Form three

Physics

Farah yasin( f.khadaad)

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:

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 law The 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 =constant

p 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.

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 Pascals

V1 = 50 cm3

p2 = 2.0 x 105 Pascals

V2 = ?

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,

1. 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

a. 50kp

b. 40kp

Exercise

The new pressure in the syringe is 2.5 × 105 Pa

Volume and temperature relationship of a gas (Charles' law)

Charles' law The 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.

Converting the recorded temperatures into the Kelvin scale and plotting the volume (V) against the absolute temperature (T) gives a straight line This 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 = 333K

V2 =?

V1/T1 = V2/T2

𝒗𝟐 =𝒗

𝟏𝑻𝟐

𝑻𝟏

1. 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

𝒗𝟐 = 𝟑𝟎×𝟑𝟑𝟑

𝟑𝟎𝟑 = 32.97𝑐𝑚3

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

𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆

𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆= 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 = 423K

p2 =?

p1/T1 = p2/T2

Therefore,

𝒑𝟐 =𝟏.𝟎×𝟏𝟎𝟓×𝟒𝟐𝟑

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

𝑻𝟏 (equation2)

From equation two , 𝒗𝟏 =𝒗

𝟐𝑻𝟏

𝑻𝟐 , substituting in equation1,we obtain

𝒗𝟐 =𝒑

𝟏𝑽𝟏𝑻𝟐

𝒑𝟐 𝑻𝟏 =

𝒑𝟏𝒗𝟏

𝑻𝟏=

𝒑𝟐𝒗𝟐

𝑻𝟐

Or

𝒑𝟏𝒗𝟏

𝑇1

= constant

Where,

p = the pressure of the gas

V = the volume the gas occupies

T = 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:

𝒑𝟏𝒗𝟏

𝑻𝟏=

𝒑𝟐𝒗𝟐

𝑻𝟐

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 Pa

T1 = 20°C = 20 + 273 = 293 K

V1 = 300 cm3

p2 = ?

T2 = 30°C = 30 + 273 = 303 K

V2 = 300 cm3

Using:

𝒑𝟏𝒗𝟏

𝑻𝟏

=𝒑𝟐𝒗𝟐

𝑻𝟐

Rearranging for P2

𝒑𝟐 =𝒑𝟏𝒗𝟏𝑻𝟐

𝒗𝟐𝑻𝟏

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

𝒑𝟐 =𝑻𝟐𝒑𝟏

𝑻𝟏

𝒑𝟐= 𝟏.𝟕×𝟏𝟎𝟓×𝟑𝟎𝟑

𝟐𝟗𝟑 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

𝒑𝟏𝒗𝟏

𝑻𝟏=

𝒑𝟐𝒗𝟐

𝑻𝟐

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 wall s

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

Putting in a constant (R ) we get

P = 𝒏𝑹𝑻

𝑽 (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 =𝒎

𝑴

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 = 𝒏𝑹𝑻

𝑽 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.K

And P is given as 2atm

Volume V is given = 2 L

Temperature of the gas is given in °C, which should be converted in to K.

So 25°C = 25+273 = 298K

Number of moles is to be calculated

n = PV/RT

n = 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.K

P = 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)