eu3c7 by adel khamis

Unit Three -340- Chapter seven

Mention when the following values vanish

Previous Exams:

1. (Azhar 2002) Average kinetic energy of gas molecules(1).

Which of the following statements are right and which are wrong?

Rewrite the incorrect statements in a correct form

Previous Exams:

1. (Egypt 91) Two different gases have the same temperature so that the average

velocity of their molecules is the same(2).

Complete the following statements

1. A gas moves in a cubic container, then

a. Change in momentum for one molecule for one collision = ………..(3)

b. Number of collisions for one molecule in one second = ……….

c. Force acting on the wall by all molecules = ……….

Previous Exams:

2. (AlAzhar 90) The quantity of mater containing Avogadro’s number is called

…………(4)

3. (AlAzhar 2002) At constant temperature the kinetic energy of oxygen at

pressure 2 atmospheric pressures …………(5) the kinetic energy of Nitrogen at 8

atmospheric pressure. And that is because K.E = ………..

What will happen when

1. The temperature of a gas decreases to zero Kelvin.

2. The temperature of a gas on the Kelvin scale is doubled.

3. The temperature of a gas on the Kelvin scale is doubled and its volume is

doubled.

Exercises 2008/2009


4. The gas is compressed very slowly to increase its pressure to the double

and decreases its volume to the half.

What is meant by

Previous Exams:

1. (Azhar 92) Avogadro’s number = 6.023 x 1023 molecules.

Choose the correct answer from those between brackets and write it

in your answer paper

Evaluation book:

1. According to the kinetic theory of gases, when a gas is compressed while its

temperature is kept constant:

a) The average velocity of its molecules increases.

b) The average velocity of its molecules decreases.

c) Higher pressure is generated on the walls of its container.

d) Smaller pressure is generated on the walls of its container.

e) The density of the gas decreases.

2. Nitrogen gas in a vessel at normal temperature and pressure (NPT), one mole

of nitrogen equals 0.028 kg and avogadro’s number equals 6.023x1023 and

occupies a volume of 22.4 x 10-3 m3 Hg = 13600 kg/m3 g = 9.8 m/sec2. Then the

root mean square of velocity of the nitrogen molecules is:

a) 4.58 m/s

b) 450 m/s

c) 540 m/s

d) 490 m/s

e) 493 m/s

Exercises 2008/2009


3. The change in momentum of a nitrogen molecule in each collision

perpendicular to the walls of the container is:

a) 4.58 x 10-23 kg m sec-1

b) 5.58 x 10-23 kg m sec-1

c) 4.58 x 10-24 kg m sec-1

d) 4.58 x 10-22 kg m sec-1

e) 5.58 x 10-22 kg m sec-1

4. At zero Kelvin [The gas pressure vanishes and mass increase – The gas

volume vanishes at constant pressure – The mass of the gas vanishes at constant

pressure] (6).

5. At (-273) degree Celsius [the kinetic energy of the molecule vanishes – the

root mean square velocity of the molecule increase – the momentum of the

molecule constant] (7).

6. If the temperature of gas on the Kelvin scale is double then:

a) The average square velocity of the molecule is doubled.

b) The kinetic energy of the molecule is doubled.

c) The root mean square velocity is doubled.

7. If the temperature of a gas on Kelvin scale is doubled and its volume is

doubled then:

a) The density of the gas is doubled.

b) The gas pressure is doubled.

c) The average square of velocity is doubled.

8. If a gas is compressed very slowly to increase its pressure to the doble and

decrease its volume to the half then the mean kinetic energy of its molecules:

Exercises 2008/2009


a) Decrease to the half.

b) Increase the double.

c) Remains constant.

Additional questions:

9. When the pressure of a certain mass of a gas is increased to double its value at

constant speed, then its density [decreased to half its value – increased to double

its value – increased to four times its value](8).

10. When the temperature of Oxygen gas is increased from 100K to 400K, the

root mean square speed of its molecules [increased to 4 times its value – increased

to double its value – decreased to half its value](9).

