Main Office: No. 25-01, 35-01 Jalan Austin Heights 3/1, Taman Mount Austin (Tel: 07-3616538)Branch: No. 15A, 17A, 21A & 41A Jalan Indah 16/12, Taman Bukit Indah (Tel: 07-2349168)

Name :Class :A2 Physics 1

Subject :A2 PhysicsLesson No :4

Chapter :11 Ideal GasesDate :5/7/14 (Sat)

Time :11.30pm-1.30pm

Topic :11.1 Ideal gas, pV=mRT, kinetic theory, Brownian motion inference, pressure of gas, & kinetic energy of gas

Ideal Gas

1. Ideal gas is a hypothetical gas that obeys the equation of state, pV = nRT at all values of pressure, volume and temperature.

2. The ideal gas is defined by a set of basic assumption in the kinetic theory of gasses.

3. No real gas will behave exactly like an ideal gas.

4. This idealized gas model works best at very low pressures and high temperatures, when the gas molecules are far apart and in rapid motion.

5. The internal energy of an ideal gas is only the kinetic energy of the molecules. It has no potential energy. (no intermolecular force no potential energy).

Equation of State

p= pressure of the gas, PaV= volume of the gas, m3n= amount of gas, molR=molar gas constant, 8.31 J mol-1 K-1 (CASIO CONST27)T=absolute temperature of the gas, K

* T must always be in KELVIN!

1. One mole has 6.02 x 1023 molecules (or atoms). That number is known as the Avogadros constant, NA. It is the number of atoms in 0.012kg of carbon-12.

2. whereN is the number of molecules.

is defined as Boltzmanns constant, k. It has a value of 1.38 x 10-23 J K-1.

So, . This is simply another form of the equation of state.ConstantRelationLaw

temperature,T

Boyles Law

pressure,p

Charless Law

volume,V

Pressure Law

*The mass is constant for all three.

A levels Practice 11. Find the volume of I mole of ideal gas at standard atmospheric pressure (101.3 x 103 Pa) and temperature (273 K). (Molar gas constant, R = 8.31 JK-1mo1-1)

2. A closed vessel of volume 0.250 m3 contains hydrogen (H2) gas at pressure of 2.00 x 105 Pa and temperature 300 K. What is the mass of hydrogen gas?

3. [Q2/J2011/42] (a) State what is meant by a mole.

(b)Two containers A and B are joined by a tube of negligible volume, as illustrated inFig. 2.1.

The containers are filled with an ideal gas at a pressure of 2.3 x 105 Pa.The gas in container A has volume 3.1 x103 cm3 and is at a temperature of 17 C.The gas in container B has volume 4.6 x103 cm3 and is at a temperature of 30 C.Calculate the total amount of gas, in mol, in the containers.amount = ........................................ mol [4]

4. [Q2/J2011/41](a) State what is meant by the Avogadro constant NA. [2]

(b) A balloon is filled with helium gas at a pressure of 1.1 x105 Pa and a temperature of25 C. The balloon has a volume of 6.5 x104 cm3. Helium may be assumed to be an ideal gas. Determine the number of gas atoms in the balloon. [4]

5. [NOV03/4/3a] The volume of some air, assumed to be an ideal gas, in the cylinder of a car engine is 540cm3 at a pressure of 1.1 x 105 Pa and a temperature of 27 C. The air is suddenly compressed, so that no thermal energy enters or leaves the gas, to a volume of 30 cm3. The pressure rises to 6.5 x 106 Pa.

(a)Determine the temperature of the gas after the compression. [3]

6. [JUN02/4/3b] Two insulated gas cylinders A and B are connected by a tube of negligible volume as shown in Fig. 3.1.

Each cylinder has an internal volume of 2.0 x 10-2 m3. Initially, the tap is closed and cylinder A contains 1.2 mol of an ideal gas at temperature of 37 C. Cylinder B contains the same ideal gas at pressure 1.2 x 105 Pa and temperature 37 C.

(i)Calculate the amount, in mol, of the gas in cylinder B. [3]

(ii)The tap is opened and some gas flows from cylinder A to cylinder B. Using the fact that the total amount of gas is constant, determine the final pressure of the gas in the cylinders.

Kinetic Theory of Gases

1. It is a simplified model that explains the physical properties of gases in terms of the motion of its constituent particles.

2. Assumptions:

All gases are made up of a large number of molecules. All molecules are in random motion The volume of the molecules themselves is negligible compared to the volume occupied by the gas. The intermolecular forces between moleculs are negligible except during collisions. Collisions between molecules or between molecules and the walls of the container are perfectly elastic and of negligible duration

Brownian Motion Inference[footnoteRef:1] [1: ]

From a Brownian motion experiment (e.g. smoke particles in AS) the evidence for movement of molecules can be inferred.

The haphazard and random motion of smoke particles is due to unbalanced collision rates by air molecules which move randomly in all directions with different speeds.

Pressure of Gas

1. When gas molecules rebound from walls of container, change in momentum gives rise to force (Newtons 2nd law).2. The molecules and the wall feel the same amount of force but in opposite direction (Newtons 3rd law). 3. Many molecules are colliding with the walls at any given moment. 4. Pressure is the average force over the area in contact.

Derivation of &1. An expression for the pressure exerted by an ideal gas can be derived using the kinetic theory of gases.2. Suppose an ideal gas in a cube of sidesl contains N molecules each of mass m as shown in Figure 9.1. A typical molecule moves with velocity c, which can be resolved into components u, v and u in the three directions Ox, Oy and Oz respectively as shown.

