6.6 Molecular Kinetic Theory Model

Kinetic theory assumes that collisions between particles and between particles and the walls of their container are perfectly elastic. 

Describe the term elastic collision, and write down three more assumptions about the properties and behaviour of gas molecules which are used in the kinetic theory to derive an expression for the pressure of an ideal gas.

A gas collimator can be used to measure the average velocity of particles in a gas. The collimator in Figure 1 maintains a gas at a constant temperature. Pressure and volume are also controlled. 

Figure 1

Explain why the double aperture is necessary in order to measure the velocity of the particles in the collimator.

A scientist is measuring the average velocity of a sample of dichlorodifluoromethane, a CFC used as a fire retardant. At a temperature of 21 °C, the dichlorodifluoromethane molecules have an average velocity, , of 139 m s^-1, with an uncertainty of 1%. 

Dichlorodifluoromethane has a molecular mass of 121 g mol^-1. 

Determine whether the dichlorodifluoromethane behaves as an ideal gas.

Describe the conditions in which the experiment should be run such that the gases tested will behave most like an ideal gas.

An ideal gas is expanded whilst maintaining at constant pressure of 5 × 10⁵ Pa. The graph below in Figure 1 shows the relationship between pressure p and volume V for this change. 

Figure 1

The change in the internal energy of the gas during this expansion is 1000 J. 

Calculate the amount and the direction of thermal energy transferred during the change.

Explain in terms of the kinetic theory how the pressure of the gas in the cylinder can remain the same when the temperature of the gas and the volume of the container are both increased.

The gas in (a) has a temperature of 120 °C after its expansion.

A car engine contains cylinders in which fuel is pressurised using pistons. As the piston moves up and down, the height of the cylinder h within which the gas is changes, as seen in Figure 1.

Figure 1

A quantity of 0.1 mol of gas enters an engine, at the Intake section of the cycle, at a pressure of 1.05 × 10⁵ Pa and a temperature of 27°C. Assume the gas to be ideal.

If the diameter of the piston is 90 mm. Calculate the current height of the cylinder, h.

During compression, the volume of the gas is reduced to one twentieth of its original volume, and the pressure rises to 7.0 × 10⁶ Pa.

Calculate the temperature of the gas immediately after the compression.

Upon compression the gas ignites and the rapid expansion causes the piston to be forced down again. The chemical energy released in the combustion is equal to 45 MJ kg^-1, and is converted to the kinetic energy of the car. 

The molar mass of the diesel fuel is 200 g mol^-1 

Calculate the total energy of the fuel in the piston immediately before ignition.

A different cylinder under testing contains n moles of the ideal gas. Initially the pressure in the cylinder is p and the volume occupied by the gas is V. The cylinder is opened and 3 × 10²² gas molecules are released. This causes a 50% increase in the volume and a 50% decrease in the pressure in the cylinder. The average kinetic energy of the gas molecules is unchanged. 

Calculate the value of n.

Within a rectangular box of dimensions , , , N gas molecules each of mass m, are all assumed to move parallel to the x-direction with speed  and make elastic collisions at the ends. One such particle is shown in Figure 1. 

Figure 1

Show that the average force, F, on the shaded face is given by:

            F =

The motion of the particles within the box is, in reality, random and in three dimensions. The average velocity of the particles, , can be determined from the sum of the velocity in each direction:

             =  

When dealing with a large number of molecules the mean square value of movement vectors of the molecules with totally random motion will show no preferred direction, therefore:

And the root mean square velocity, , is given by: 

Use the information above, and your answer to part (a), to derive the equation:

            pV = Nm

Discuss how experimental evidence informs our understanding of the behaviour of a gas, and whether models, such as the kinetic theory of gases, represent real life situations. In your answer you should: 

• Explain how experiments into the gas laws formed conclusions about the behaviour of a gas
• Explain differences between the behaviours of real and ideal gases, and describe some of the assumptions on which the kinetic theory of gases is formed
• Comment on whether empirical evidence or models, such as this, better describe reality. 

You may wish to draw a diagram to clarify your answer.

The quality of your written communication will be assessed in your answer.

The mass of one mole of helium molecules is 4.00 g mol^–1. 

Calculate the typical speed of helium molecules at 208 K.

The helium gas has an initial pressure p. The temperature, T of the helium molecules increases so that the root mean square (r.m.s) speed is doubled. 

What will the new pressure be, in terms of p, if the volume remains constant?

State three assumptions used in the derivation of 

         pV =

Explain in terms of the kinetic theory why the pressure of a gas in a cylinder falls when gas is removed from the cylinder.

State four assumptions made in the molecular kinetic theory model of an ideal gas.

A gas of N molecules, each with mass m and root mean square speed  in a volume V have a pressure p. 

By using the kinetic theory model equation pV = , derive the equation for the kinetic energy of a single gas molecule.

Helium is a monatomic gas. Therefore, all the internal energy of the molecules may be considered to be translational kinetic energy only. 

Molar mass of helium = 4.0 × 10^–3 kg mol^–1 

Calculate the internal energy of 2.5 g of helium gas at a temperature of 52 K.

At what temperature would the internal energy of 2.5 g of helium gas be equal to a tennis ball with a kinetic energy of 160 J.

A single gas molecule of mass m is moving in a rectangular box with a velocity of  in the positive x-direction, as shown in Figure 1. The molecule moves backwards and forwards in the box, striking the shaded end faces normally. 

Figure 1

Altogether, there are N molecules in the rectangular box. 

In terms of  and  determine the time interval, t, between collisions with a shaded face. 

Show that the change in momentum per collision with a shaded face is .

(i)         Show that the average force, F, on the shaded face is given by    

(ii)        State one assumption made.

In a better model of molecular motion in gases, molecules of mean square speed are assumed to move randomly in the box. 

By first obtaining an expression for  in terms of the mean square speed, , show that a better expression for the average force, F, is  

Hence, derive the kinetic theory of gases equation pV = .

Figure 1 below shows smoke particles suspended in air. The arrows indicate directions in which the particles are moving at a particular time. 

Figure 1

Particles in the gas are observed to move with Brownian motion. 

State two conclusions about smoke particles and their motion resulting from this observation.

A sample of air has a density 2.74 kg m^–3 at a pressure of 1.50 × 10⁵ Pa. Each air molecule has a mean kinetic energy of 0.05 eV.

Calculate the temperature of the air under these conditions.

Calculate the root mean square (r.m.s) speed of the air molecules.

Explain why, when the mean kinetic energy of the molecules increases to 2.00 eV, some of the molecules will have speeds much less than that suggested by the value you calculated in part (c).