11. The ratio between average K.E. of nitrogen molecule at 27C and average K.E.

of Hydrogen molecule at the same temperature is [les than one – equal one - more

than one] (10).

Previous Exams:

1. (Egypt 95) When a gas is compressed slowly at constant temperature such that

its pressure is doubled and its volume becomes half its original one, the average

velocity of its molecules will be [doubled – does not changed – decreased to its

half].

2. (Egypt 99) The ratio between the root mean square velocities of Hydrogen gas

molecules at 200C to the root mean square velocity of Nitrogen gas molecules at

the same temperature [is larger than one - is smaller than one - equals one - there

is no relation between them].

3. (Azhar 2002) At constant temperature, when the pressure of gas increases its

density [decreases – increases – remains constant]

Exercises 2008/2009


Give reasons

Evaluation book:

1. The absolute zero is the temperature at which the kinetic energy of gas is

vanished.

2. The velocity of gas molecules is constant although they collide with one

another and collide with the walls of the container(11).

3. The root mean square velocity of the gas molecules and its K.E. does not

depend on its pressure(12).

Additional questions:

4. Gases are compressible.

5. Kinetic energy of two molecules of different gases is the same at the same

temperature while their velocities are variable.

Previous Exams:

6. (Egypt 99) In spite of the relation between the gas pressure (P) and the mean

square velocity of its molecules (V2) is given by P = 1/3 V2 where () is the gas

density, but the root mean square velocity of gas molecules does not depend on its

pressure at constant temperature.

7. (Azhar 2002) There is no atmosphere at the moon.

Essay questions

School Book:

1. State the main postulates of the kinetic theory of gases.

2. On the basis of the postulates of the kinetic theory of gases, show how to

prove that the gas pressure P is given by the relation , where ρ: is the gas

density and v2 is the mean square speed of its molecules.

Exercises 2008/2009


3. Using the previous relation, show how to find expression for each of the

following:

a) The root mean square speed of the gas molecules.

b) The concept of the gas temperature.

c) The average kinetic energy of a free particle.

4. A uniform cubic vessel of side length “l” has gas whose molecule has

mass m moving in the x direction with velocity vx, and collides with the walls of

the vessel in perfectly elastic collisions:

a) What is the linear momentum of the molecule before collision?

b) What is the linear momentum of the molecule after collision?

c) What is the change in linear momentum of the molecule on collision?

d) What is the distance traveled by the molecule before the next collision

with the walls of the vessel?

e) What is the number of the collisions with the walls of the vessel per

second made by the molecule?

f) What is the total change in linear momentum of one molecule per

second due to its successive collisions with the walls of the vessel?

g) What does the above quantity represent?

h) If NA is the number of the gas molecules in the container, what will be

the total force acting on the internal surface of the vessel?

4. Suppose that the atoms of helium gas have the same average velocity as the

atoms of oxygen gas, which of them has a higher temperature and why?

Evaluation Book:

5. Using the relation prove boyle’s law.

Exercises 2008/2009


6. Find the value and the unit of the general gas constant (R), and boltzman

constant. Given that NA = 6.023x1023 molecules

7. A container made of plastic has thin walls and confined a quantity of dry

air of pressure that equals to the atmospheric pressure, explain according to the

kinetic theory of gases why the container walls will compressed from some sides

when it placed in a freezer of a certain refrigerator of very low temperature.

8. Which graph represents each of the following relation:

a) Volume of gas and temperature

(C) at constant pressure.

b) Specific heat of a metal and its

mole’s mass.

c) Pressure of a gas and temperature (K) at constant volume.

9. Prove that the average square root velocity of a molecule of a gas =

10. Find at which temperature a volume of a certain mass of a gas at steady

pressure:

a) Is double of its volume at 0C.

b) Is reduced to half main volume at 0C.

Previous Exams:

11. (August 2000) Which of the following graphs illustrates the relation

between each of the following:

a) The relation between the pressure at a point inside a lake and the depth of

the point from its surface.

