Question

Give the laws of Boyle, Charles, Gay-Lussac and Dalton by writing the postulates of kinetic theory of gases.

Answer 1

The main assumptions of kinetic theory of gases are :
1. A gas is made up of a large number of sub-microscopic particles called atoms.
2. Gases consist of particles in constant random motion. They continue to move in a straight line until they collide with each other on the walls of their container.
3. Gas pressure is due to the moleiMIeS colliding with the walls of the container. All of these collisions are perfectly elastic. This means that there is no change in energy of either the particles or the wall upon collision. So, no energy is lost or gained from collisions.
4. No molecular forces are at work. The potential energy of molecules is zero. So, whole of the energy. in an ideal gas is kinetic energy only. There is no attraction or repulsion between the particles.

5. The time taken for the collision is negligible as compared with the time between collisions.
6. The kinetic energy of a gas is a measure of its kelvin temperature. Each gas molecule has different speed but the temperature and kinetic energy of gas refer to the average of these speeds.
7. The average kinetic energy of an atom of gas is directly proportional to the temperature. An increase in temperature increases the speed in which the gas molecules move.
8. All gases at a given temperature have same average kinetic energy.
9. fighter molecules of a gas move faster than the heavier molecules.

According to the kinetic theory, the average kinetic energy of molecules of a gas is directly proportional to the absolute temperature of the gas. So, from equation(1).

i.e., the product of pressure and volume of gas at constant temperature is constant. This is Boyle’s law.

From equation (1), if pressure (P) of a gas is constant, then
V ∝ T …(2)
i. e., for a gas of a given mass, volume of gas is directly proportional to the temperature at constant pressure. This is Charles’ law.

So, at constant volume for a gas of given mass, the pressure of a gas is directly proportional to the temperature of gas. This is Gay Lussac’s law.

According to Dalton’s law of partial pressure, in thermal equilibrium state, the total pressure of each gas is equal to the sum of the different pressures of each gas for a mixture of unreactive gases filled in a container, i.e., if P₁,P₂,P₃,… respectively are the different pressure of gases. Then, the total pressure (P)
P = P₁ + P₂ + P₃ +…. 

Similarly, we can write equation for other gases. Since the mixture of all- the gases are in thermal equilibrium state, i.e., the temperature is same. Then, average kinetic energy of each molecule of a gas will be equal, i.e.,

If the total pressure of container is P and the total number of molecules are (n₁ + n₂ + n₃ +…), then the total applied pressure,

This is Dalton's law of partial pressure.

Answer 2

Gas Laws

During high pressure or high-temperature conditions, a tyre inflated with air is at the risk of bursting. Or while climbing a mountain you start feeling problems to inhale? Why is it so? With changing physical conditions the behaviour of gaseous particles also deviates from their normal behaviour. The behaviour of a Gas can be studied by various laws known as the Gas laws. Let us see more!

The Gas Laws

All gases generally show similar behaviour when the conditions are normal. But with a slight change in physical conditions like pressure, temperature or volume these show a deviation. Gas laws are an analysis of this behaviour of gases. The variables of state like the Pressure, Volume and Temperature of a gas depict its true nature. hence gas laws are relations between these variables. Let us study more about the important gas laws!

Boyle’s Law

Boyle’s law states the relationship between volume and pressure at constant temperature and mass. Robert Boyle experimented on gases to study the deviation of its behaviour in changed physical conditions.

It states that under a constant temperature when the pressure on a gas increases its volume decreases. In other words according to Boyle’s law volume is inversely proportional to pressure when the temperature and the number of molecules are constant.

p  \(\propto\) 1/V

p = k₁ 1/V

k₁ here is a proportionality constant, V is the Volume and p is the pressure. On rearranging, we get: k₁= PV. Now, if a fixed mass of gas undergoes an expansion at constant temperature then the final volume and pressure shall be p₂ and V₂. The initial volume and initial pressure here is p₁ and V₁ then according to Boyle’s law: p₁×V₁ = p₂×V₂ = constant (k₁)

p₁/p₂  =  V₂/V₁

So according to Boyle’s law, if the pressure is doubled then at constant temperature the volume of that gas is reduced to half. The reason being the intermolecular force between the molecules of the gaseous substance. In a free state, a gaseous substance occupies a larger volume of the container due to the scattered molecules.

