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4.4 Gas Law

Gas pressure, temperature and volume in terms of  gas molecules.

The kinetic theory of gases was proposed to explain the gas laws. The basic assumptions are:

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1. All gases are made up of a very large number of tiny molecules.

2. These molecules are constantly moving around randomly at high speeds.

3. The molecules collide elastically with anything they meet. If they hit the inner walls of the container , they bounce off again at the same speed.

4. The molecules are so small and so far apart that they almost never collide with each other. So the volume of the gas molecules themselves is negligible with the volume of container, that is, almost all the gas is empty space.

5. They do not exert any forces on each other , but move randomly..There is no intermolecular attractive forces. Intermolecular forces of repulsion act only during collisions between molecules; the duration of collisions is negligible compare compared with the time interval between collisions.

How the gas pressure is produced?

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Based on the assumptions of kinetic theory of gases , molecules of a gas will occupy the entire available space and collisions occur between molecules and the walls of container.

Imagine a molecule of mass m approaching one wall with velocity , v .

Its momentum = mv.

It rebounds with velocity (-v) because it experiences an elastic collision.

Its momentum now is – mv.

So the change of momentum = 2mv

According to Newton’s second law of motion ,

force is exerted on the wall of container because force is the rate change of momentum

( F = change of momentum )

               time

As the result gas pressure is produced because  by the definition of pressure;

Pressure is Force per unit area

( P = F )

         A

Hence the gas pressure  in the container is the total force , produced by the collision between molecules and the walls of container.

The higher the average velocity of the molecules in the gas, the  greater pressure exerted by the gas.

What happen when a gas is heated?

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As a gas is heated , the molecules move faster because the kinetic energy of  the gas molecules is proportional to the temperature of the gas. As the result the pressure of the gas increases if the volume of the gas is fixed.

Therefore a fixed mass of a gas in a container has three characteristics ,i.e pressure, volume and temperature. The relationship between these characteristics can be explained by the three gas laws.

The three gas laws are shown in the following table.

Gas law

Relationship

constant

Appilcation

Boyle

P a 1

V

or

P1V1 = P2V2

T

1.The bubbles formed by a fish expand as they floats towards the surface.

2. Bicycle pump

Charles

V a T

V1 = V2

T1 T2

P

1. Hot- air balloon

Pressure

P a T

P1 = P2

T1 T2

V

1.Car tyres after a long drive become very firm.

Boyle’s Law

Boyle’s law states that “ For a fixed mass at constant temperature, the pressure of gas is inversely proportional to its volume”

Or

P α 1 if T   constant

      V

Where           P = pressure , V = volume and        T = temperature

Or

P = k

      V

PV = k

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Example 1

A mixture of air and petrol vapour is drawn into the cylinder of a car engine when the cylinder volume is 120 cm3 . Its pressure is then 1.0 atm. The valve closes and mixture is compressed until its  volume is 15 cm3. What is its pressure now ?

Solution

Example 2

An air bubble has a volume 2.0 cm3 at a depth 40m in the sea. What is the volume of the air bubble when it reaches the surface of the sea water.

[ Assume the atmospheric pressure is equivalent to 10 m of water ]

Solution

Example 3

A 500 cm 3 beaker is inverted and immersed in water  at a depth of 2.5 m. What is the volume of water in the beaker at the depth of the water?

[ Atmospheric pressure= 10 m of water ]

Solution

Example 4

A balloon is filled by a gas at atmospheric pressure. The balloon is later immersed in water until its volume becomes clip_image002[3] of its initial volume. What

is the depth of the balloon?

[ Atmospheric pressure= 10 m of water ]

Solution

Example 5

Figure (a) shows a glass tube closed at one end with 15 cm  of mercury in a inverted position . The length of air column is 30 cm. Figure (b) shows the tube at a horizontal positions with the length of air column is L.

[ Atmospheric pressure = 75 cm Hg ].

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What is the length , L?

Solution

Example 6

The diagram shows 10 cm of air column trapped in a glass tube by 5 cm of mercury.

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Later mercury is added into the glass tube until the length of the air trapped becomes 8 cm. What is the new length of the mercury column?

[ The atmospheric pressure = 75 cm Hg ]

Solution

Example 7

Figure (a) shows two containers W and X which are separated a pipe. At the beginning , the container W of volume 150 cm3 contains  gas at  a pressure of 200 kPa. The container X of volume 50 cm3 contains a vacuum.

Figure (b) shows the pipe is opened.

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Figure(a)                                           Figure(b)

What is the pressure in both the containers W and X attain equilibrium.

Solution

Boyle’s Law and the kinetic theory of gases.

Imaging a sample of  gas being compressed , with the temperature staying constant.

The average kinetic energy of the molecules of the gas remains unchanged but they are  now confined to a smaller space.

The molecules are squeezed closer together. As a result , the frequency of collisions between the molecules and the walls of the container increases .

Therefore , the force increases resulting in a corresponding increase in the pressure of the gas.  P­= F­

A

i.e. as volume decreases , pressure increases

To investigate the relationship between the pressure ,P and volume,V of gas at constant temperature.

Hypothesis : When the gas pressure continues to increase , its volume will continue to decrease.

Variables :

Manipulated ; Pressure of air trapped

Responding ; Volume of air

Constant ; Mass and temperature of air inside the

syringe.