Exercises 2008/2009


b) The relation between the volume of a fixed mass of a gas and its pressure

at constant temperature.

c) The relation between the angles of incidence of light rays on one side of a

triangular prism and their angles of deviation.

d) The relation between the volume of a fixed mass of a gas and its

temperature on Celsius scale at constant pressure.

12. (Egypt 97, Egypt 2002) The gas pressure is determined according to the

kinetic theory of gases from the relation and from the general gas law

for one mole PV = RT and given the relation between Boltzmann’s constant and

the general gas constant Prove that the average kinetic energy of a

molecule of gas is proportional to its absolute temperature.

13. (Egypt 97) Given that the force by which each molecule of a gas acts on

the internal surface of a spherical vessel per second equals (mv2)/l , where, m is

the molecule mass, v2 is the average square speed of the molecules, l is the length

of the cubic vessel, then prove that P = 1/3 v2 where P is the gas pressure and

the density of the gas.

14. (August 99, Egypt 2000, August 2001) If you know that the pressure of a

gas using kinetic theory of gases is given by P = 1/3 V2, where () is the gas

density and (v2) is the mean square velocity of its molecules, prove that the kinetic

energy of gas molecules vanishes at zero Kelvin.

15. (Egypt 94, August 96, August 2002) Write down the postulates of the

kinetic theory of gases, and then prove that the average kinetic energy of a gas

molecule is directly proportional to its absolute temperature.

16. (Egypt 93) Each of two identical quantities of a certain gas is placed in a

cylinder provided with a movable press. The gas pressure in the first cylinder (at

constant temperature) is doubled and the temperature of the gas in the second

Exercises 2008/2009


cylinder is raised (at constant pressure) to its double value on Kelvin scale.

Mention and explain what happens to the average kinetic energy of the gas

molecules in each cylinder.

17. (Egypt 92) A gas molecule of mass m. moves with velocity v to be

incident on the walls of a cubic container of length “l”. If the angle of incidence is

Ф, Show that the total change in the normal component of the momentum of one

molecule per second =mv²/l

18. (Al-Azhar 1991) Describe an experiment, which can be used to determine

the absolute zero.

19. (Egypt 2001) A gas molecule of mass (m) is moving with velocity (v) in

a direction perpendicular to the internal surface of a uniform cubic vessel of length

(l). Prove that the total change in the momentum of this molecule per unit time =

mv2/r

20. (Azhar 2002) Molecule of mass (m) moves with velocity (v) along the

diameter of spherical container. Find the force of that molecule at the wall of the

container.

21. (Egypt 92) For a given quantity of gas, its volume pressure and

temperature are recorded at certain conditions and the recorded at other conditions

as indicated in the corresponding table .you have to choose throughout the five

groups A, B, C, D and E the most suitable one for each of the following (knowing

that each group may be used once or more or not used at all):

Gas information A B C D E

P1, atm, pressure 2 2 2 2 2

V1, liter 4 4 4 4 4

T1, cº 27 27 27 27 27

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P2, atm, pressure 1.8 2 4 2 1

V2, liter 4 8 3 5 8

T2, Kelvin 270 600 450 375 300

a) The general gas law.

b) The constancy of gas density.

c) The constancy of the root-mean-square velocity of the molecules.

d) The increase of the rate of collisions of gas molecules with the wall of the

container.

e) Boyle’s law.

f) Pressure law.

Problems

School Book:

1. Hydrogen gas in a vessel at NTP calculate the root mean square speed of

its molecules. (NA = 6.02 x 1023, Mass of Hydrogen mole = 0.002 kg) then

calculate the change in the momentum of the hydrogen molecule in the previous

problem on each impact perpendicular to the walls of the vessel?

[1844.9 m/s, 1.23x10-23 kg.m/s]

2. What is the change in linear momentum of the hydrogen molecule in the

above problem on each impact perpendicular to the walls of the vessel?

