Form threePhysics

Kinetic Theory of Gases The kinetic theory of gases explains the the relationship between the pressure,

temperature and volume of gases base on the following assumptions:

Farah yasin( f.khadaad)

The gas consists of very small particles, each of which has a mass. These particles are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the

container. All these collisions are perfectly elastic. There are forces of attraction between particles of matter. These attraction forces will

increase as the distance between the particles becomes closer. The average kinetic energy of the gas particles depends only on the temperature of the

system. The higher the temperature, the higher the kinetic energy of the particles.Pressure and volume relationship of a gas (Boyle's law)Boyle's lawThe relationship of a gas with pressure and volume was developed by the scientist Robert Boyle at around 1660 and is known as Boyle’s Law.Boyle’s law states:"For a fixed mass of gas, at a constant temperature, the product (pressure x volume) is a constant."

Pressure x Volume =constantp x V = constant

A sealed cylinder with no leaks contains a fixed mass of a gas kept at a constant temperature. The gas pressure is created by the collision of the moving gas particles with each other and against the walls of the cylinder.

The relationship between pressure and volume

By plotting the recorded values of pressure (p) against volume (V) a curve is produced.

A balloon contains 6m3 of helium at a pressure of 100kp, as the balloon rises through the atmosphere the pressure falls and the balloon expends, assuming that the atmosphere does not change, what is the volume of the balloon when the pressure has fallen to 50kp 40kp

Exercise

We can see from the values that when the pressure is doubled the volume is halved. If the pressure was to increase by 3 the volume would decrease to a third.

Thus, the volume is inversely proportional to the pressure. By plotting pressure (p) against the reciprocal of the volume (1/V) a straight line is obtained the gradient of which is the constant in Boyle’s Law.

Example:

Using the example of the sealed cylinder above, the volume of gas at the start is 50 cm3with a pressure of 1.2 x 105 Pascals. The piston is pushed slowly into the syringe until the pressure on the gauge reads 2.0 x 105 Pascals. What is the volume of gas?

Solution:

We KNOW

p x V = constant

therefore,

p1 x V1 = p2 x V2

p1= 1.2 x 105 PascalsV1 = 50 cm3

p2 = 2.0 x 105 PascalsV2 = ?

p1 x V1 = p2 x V2

v2=p

1v1

p2

v2=1 ⋅2×105×50

2×105

v2=30cm3

Example2

A sealed syringe contains 10 x10-6 m3 of air at 1 × 105 Pa. The plunger is pushed until the volume of trapped air is 4 x 10-6 m3. If there is no change in temperature what is the new pressure of the gas?

Solution

Therefore,

The new pressure in the syringe is 2.5 × 105 Pa

Volume and temperature relationship of a gas (Charles' law)Charles' lawThe relationship between the volume and temperature of a gas was first put forward by the French scientist Jacques-Alexandre-César Charles at around 1787 and is known as Charles’ Law.Charles’ law states:

"For a fixed mass of gas, at a constant pressure, the volume (V) is directly proportional to the absolute temperature (T)."

Volume α Temperature volume

Temperature=constant

A sealed cylinder with no leaks contains a fixed mass. In order to keep the gas pressure constant the piston is allowed to move freely so that the internal pressure created by the gas particles can equal the constant external pressure. If the internal pressure increases the piston will move up to allow the pressure to equalize.

The relationship between temperature and volume for a gas

By plotting the recorded values of volume (V) against temperature (T) a straight line is produced. We can see from the values that the gas expands uniformly with temperature.

A gas in fixed container is at a pressure of 4 atm and the temperature of 27oC what will be its pressure if it is heated to a temperature of 177oC ? Exercise

Converting the recorded temperatures into the Kelvin scale and plotting the volume (V) against the absolute temperature (T) gives a straight lineThis shows the volume of the gas is directly proportional to the absolute temperature of the gas. Doubling the temperature will double the volume. The gradient of the slope is the constant in Charles’ Law.Example:

Using the example of the sealed cylinder above, the volume of gas at the start is recorded as 30 cm3 with a temperature of 30°C. The cylinder is heated further till the thermometer records 60°C. What is the volume of gas?

