derive any four gas law from kinetic gas eqution
Answers
Explanation:
Boyle’s Law
Boyle’s law states the relation between volume and pressure at constant temperature and mass. Robert Boyle conducted an experiment 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 Vt is the volume at t° C then,
Vt = V0 +t/273.15 V0 ⇒ Vt = V0 (1+ t/273.15 )
⇒ Vt = V0 (273.15+ t/273.15 )
For the purpose of measuring 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 [ Vt = V0 (273.15+ t/273.15 ) ] if we take the values Tt = 273.15+t and T0 = 273.15 then:
Vt = V0 ( Tt / T0 )
which implies Vt/V0= ( Tt / 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
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