[1.224x10-23 Kg.m.sec-1]

3. (Egypt 95) Calculate the average kinetic energy of a free electron at 27C

(K = 1.38 x 10-23 J/k).

[6.21x10-21 J]

Exercises 2008/2009


4. Using the data given in the previous problem, find the root mean square

speed of a free electron if its mass is 9.1x10-23 Kg

[1.168x105 m/sec]

5. Find the ratio between the root mean square speed of the molecules of a

certain gas at temperature 6000K (sun’s surface) and that at temperature 300K

(Earth’s surface).

[4.472]

1 (?) At temperature equals to zero degree Kelvin.

2 (?) Wrong, Two different gases have the same temperature so that the average

Kinetic energy of their molecules is the same.

3 (?) 2mv

4 (?) Mole

5 (?) Equal to

6 (?) The gas volume vanishes at constant pressure.

7 (?) The kinetic energy of the molecule vanishes.

8 (?) Increased to double its value.

9 (?) Increased to double its value.

10 (?) Equals to one.

11 (?) Because the molecules make elastic collisions with the walls of the

container.

12 (?) Because the gas pressure changes the volume and the density but the

molecules velocity remains constant when the temperature remains constant.

Exercises 2008/2009


6. Calculate the average kinetic energy and root mean square of the velocity

of a free electron at 300ºK, where Boltzmann’s constant = 1.38 x 10-23 J/K, the

mass of electron is 9.1x10-31 Kg.

[1.17x105 m/s]

7. An amount of an ideal gas has a mass of 0.8 x10-3 kg, a volume of

0.285x10-3 m3 at a temperature of 12º C and under pressure of 105 N/m2. Calculate

the molecular mass of the gas where the universal gas constant equals 8.31 J/K.

8. Calculate the mean kinetic energy of an oxygen molecule at a

temperature 50ºC, where Boltzmann’s constant is equal to 1.38x10-23 J/K.

9. If the temperature at the surface of the Sun is 6000ºK, find the root mean

square speed of hydrogen molecules at the surface of the sun, knowing that the

hydrogen is in its atomic state. Its atomic mass = 1, Avogadro’s number (NA) =

6.02 x 1023, and Boltzmann’s constant = 1.38 x 10-23 J/K.

Evaluation Book:

10. Calculate the number of molecules in 20 liters of a gas at 1495.8x105

N/m2, 27C. if Avogadro’s number is 6.023x1023 molecules and Boltzman’s

constant is 1.38x10-23 J/K

[7.226x1026]

11. Nitrogen gas in a vessel at normal temperature and pressure (N.T.P) one

mole of nitrogen equals 0.028 kg and Avogadro’s number equals 6.02 x 1023 and

occupies a volume of 22.4 liter, find the root mean square of velocity of the

nitrogen molecules, and the change in momentum of a nitrogen molecule in each

collision perpendicular to the walls of the container.

[493 m/s, 4.58 x10-23kg.m/s]

Exercises 2008/2009


12. At a certain temperature the R.M.S speed of Oxygen molecule is 500 m/s.

Calculate the R.M.S speed of Hydrogen whose atomic mass is 1/16 that of

Oxygen at the same temperature.

[2000 m/s]

13. If the temperature of the external outer space is about 3.4K and it

contains 1 Hydrogen atom/m3. Find the gas pressure given that K = 1.38 x 10-23

J/K.

[4.692 x 10-23 N/m2]

14. Calculate in Celsius degrees the degree at which the root mean square

velocity of oxygen gas molecules becomes twice its value at 27C.

[927C]

15. Calculate the number of molecules in unit volume of Oxygen gas at SPT

if root mean square speed of its molecules at STP = 4.62 x 102 m/s, mass of one

molecule 52.8 x 10-27 kg.