When a pressure is applied to the gaseous substance, these molecules come closer and occupy a lesser volume. In other words, the pressure applied is directly proportional to the density of the gas. Boyle’s law can be graphically represented  as follows:

Charle’s Law

Jacques Charles in 1787 analyzed the effect of temperature on the volume of a gaseous substance at a constant pressure. He did this analysis to understand the technology behind the hot air balloon flight. According to his findings, at constant pressure and for constant mass, the volume of a gas is directly proportional to the temperature.

This means that with the increase in temperature the volume shall increase while with decreasing temperature the volume decreases. In his experiment, he calculated that the increase in volume with every degree equals 1/273.15 times of the original volume. Therefore, if the volume is V₀ at 0° C and V_t is the volume at t° C then,

 V_t = V₀ +t/273.15 V₀   ⇒   V_t  =  V₀ (1+ t/273.15 )

⇒    V_t  =  V₀ (273.15+ t/273.15 )

To measure the observations of gaseous substance at temperature 273.15 K, we use a special scale called the Kelvin Temperature Scale. The observations of temperature (T) on this scale is 273.15 greater than the temperature (t) of the normal scale.

T= 273.15+t

while, when T = 0° c then the reading on the Celsius scale is 273.15. The Kelvin Scale is also called Absolute Temperature Scale or Thermodynamic Scale. This scale is used in all scientific experiments and works. In the equation [ V_t  =  V₀ (273.15+ t/273.15 ) ] if we take the values T_t = 273.15+t and T₀ = 273.15 then:

V_t = V₀ ( T_t / T₀ )

which implies V_t/V₀=  ( T_t / T₀ ), which can also be written as:

V₂/V₁=  T2/ T₁ 

or V₁ /T₁ = V₂ / T₂

V/T = constant = k₂ 

Therefore, V= k₂ T

The graphical representation of Charles law is shown in the figure above. Its an isobar graph as the pressure is constant with volume and temperature changes under observation.

Gay-Lussac’s law

Also referred to as Pressure-Temperature Law, Gay Lussac’s Law was discovered in 1802 by a French scientist Joseph Louis Gay Lussac. While building an air thermometer, Gay-Lussac accidentally discovered that at fixed volume and mass of a gas, the pressure of that gas is directly proportional to the temperature. This mathematically can be written as: p \(\propto\) T

⇒  p/T = constant= k₃ 

The temperature here is measured on the Kelvin scale. The graph for the Gay- Lussac’s Law is called as an isochore because the volume here is constant.

Avogadro’s Law

Amedeo Avogadro in 1811 combined the conclusions of Dalton’s Atomic Theory and Gay Lussac’s Law to give another important Gas law called the Avogadro’s Law. According to Avogadro’s law, at constant temperature and pressure, the volume of all gases constitutes an equal number of molecules. In other words, this implies that in unchanged conditions of temperature and pressure the volume of any gas is directly proportional to the number of molecules of that gas.

Mathematically, V \(\propto\) n

Here, n is the number of moles of the gas. Hence, V= k₄n

The number of molecules in a mole of any gas is known as the Avogadro’s constant and is calculated to be 6.022 * 10²³. The values for temperature and pressure here are the standard values. For temperature, we take it to be 273.15 K while for the pressure it equals 1 bar or 10⁵ pascals. At these Standard Temperature Pressure (STP) values, one mole of a gas is supposed to have the same volume. Now, n = m/M

According to Avogadro’s equation: V= k₄ (m/M)

M=k₄(m/V)

m/V= d (density); Therefore M=k₄D

This means that at an unchanged temperature and pressure conditions, the molar mass of every gas is directly proportional to its density.

The above gas laws provide us with an indication of the various properties of gases at changed conditions of temperature, pressure-volume and mass. These laws seem trivial but these find great importance in our day to day lives. From breathing to hot air balloons and vehicle tyres the deviation in gaseous behaviour in changed conditions may affect all. So the next time you are travelling just remember the effect change in physical conditions can have!

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Answer 3

Boyle’s law is a gas law which states that the pressure exerted by a gas (of a given mass, kept at a constant temperature) is inversely proportional to the volume occupied by it. In other words, the pressure and volume of a gas are inversely proportional to each other as long as the temperature and the quantity of gas are kept constant. Boyle’s law was put forward by the Anglo-Irish chemist Robert Boyle in the year 1662.

For a gas, the relationship between volume and pressure (at constant mass and temperature) can be expressed mathematically as follows.