Apparatus/ material : A 100 cm3 syringe, ruler, weight, clip , retort stand.

Arrangement of apparatus:

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Procedure:

Measure the distance between the 0 cm3 and 100 cm3 marked  are marked onto the syringe scale by using a ruler = L

Calculate the cross-sectional area, A of the piston ,

A = 100

         L

Record the atmospheric pressure when the position of the piston at  0 cm3 marked = Po

A weight of mass m is placed onto the piston.

Record the reading of the syringe = V

Calculate the pressure of the trapped air , P

P = Po + mg

                A

The experiment is repeated for 5 times with different  value of m.

Tabulate the data:

P

V

Analyse the data:

Plot a graph V against P

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Charles’s Law

Charles’s law states that “ For a fixed mass at constant pressure, the volume  of gas is directly proportional to its absolute temperature”

V aT                    Where V = Volume

V = KT                                       T = Absolute temperature

V = K

T

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Example 8

A sample of gas has a volume 100 cm3 at 20o C. To what temperature would you have to heat if you wanted to double the volume to 200 cm3 .

Solution

Example 9

The diagram shows a glass tube containing some trapped air inside it. At 17o C , the vertical column of trapped air is 29 cm.

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What is the vertical column of trapped air at a temperature of 57oC ?

Solution

Charles’s Law and the kinetic theory of gases.

In fixed mass of gas at constant pressure, the frequency of collisions between the gas molecules and the walls container is constant.

As the gas is heated , the molecules move faster. They collide with the walls more frequently and at greater speed.

So they exert a larger pressure on the walls of the containers.

As a result , the gas will expand if it is able to. This allows molecules to spread out a little which reduces the number of collisions per second with each unit area of the walls.

The gas continues to expand until the pressure is back to its original value.

i.e. if temperature is increased but pressure stays the same, the volume must increase.

To investigate the relationship between the temperature ,T and volume,V of gas at constant pressure.

Hypothesis :  When the temperature of a gas increases , its volume increases too.

Variables :

Manipulated ; Air temperature

Responding ; Volume of air

Constant ; Mass and air pressure in the capillary tube.

Apparatus/ material : thermometer, capillary tube , beaker,retort stand, Bunsen burner, tripod stand, wire gauze, ruler, sulphuric acid ,water and ice. Arrangement of apparatus:

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Procedure:

The internal cross-sectional area of the capillary tube is recorded  = A

Ice is placed into the water and these are continuously stirred .

The temperature , q . of water  , and  the vertical column , L  of trapped air are recorded and measured.

Calculate the volume , V of trapped air V = AL

The experiment is repeated for 5 times with different  value of q

Tabulate the data:

q

V

Analyse the data:

Plot a graph V against q

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The Pressure Law

The pressure law states that “ For a fixed mass at constant volume, the pressure  of gas is directly proportional to its absolute temperature”

P aT                    Where P = Pressure

P = KT                                      T = Absolute temperature

P = K

T

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Example 10

A motorist blows up her car tyres to a pressure of 5.4 atm on a cold morning when the temperature is -3o C. What will be the pressure in the tyres on a hot day if the temperature is 27o C

Solution

Example  11

A  cylinder of oxygen at 27o C has a gas pressure at 3 x 106 Pa. What is the temperature of the oxygen if the cylinder is cooled and the new pressure of the gas is 2.73 x 106 Pa.

Solution

The pressure Law and the kinetic theory of gases.

The sample of gas is kept at constant volume.

As the temperature of the gas rises, its molecules move more rapidly.

As the result, they collide with the walls of the container at higher frequency , the change of momentum is greater ,and so the force they exert on the walls is larger.

As a consequence, the force and hence the pressure increases.

P­= F­

     A

i.e. as temperatures increases , pressure increases

To investigate the relationship between the temperature ,T and pressure,P of gas at constant volume.

Hypothesis :  When the temperature of a gas increases , its pressure increases too.

Variables :

Manipulated ; Temperature of the trapped air

Responding ;

Constant ; Mass and volume of trapped air

Apparatus/ material : thermometer, round flask , beaker,retort stand, Bunsen burner, tripod stand, wire gauze, ruler, Bourdon gauge,rubber tube,wooden block,  water and ice.

Arrangement of apparatus:

clip_image028

Procedure:

The mixture of water and ice is stirred continuously until the temperature of the bath is steady.

By using thermometer  the temperature of the trapped air is recorded , q

By using Bourdon Gauge   the pressure of the trapped air is recorded , P

The experiment is repeated for 5 times with different  value of q

Tabulate the data:

q

P

Analyse the data:

Plot a graph P against q

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Absolute Temperature,T

The Kelvin scale is known as the absolute temperature scale.

q o C = ( q + 273 ) K

The absolute zero temperature of -273 oC  or 0 K is the lowest possible temperature that could be attained.

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Based on  Charle’s Law and the pressure law, at the absolute zero temperature the volume and the pressure of the gas become zero.

If  absolute zero temperature is related to the kinetic energy of molecules, then we might expect that there would be a temperature  where  the molecules would be stationary  and their kinetic energy would be zero. At absolute zero the kinetic energy of molecules is a minimum. No object can be cooled to a lower temperature than this.

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