[2.699 x 1025 molecule /m3]

16. Calculate the number of molecules of a liquid vapor occupying a volume

0.277 liters at (-73C) under pressure 3x10-12 Pascal given that Avogadro’s number

= 6x1023 molecules/mole and universal gas constant 8.31 J/K

[3x105 molecules]

17. If the center of the sun consists of gases of average molecular mass 0.7

gm, density 9x104 kg/m3, Pressure 1.4x1016 N/m2. Calculate the temperature at the

center.

[7.89x1030 K]

Exercises 2008/2009


18. A quantity of gas of mass 0.8 gm occupies 0.285 liter at 12C, Pressure

105 N/m2 calculate the molecular mass of the gas given universal gas constant is

8.31 J/K.

[66.48]

19. Calculate the number of molecules per unit volume under pressure 105

N/m2; if the mass of one molecule is 3x10-26 kg, its root mean square speed is 400

m/s.

[6.25x1025]

20. The following table illustrates the relation between the average kinetic

energy of gas molecule and absolute temperature; represent the relation plotting

average kinetic energy on the ordinate and absolute temperature on the abscissa.

Average K.E (J) 6.2x10-21 8.3x10-21 12.5x10-21 14.5x10-21 16.6x10-21

Temperature (K) 300 400 600 700 800

From the graph find:

a. The average kinetic energy of the gas molecule at 500K.

b. The value of the universal gas constant if Avogadro’s number

6.023x1023 molecule.

21. The following table illustrates the relation between the mean square

velocity of gas molecules (v2)at different temperatures:

T (C) -173 -123 -73 -23 27 77 127

v2 x 104 m2.s-2 9 13.5 18 22.5 27 31.5 36

Plot a graph relating the temperature on the Kelvin scale on the abscissa and

the means square velocity on the ordinate. From the graph find:

a) The relation between v2 and t.

Exercises 2008/2009


b) The mass of one molecule of this gas (Boltzman constant is

1.38 x 10-23 j/k)

c) The means of square velocity of the gas at 0K and also the

kinetic energy of the gas at this temperature.

Previous Exams:

22. (August 97) Given that the root mean square speed of the molecules of

gas at SPT is 5x102 m/s and the mass of each molecule of such a gas is 55x10 -27

kg. Find the number of molecules of such a gas per unit volume.

[2.2 x 1025 molecules]

23. (Egypt 96) Deduce the general gas law, and find the value of the

universal gas constant R (at N.T.P, the pressure = 0.76 m Hg, the temperature =

0C and the mole of gas occupies 22.4 litters), knowing that the density of

mercury is 13600 kg / m3 and g = 9.8 m/s2.

[R=8.31 J/K]

24. (Egypt 96) Calculate the root mean square speed of the molecules of

carbon dioxide gas at 27C, given that at NTP the density of carbon dioxide is 1.96

kg / m3 and the mole of the gas occupies 22.4 litters. (Avogadro’s number =

6.02x1023, Boltzman’s constant = 1.38x10-23 J/K)

[412.76 m/s]

25. (Egypt 90, Egypt 91) The mass of a sample of a gas is 3.2 x 10-3 kg and it

occupies 2.24 liter at STP. Find the square of the average velocity of the gas

molecule in the considered sample at 100C.

[(539.123 m/s)2]

26. (Egypt 96) If the density of nitrogen gas at pressure 0.76 m Hg and at

temp. 0C is 1.25 kg/m3. Calculate the root mean square speed of nitrogen at 0C

and at 300K.

Exercises 2008/2009


[493.05 m/s, 516.8 m/s]

27. (Azhar 92) Under similar conditions of pressure and temperature the

density of hydrogen is 0.09 Kg/m3 and that of nitrogen is 1.25 kg/m3. Calculate the

r.m.s speed of nitrogen and the change in momentum for each collision with the

wall given that r.m.s speed of hydrogen is 1.8 x 103 m/s and the molecular mass of

nitrogen is 4.6 x 10-26 kg.

[483 m/s, 4.444 x 10-23 kg m/s]

Exercises 2008/2009


Model Answers:

Exercises 2008/2009

eu3c7 by adel khamis

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