P ∝ (1/V)

Where P is the pressure exerted by the gas and V is the volume occupied by it. This proportionality can be converted into an equation by adding a constant, k.

P = k*(1/V) ⇒ PV = k

The pressure v/s volume curve for a fixed amount of gas kept at constant temperature is illustrated below.

It can be observed that a straight line is obtained when the pressure exerted by the gas (P) is taken on the Y-axis and the inverse of the volume occupied by the gas (1/V) is taken on the X-axis.

Formula and Derivation

As per Boyle’s law, any change in the volume occupied by a gas (at constant quantity and temperature) will result in a change in the pressure exerted by it. In other words, the product of the initial pressure and the initial volume of a gas is equal to the product of its final pressure and final volume (at constant temperature and number of moles). This law can be expressed mathematically as follows:

P₁V₁ = P₂V₂

Where,

• P₁ is the initial pressure exerted by the gas
• V₁ is the initial volume occupied by the gas
• P₂ is the final pressure exerted by the gas
• V₂ is the final volume occupied by the gas

This expression can be obtained from the pressure-volume relationship suggested by Boyle’s law. For a fixed amount of gas kept at a constant temperature, PV = k. Therefore,

P₁V₁ = k (initial pressure * initial volume)

P₂V₂ = k (final pressure * final volume)

∴ P₁V₁ = P₂V₂

This equation can be used to predict the increase in the pressure exerted by a gas on the walls of its container when the volume of its container is decreased (and its quantity and absolute temperature remain unchanged).

Examples of Boyle’s Law

When a filled balloon is squeezed, the volume occupied by the air inside the balloon decreases. This is accompanied by an increase in the pressure exerted by the air on the balloon, as a consequence of Boyle’s law. As the balloon is squeezed further, the increasing pressure eventually pops it. An illustration describing the increase in pressure that accompanies a decrease in the volume of a gas is provided below.

If a scuba diver rapidly ascends from a deep zone towards the surface of the water, the decrease in the pressure can cause the gas molecules in his/her body to expand. These gas bubbles can go on to cause damage to the diver’s organs and can also result in death. This expansion of the gas caused by the ascension of the scuba diver is another example of Boyle’s law. Another similar example can be observed in the deep-sea fish that die after reaching the surface of the water (due to the expansion of dissolved gasses in their blood).

Charles law also sometimes referred to as the law of volumes gives a detailed account of how gas expands when the temperature is increased. Conversely, when there is a decrease in temperature it will lead to a decrease in volume.

When we compare a substance under two different conditions, from the above statement we can write this in the following manner:

V₂/V₁=T2/T1

OR

V1T2=V2T1

This above equation depicts that as absolute temperature increases, the volume of the gas also goes up in proportion.

In other words, Charle’s law is a special case of the ideal gas law. The law is applicable to the ideal gases that are held at constant pressure but the temperature and volume keep changing.

Charles Law Everyday Examples

Here are some of the examples by which you can understand Charle’s law very easily.

In winters as the temperature decreases, when u take a basketball outside in the ground the ball shrinks. This is the only reason why to check the pressure in the car tier’s when to go outside in the cold days. This is also the case with any inflated object and explains why it’s a good idea to check the pressure in your car tires when the temperature drops.

If you overfill a tube that is placed on a pool on a hot day, it can swell up in the sun and burst. Similarly, as the turkey cooks, the gas inside the thermometer expands until it can “pop” the plunger. Pop-up turkey thermometers work based on Charles’ law. Another common application can be seen in the working of a car engine.

Charles Law Formula

Charle’s Law formula is written as,

V_I /T_I=V_F /T_F

Where V_I=Initial volume

V_F=Final volume

T_I=Intial absolute temperature

T_F=Final absolute temperature

Here we should remember that the temperatures are absolute temperatures that are measured in Kelvin, not in ⁰F or ⁰C.

Derivation of Charles Law

As we are aware of the fact that, at constant pressure, the volume of the fixed amount of the dry gas is directly proportional to absolute temperature according to Charle’s law. We can represent the states in the following manner.

V∝T

Since V and T are varying directly, we can equate them by making use of the constant k.

V/T=constant =k

In this, the value of k depends on the pressure of the gas, the amount of the gas and also the unit of the volume.

V*T=k——-(1)

Let us consider V1 AND T1 to be the initial volume and the temperature respectively of an ideal gas.