Solution:

We know

V/T = constant

therefore

V1/T1 = V2/T2

V1 = 30 cm3

T1 = 30°C = 30+273 = 303K (remember to convert from Celsius to Kelvin)T2 = 60°C = 60+273 = 333KV2 =?

V1/T1 = V2/T2

v2=v

1T2

T 1

v2= 30×333

303= 32.97c m3

 Pressure and temperature relationship of a gas (pressure law)

Imagine a gas is trapped inside a container, which has a fixed size (its volume cannot change).

If the gas is heated the particles will gain kinetic energy which will make them move faster. This means they will collide with the walls more frequently.

This causes the force on the walls of the container to increase and so the pressure increases.

The pressure law states:

"For a fixed mass of gas, at a constant volume, the pressure (p) is directly proportional to the absolute temperature (T)."

Pressure α Temperature

pressureTemperature

=constant

A sealed cylinder with no leaks contains a fixed mass. The volume of the gas is kept constant by using a cylinder with a fixed roof capable of withstanding high pressures. The gas pressure is created by the collision of the moving gas particles with each other and against the walls of the cylinder.

The relationship between temperature and pressure for a gas

Heat energy is applied to the cylinder and the temperature of the gas increases. The average velocity of the gas particles increases resulting in an increase in the rate of collisions and the

average force per collision. Because the areas of the walls are kept constant, the force per unit area increases resulting in an increase in pressure.

Plotting the pressure (p) against the absolute temperature (T) gives a straight line

This shows the pressure of the gas is directly proportional to the absolute temperature of the gas. Doubling the temperature will double the volume. The gradient of the slope is the constant in Charles’ Law.

Example:

Using the example of the sealed cylinder above the pressure of gas is recorded as 1.0 x 105N/m2 at a temperature of 0°C. The cylinder is heated further till the thermometer records 150°C. What is the pressure of the gas?

Solution:

We know

p/T = constant

therefore,

p1/V1 = p2/V2

p1 = 1.0 x 105 N/m2

T1 = 0°C = 0+273 = 273K (remember to convert from Celsius to Kelvin)

T2 = 150°C = 150+273 = 423Kp2 =?

p1/T1 = p2/T2

Therefore,

p2=1.0×1 05×423

273 1.54×105N /m2

Exercise

1. An iron cylinder containing gas with pressure 200kPa when it is kept is a room of temperature 27°C. What is the pressure of the gas when the cylinder is located outdoor where the temperature is 35°C.

2. A car tyre contains air at 1.25 × 105 Pa when at a temperature of 27°C. Once the car has been running for a while the temperature of the air in the tyre rises to 42°C. If the volume of the tyre does not change what is the new pressure of the air in the tyre?

The Gas Equation

The three gas laws give the following equations:

pV = constant (when T is kept constant) Boyle’s Law

V = constant (when p is kept constant)T

Charles’ Law

P = constant (when V is kept constant)T

Pressure Law

These 3 equations are combined to give the gas equation:

Any two of the three gas laws can be used to derive a general gas law proceed as follows

Suppose we have fixed mass of gas temperature T1 occupying a volume1 the pressure on the gas is changed from p1 to p2 at constant temperature we can find the new volume V2 as

v2=p

1v1

p2

(equation1)

The pressure is kept constant at p2, the new volume V2 can be foun as

v2=v

1T2

T 1

(equation2)

From equation two , v1=v

2T1

T 2

, substituting in equation1,we obtain

v2=p

1V1T2

p2T1

= p1 v1

T1

=p2 v2

T 2

Or

p1 v1

T1

=constant

Where,

p = the pressure of the gasV = the volume the gas occupiesT = the gas temperature on the Kelvin scale

From this equation we know that if a fix mass of gas has starting values of p1, V1 and T1, and then some time later has value p2, V2 and T2, the equation can be written as:

p1 v1

T1

=p2 v2

T 2

Example1:

Sabah pumps up her front bicycle tyre to 1.7 x 105 Pa. The volume of air in the tyre at this pressure is 300 cm3. She takes her bike for a long ride during which the temperature of the air in the tyre increases from 20°C to 30°C. Calculate the new front tyre pressure assuming the tyre had no leaks and so the volume remained constant?