3. Consider the motion along Ox. The molecule has momentum mu in the direction of Ox. When it strikes the wall A of the cube elastically, it rebounds with momentum -mu as shown in Figure 9.2. So the change of momentum = mu - (-mu) = 2mu.

4. After the rebounce, the molecule moves to the other side of the cube and back before bouncing off wall A again. The time taken for this round trip of distance 2l is 2l/u. Hence, the rate of change of momentum due to one molecule

5. Newton's Second Law of Motion states that force equals rate of change of momentum. Therefore, the total force on wall A due to the impact of all N molecules in the cube is given by

whereu1, u2,uN, are the Ox components of the velocities of molecules 1, 2, ..., N6. The pressure on wall A is

7. If represents the mean value of the squares of all the velocity components in the Ox direction, then

8. Since , the pressure on wall A becomes

9. For any molecule, the application of Pythagoras' theorem twice to Figure 9.3 shows that c2 = u2+ v2 + w2. This is also true for the mean square values.

10. However, since the number of molecules N is large and the molecules move randomly, it follows that the mean values for u2, v2 and w2 are equal. Therefore,

11. Since l3 = volume of the cube = volume of gas = V, so

12.

Since = , hence .

13. We can also express pressure as p = , where n is the number of molecules per unit volume.

14. In summary,

wherep is the density of gas, n is the number of molecules volume and m is the mass of one molecule.Molecular Kinetic Energy1. If M is the mass of one mole of a gas and V. is its volume, then

density .

From the equation

Also and where is Avogadro number and m is the mass of a gas molecule.

Therefore

Mean translational kinetic energy of the gas molecules,

=

where k = = 1.38 x 10-23 J K-1 is Boltzmann's constant.

2. The mean translational kinetic energy is proportional to the absolute temperature T.

3. From this, we can see that the meantranslational kinetic energy, of a single molecule is proportional to T. (Total Ek = U = )

4. Since the internal energy of an ideal gas is solely the total kinetic energy, it means if the temperature changes, the internal energy must change, and vice-versa.

5.

Unless is given, you CANNOT use it directly. You have to derive from and pV = NkT. (but you can use T)

Molecular Speeds in a Gas

1. In any gas, the molecules randomly collide with each other. In these collisions, some molecules gain energy (in another word, speed) while others lost it. As a result, at any instant, the molecules have a range of speeds, as shown in the distribution graph below.

2. The temperature and pressure of a gasdepends, not on the average speed of the molecules, but in the average of (speed)2, as can be seen from the equations derived.

3. Mean square speed is the average of (speed)2 for all the molecules, .

4. Its square root is called root mean square speed, .

A levels Practice 2

1. [Nov11/41/2](a)One assumption of the kinetic theory of gases is that gas molecules behave as if they are hard, elastic identical spheres.State two other assumptions of the kinetic theory of gases. [2]

(b)Using the kinetic theory of gases, it can be shown that the product of the pressurep and the volume V of an ideal gas is given by the expression

wherem is the mass of a gas molecule.

(i)State the meaning of the symbol

1.N, [1]

2.< c2> [1]

(ii)Use the expression to deduce that the mean kinetic energy of a gas moleculeat temperature T is given by the equation

wherek is a constant. [2]

2. [JUN08/4/2(a)] (a) Explain qualitatively how molecular movement causes the pressure exerted by a gas. [3]

3. [JUN04/4/2] The pressure p of an ideal gas is given by the expression .

(a) Explain the meaning of the symbol . [2]

(b)The ideal gas has a density of 2.4 kgm-3 at a pressure of 2.0 x 105 Pa and a temperature of 300 K.

(i)Determine the root-mean-square (r.m.s.) speed of the gas atoms at 300 K. [3]

(ii)Calculate the temperature of the gas for the atoms to have an r.m.s. speed that is twice that calculated in (i). [3]

Homework4. [NOV08/4/5b] (b) For a fusion reaction to occur, the deuterium nuclei must come into contact. Assuming that deuterium behaves as an ideal gas, deduce a value for the temperature of the deuterium such that the nuclei have r.m.s. speed equal to the speed calculated in (a).(v = 2.5 x 106 m s-1) [4]

5. [JUN02/4/3a] (a) (i) The kinetic theory of gases leads to the equation

Explain the significance of the quantity . [2]

(ii)Use the equation to suggest what is meant by the absolute zero of temperature. [1]

6. [JUN08/4/2b,c] (b) The density of neon gas at a temperature of 273 K and a pressure of 1.02 x 105 Pa is 0.900 kg m-3. Neon may be assumed to be an ideal gas. Calculate the root-mean-square (r.m.s.) speed of neon atoms at

(i)273 K, [3]

(ii)546 K. [2]

(c)The calculations in (b) are based on the density for neon being 0.900 kg m-3. Suggest the effect, if any, on the root-mean-square speed of changing the density at constant temperature. [2]

7. [JUN10/41/2b,c].The pressure p of an ideal gas is given by the expression where is the density of the gas.

(i)State the meaning of the symbol . [1]

(ii)Use the expression to show that the mean kinetic energy . Explain any symbols that you use. [4]

(c)Helium-4 may be assumed to behave as an ideal gas. A cylinder has a constant volume of 7.8 x 103 cm3 and contains helium-4 gas at a pressure of 2.1 x 107 Pa and at a temperature of 290 K. Calculate, for the helium gas,

(i)the amount of gas, [2]

(ii)the mean kinetic energy of the atoms, [2]

(iii)the total internal energy. [3]

12