Then we can write equation (1) as

V1/T1=k——-(2)

After it lets change the temperature of the gas to T2. Alternatively, its volume changes to V2 then we can write

V2/T2=k——–(3)

Equating the above equations that is equation 2 and 3, we get

V1/T1=V2/T2

OR

V1T2=V2T1

You are unaware of the fact that, on heating up a fixed amount of gas, that is, by increasing the temperature the volume also increases. Similarly lowering the temperature, the volume of the gas decreases. And at 0-degree centigrade, the volume of the also increases by 1/273 of its original volume for a unit degree increases in temperature.

It is important to know, as already discussed above that the unit of temperature must be in Kelvin not in Celcius or Fahrenheit for solving the problems related to Charle’s law. The temperature in Kelvin is also known as the absolute temperature scale. For converting the temperature in Celcius to Kelvin, you add 273 to the temperature in the Celsius scale.

According to Charles’ Law which states that the volume (V) of the gas is directly proportional to its temperature (T)  which must be in Kelvin.

When the temperature changes one unit of the Kelvin scale it equals to a change in one Celsius degree. Remember always that 0 on the Kelvin scale means -273 or “Absolute Zero”.

The Density of the gas is inversely proportional to the temperature in the Kelvin when it is at a constant mass and pressure.

Graphical Representation Of Charles Law

ISOBAR- Graph between V and T at constant pressure is known as isobar or bioplastics and it always gives a straight line. A plot of V versus T (°C) at constant pressure is a straight line at – 273.15°C. -273.15-degree Celcius is the lowest possible temperature.

Avogadro’s Law – N α V at Constant Pressure:

When there is a greater number of particles it increases the collisions and the pressure. If the pressure is to remain constant, the number of collisions can be reduced only by increasing the volume.

At constant pressure, the volume is proportional to the amount of gas.

Answer 4

Or more simplified:

Assumptions of Kinetic Theory of Gases 

1. Every gas consists of extremely small particles known as molecules. The molecules of a given gas are all identical but are different from those of another gas.

 2. The molecules of a gas are identical spherical, rigid and perfectly elastic point masses. 

3. Their molecular size is negligible in comparison to intermolecular distance (10-9 m).

 4. The speed of gas molecules lies between zero and infinity (very high speed). 

5. The distance covered by the molecules between two successive collisions is known as free path and mean of all free path is known as mean free path.

 6. The number of collision per unit volume in a gas remains constant. 

7. No attractive or repulsive force acts between gas molecules.

 8. Gravitational to extremely attraction among the molecules is ineffective due to small masses and very high speed of molecules. 

Gas laws :

Assuming permanent gases to be ideal, through experiments, it was established that gases irrespective of their nature obey the following laws. 

Boyle’s Law:

 At constant temperature the volume (V) of given mass of a gas is inversely proportional to its pressure (p), i.e., V ∝ 1/p ⇒ PV = constant For a given geas, p1V1 = p2V2 Charles’ Law At constant pressure the volume (V) of a given mass of gas is directly proportional to its absolute temperature (T), V ∝ T ⇒ V / T = constant For a given gas, V1/T1 = V2/T2 At constant pressure the volume (V) of a given mass of a gas increases or decreases by 1/273.15 of its volume at 0°C for each 1°C rise or fall in temperature. Volume of the gas at t°Ce Vt = V0 (1 + t/273.15) where V0 is the volume of gas at 0°C. 

Gay Lussacs’ or Regnault’s Law :

At constant volume the pressure p of a given mass of gas is directly proportional to its absolute temperature T, i.e., p ∝ T ⇒ V/T = constant For a given gas, p1/T1 = p2/T2 At constant volume (V) the pressure p of a given mass of a gas increases or decreases by 1/273.15 of its pressure at 0°C for each l°C rise or fall in temperature. Volume of the gas at t°C, pt = p0 (1 + t/273.15) where P0 is the pressure of gas at 0°C. 

Avogadro’s Law:

 Avogadro stated that equal volume of all the gases under similar conditions of temperature and pressure contain equal number of molecules. This statement is called Avogadro’s hypothesis.

 According to Avogadro’s law

 (i) Avogadro’s number The number of molecules present in 1g mole of a gas is defined as Avogadro’s number. NA = 6.023 X 1023 per gram mole 

(ii) At STP or NTP (T = 273 K and p = 1 atm 22.4 L of each gas has 6.023 x 1023 molecules.

 (iii) One mole of any gas at STP occupies 22.4 L of volume.

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