Solution:

p1 = 1.7 x 105 PaT1 = 20°C = 20 + 273 = 293 KV1 = 300 cm3

p2 = ?T2 = 30°C = 30 + 273 = 303 KV2 = 300 cm3

Using:

p1 v1

T1

=p2 v2

T 2

Rearranging for P2

p2=p1 v1T2

v2T 1

We know V1 = V2, therefore the equation can be simplified to

p2=T 2 p1

T 1

p2= 1.7×105×303293

p2 = 1.76×105Pa

Example2

A gas at 27°C expends from a volume of 5.0 m3 to 7.5 m3 at constant pressure, find its final temperature?

Solution

p1 = p2

v1 = 5.0 m3 and v2 = 7.5 m3

T1 = 27°C = 27 + 273 = 300 K

p1 v1

T1

=p2 v2

T 2

T2= 7.5m3×300 k

T2 ¿7 .5×300 k

5 = 450k or 177°C

Ideal gas law

The gas laws tell us that for fixed amount of ideal gas

PV/T= constant

If we fix the volume and temperature of a container of gas and double the amount of gas in the container, the pressure doubles because there twice as many particles to collide with the walls of the container.

Pressure is therefore proportional to the amount of gas, n, measured in moles (Mole)

Pressure α amount of gas

P α n

Since Pα 1/V and P α T then it follows that P = nT/V

Putting in a constant (R ) we get

P = nRTV

(this equation is called Ideal gas equation)

R is the universal gas constant, and has a value of (8.31jmol-1k-1 or(R= 0.08206L.atm/mol.K )

The number of moles of a substance can be found from its mass(m)in gram and its molar mass(M) in g/mole

n= mM

Example 1

in an exprement 11.0g of Carbon dioxide( CO2) occupies a volume of 10.0dm3at a temperature of 350.0K . find the pressure exerted by the gas?

Solution

The molar mass of CO2 = 44.0 g/mole

The number of moles of gas n = 11.0/44.0 =0.250 mol

T= 350 K; V = 0.010m3

P = nRTV

P = 0 .250×8 .31×350

0 .010 = 72.7kp

Example2

Find the number of moles of a gas occupying a volume of 2L at 25°C and the pressure of the gas is 2atm.

Solution

Here we use the R value as 0.0821atmL/mol.KAnd P is given as 2atmVolume V is given = 2 LTemperature of the gas is given in °C, which should be converted in to K. So 25°C = 25+273 = 298KNumber of moles is to be calculatedn = PV/RTn = 0.163 mol

Example3

If there is 5.0g of CO2 gas in a 10 L cylinder at 25oC, what is the gas pressure within the cylinder?

Solution

We are given the quantity of CO2 in grams but to use the ideal gas law, we must express the quantity in moles. Therefore we must first convert grams of CO2 to moles, and then use this value in the ideal gas law.

To convert from grams to moles we use the conversion factor 1 mol CO2 = 44g.

We now use this value in the ideal gas equation to solve for the pressure of the gas.

P = nRTV P = [0.11 mol CO2][298K]10L × 0.0821L.atmmol.KP = 0.27atm

Exercise

1. How much gram of nitrogen gas will exert a pressure of 250mm of Hg in a 2 liter container at 273K (0.84g)

2. One mole of CH4 gas occupies 20.0L at 1.00atm pressure. What is the temperature of the gas in kelvin? (244K)