentration, temperature, etc. gradients are not too much big). In consequence, can supjpose that is local thermodynamic equilibrium so speed distribution in any volume element (macroscopic) of medium is Maxwellian, although density, temperature and macroscopic velocity change the position (Duderstadt Martin, 1979 Schwabl, 2002 Bhatnagar et al., 1954 Pai, 1981 Maxwell, 1997 Sued, 2001 Succi, 2002 Cercignani, 1975 Lebowitz Montroll, 1983). 

Distribution functions are usually first met in physical chemistiy when the crude treatment of molecular velocities in the kinetic theory of gases (all the molecules taken as having the same mean speed) is replaced by Maxwell s seminal equation showing that the number of molecules having velocities between narrow limits depends very much on what velocities are chosen. This is shown in Fig. 9.1. Thus, this first and basic distribution law of Maxwell, the distribution of velocities, gives an unexpected result (the nonsymmetrical nature of the distribution), which still causes us to think, more than a century after its publication. 

All scientists know Maxwell s law concerning the fractions of molecules present at a certain speed and how this fraction varies with their speed. This is the best-known distribution law. State other distribution laws and represent each graphically in an electrochemical context. (Bockris)... 

If the laboratory speeds 0, are distributed according to Maxwell s law, relative (and also center-of-mass) speeds are likewise,... 

Exercise. Maxwell s derivation of the velocity distribution in a gas was based on the assumptions that it could only depend on the speed v, and that the cartesian components are statistically independent. Show that this leads to Maxwell s law. 

In section 2 the distribution of speeds among the molecules of a gas was discussed. According to Maxwell s law the fractional number of molecules possessing energies corresponding to velocity components between u and u + du is given by the expression... 

This result was obtained by Maxwell in 1860 and is called the Maxwell s distribution of molecular velocities. It is customary to writep(c)dcas dNIN, where /Vis the total number of gas molecules. The quantity dN/Nor p(c)dc gives the fraction of molecules with speeds between c and c + dc. The molecular mass m = Mn/NA where Mm is the molar mass and NA is the Avogadro number. Accordingly, Eq. 3.7 may also be written as... 

What does the Maxwell speed distribution curve tell us Does Maxwell s theory work for a sample of 200 molecules Explain. 

The main aspect of kinetic theory is that the molecules in a gas are in a state of continuous random motion. The speed of molecules depends on the temperature, and can have a remge of values. Maxwell s distribution curve is perfect to analyze this fact. In the Maxwell s distribution curve, the relative number of molecules are plotted against the molecular speed on the x-axis. Take a look at the curve in Figure 6-2. 

Maxwell s Distribution of Speeds We will now return to our derivation of... 

Maxwell s distribution of molecular speeds a theoretical relationship that predicts the relative number of molecules at various speeds for a sample of gas at a particular temperature. (5.7)... 

Let V represent a particle s speed, N the total number of particles,/(v) the Maxwell distribution of speeds, A a finite change, and d an infinitesimal change not density) in the following relationships. 

Knowing the Maxwell distribution of a molecule s speed (3.3.6) derive function n) where u is the molecule s relative speed u=(v/Vp ), i.e., expressed in dimensionless units. Apply this distribution to find that the particular molecule has the probabiUty w of having a speed that differs from half the most probable speed by not more than 1%. 

S FIGURE 4.27 The Maxwell distribution t again, but now the curves correspond to the speeds of a single substance inf molar fyg mass 50 g-mol 1) at different... 

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