The Kinetic Model of Gases

The basis for our discussion is the kinetic model of gases (also called the kinetic molecular theory , KMT, of gases), which makes the following three assumptions  

The size of the molecules is negligible in the sense that their diameters are much smaller than the average distance traveled between collisions. 

The assumption that the molecules do not interact unless they are in contact implies that the potential energy of the molecules (their energy due to their position) is independent of then-separation and may be set equal to zero. The total energy of a sample of gas is therefore the sum of the kinetic energies 

The kinetic model accounts for the steady pressure exerted by a gas in terms of the collisions the molecules make with the walls of the container. Each collision gives rise to a brief force on the wall, but as billions of collisions take place every second, the walls experience a virtually constant force, and hence the gas exerts a steady pressure. On the basis of this model, the pressure exerted by a gas of molar mass M in a volrnne V is  

speed might at first encounter seem to be a rather peculiar measure of the mean speeds of the molecules, but its significance becomes clear when we make use of the fact that the kinetic energy of a molecule of mass m traveling at a speed f is Ejj=jtnf which implies that the mean kinetic energy, ( it), is the average of this quantity, or mc. It follows that 

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FIGURE 4.23 In the kinetic model of gases, the molecules are regarded as infinitesimal points that travel in straight lines until they undergo instantaneous collisions. 

FIGURE 4.24 In the kinetic model of gases, the pressure arises from the force exerted on the walls of the container when the impacting molecules are deflected. We need to know the force of each impact and the number of impacts in a given time interval. 

In the kinetic model of gases, we picture the molecules as widely separated for most of the time and in ceaseless random motion. They zoom from place to place, always in straight lines, changing direction only when they collide with a wall of the container or another molecule. The collisions change the speed and direction of the molecules, just like balls in a three-dimensional cosmic game of pool. 

The kinetic model of gases is consistent with the ideal gas law and provides an expression for the root mean square speed of the molecules vnns = (3RT/M)l/2. The molar kinetic energy of a gas is proportional to the temperature. 

Do all the molecules of a gas strike the walls of their container with the same force Justify your answer on the basis of the kinetic model of gases. 

What Do We Need to Know Already Much of this chapter stands alone, but it would be helpful to review the kinetic model of gases (Section 4.10) and equilibrium constants (Section 9.2). 

In fact, the kinetic model of gases (Chapter 21) says that the pressure of a gas is equal to - — E where E is the average kinetic energy of the gas molecules-completely consistent with interpreting it as the... 

In expansion, the volume increases, meaning that the box gets bigger. Equation 9.12b tells us that the kinetic energy decreases, even as the quantum numbers remain constant. This is also consistent with what we know of adiabatic expansion and the kinetic model of gases the temperature of the sample drops on expansion, and temperature is related to the kinetic energy (T2 oc E). 

The value of A can be calculated from the kinetic model of gases Further information 7.1), and the result is... 

To find expressions for X and z, we need a slightly more elaborate version of the kinetic model of gases. The basic kinetic model supposes that the molecules are effectively pointlike however, to obtain collisions, we need to assume that two points score a hit whenever they come within a certain range d of each other, where d can be thought of as the diameter of the molecules (Fig. 7.20). The collision cross-section, a (sigma), the target area presented by one molecule to another, is therefore the area of a circle of radius d, so O = nd. When this quantity is built into the kinetic model, we find that... 

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