### PHYSICS

# Gas laws and equations in physics

**What are Gas laws ?**

To study gas behaviors, we should first of all consider three physical properties of gas, which are **pressure volume and temperature. **It is based on these three properties that the three gas laws were formulated. Note that among the three states of matter gas is most movable because of high kinetic energy they have.

**The three gas laws are
**1. Boyle’s law

2. Charles’ law

3. Pressure law or Gay Lussac’s law.

**Boyle’s law **

This is one of the gas laws, It states that the volume of a fixed mass of gas varies inversely as its pressure provided the temperature remains constant.

Boyle’s law can be mathematically stated as

Or P_{1}V_{1} =P_{2}V_{2 }, K=constant, p =pressure and V= volume

Applications of boyle’s law

1.It is used in determination of a volume of a given mass of gas at a constant temperature when pressure is known

2. It can also be used to find the pressure of a given mass off gas at constant temperature when the volume is known

**One of the gas laws also is Charles’ law**

The Charles’ law state that the volume of a fixed mass of gas is directly proportional to its absolute temperature(T), provided the pressure remains constant

Charles law can be represented in mathematical form as

V_{1 }and V_{2} are initial and final volumes while T_{1} and T_{2} are initial and final temperatures and K=constant

**Applications of Charles’ law**

1.With Charles’ law absolute zero temperature, that is the temperature at which volume of gas is theoretically zero can be determined by plotting the volume of the gas against its temperature

- The cubic expansivity of gas is also a result of Charles’ law. Therefore cubic expansivity is given as

Hence Charles’ law can be restated as the volume of fixed mass of gas at a constant pressure increase by 1/273 of its volume at 0^{o}C (Celsius or Kelvin) for every degree rise in temperature

**Pressure law or Gay Lussac’s law
**This deals with change in pressure and absolute temperature of a gas when the volume is kept constant

It states that the pressure of a fixed mass of a gas is directly proportional to its absolute temperature provided the volume is kept constant

Mathematically it is written as

P=KT, K= constant, p= pressure and T=temperature

**Absolute temperature and pressure** **In gas laws**

In gas laws if P_{0 }= gas pressure at temperature 0^{o}C and P_{Ѳ} = gas pressure at temperature Ѳ^{o}C. The pressure law states mathematical that

P_{0 }= P_{Ѳ} (1 + γѲ)

Where γ= 1/273k^{-1} =pressure cubic expansivity at constant volume

**General gas laws equation**

The general gas laws equation is derived from the combination of these three laws the boyle’s law Charles’ law and pressure law or Gay Lussac’s law

From boyle’s law PV =k at constant temperature

From Charles’ law V/T =k at constant pressure

From pressure law P/T =k constant volume

When you combine these three equation of gas laws you abstain general gas laws equation as

**Standard temperature and pressure**

The temperature and pressure of volume of gases can be converted to a standard temperature and pressure. This conversion makes it easy for possible comparison. The standard temperature is taken as 0^{o}C or 273K while the standard pressure is taken as 760mm of mercury.

**Van der waals force and equation for real gas**

**What is Vander waals force**? It is the force of attraction between gases molecule. It can also be called weak Vander waal force, because is indeed weak but not negligible.

**Vander waals equation **

Vander waals equation is given as

Where a =constant, b= volume of gas, (V-b)=volume of the space insider container and R also =constant.

** **

**Kinetic molecular theory and gas laws**

The kinetic theory state that:

(I). All matters are made of atoms and molecular

(II). The molecules are in a constant state of motion

(II). That the molecule posses kinetic energy because of it continuous motion

(IV). The molecule exert attractive force on one another

(V). The nearer the molecules to one another the greater the attraction force

**Using of kinetic theory to explain the gas laws**

** **

**Explanation of gas laws using kinetic theory
**1. Using kinetic theory to explain pressure exerted by a gas, kinetic theory considers gas as made up of large number of molecules which behave like elastic sphere. The molecule moves about in their containers with random velocities, colliding with one another and with the walls of the container. As gas molecules hit the walls of the containing vessel and rebound they experience a change in velocity and therefore change in momentum. The walls of the containers also experience forces due to change in momentum of the molecules of gas, thereby exacting pressure on the walls of the vessel.

2. **Using kinetic theory to explain Boyle’s law**

If the Volume of the vessel containing gas is reduced at constant temperature the gas molecules will have less spaces to occupy. Therefore they take less time to hit the vessel. In this case more bombardments are made on the wall per unit time and therefore the pressure exerted by the gas increases.

On the other hand when the volume of gas is increased the molecules will take more time to hit the wall, and this lead to the decrease in pressure of gas. In conclusion the pressure of a gas is therefore inversely proportional to the volume at a constant temperature( Boyle’s law)

**3 Using** **kinetic theory to explain Charles’s law**

If you Increase the temperature of gas molecule at constant pressure the monocle we gain kinetic energy and thereby increasing the velocities of the molecules. With the increase in velocities the molecules will hit the wall more frequently and this brings about the increase pressure but to keep the pressure constant the volume must increase. therefore an increase in temperature at constant pressure will lead to an increase volume (Charles’s law)

**4 using kinetic theory to explain pressure law(Gay Lussac’s Law)**

If the temperature of a given mass of gas is increased at constant volume the average kinetic energy of the molecules increases and this results in the increase in speed of molecules which make them to hit the wall of containing vessel more frequently. Consequently the pressure of this given mass of gas increases with increase in temperature at constant volume. (Gay Lussac’s Law)

Questions and answers on gas laws

- Calculate the volume of fraction change of a fixed mass of gas whose pressure is tripled at constant temperature

Answer: Using Boyle’s law

But P_{2 }= 3P_{1}, therefore

So the final volume becomes one-third of the initial volume

- The pressure of fixed mass of gas is given as 850cmHg when its volume is 20cm
^{3}. Calculate the volume of the gas when its pressure is 600cmHg

Answer: using Boyle’s law. P_{1}V_{1}=P_{2}V_{2}

P_{1}=pressure= 850cmHg, P_{2}=600cmHg, V_{1}=20cm^{3}

850 x 20 = 600 x V_{2}

- A fixed mass of gas with volume 600cm
^{3}at 0^{o}C is heated at constant pressure. Calculate the volume of the gas at 130^{o}C

Answer: Using Charles’ law

T_{1}=0 + 273 =273K, T_{2}= 130 + 273 =403K, and V1 = 600cm^{3}

- The pressure of a gas at constant volume is 100cmHg at 27
^{o}C. Calculate the pressure at 87^{o}C

Answer: using pressure law ( Gay lussac’s law)

T_{1}= 27 + 273 =300K

T_{2}= 87 + 273 =360K

P_{1}=100cmHg, so

- Some quantity of dry air has a volume of 500cm
^{3}at 40^{o}C and under a pressure of 76cmHg. At a temperature of 120^{o}C its volume becomes 600cm^{3}, calculate the pressure of the gas at this temperature.

Answer: Using gas laws general equation

P_{1}=76cmHg, p_{2}=?, V_{1}=500cm^{3}, V_{2}= 600cm^{3}, T_{1} = 40 + 273=313K, and T_{2}= 120 + 273 =393K

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