Gas Laws

What is a Perfect Gas?

Perfect gas, also called ideal gas, a gas that conforms, in physical behaviour, to a particular, idealized relation between pressure, volume, and temperature called the general gas law. This law is a generalization containing both Boyle’s law and Charles’s law as special cases and states that for a specified quantity of gas, the product of the volume v and pressure p is proportional to the absolute temperature t; i.e., in equation form, pv = kt, in which k is a constant. Such a relation for a substance is called its equation of state and is sufficient to describe its gross behaviour.

Read Books :

Fox and McDonald’s Introduction to Fluid Mechanics

A Textbook of Fluid Mechanics and Hydraulic Machines

Fluid Mechanics : By Douglas (Author)

A perfect gas or an ideal gas is a state of a substance, whose evaporation from the liquid state is completed and which obeys all the gas laws strictly under all conditions of pressure and temperature. In reality there is no ideal or perfect gas which obeys all the law of gas under all conditions of pressure and temperature. But the real gases such as oxygen, hydrogen, nitrogen and air which are ordinary difficult to liquefy; within certain range of pressure and temperature may be considered as perfect gas (or ideal gas).

The general gas law can be derived from the kinetic theory of gases and relies on the assumptions that (1) the gas consists of a large number of molecules, which are in random motion and obey Newton’s laws of motion; (2) the volume of the molecules is negligibly small compared to the volume occupied by the gas; and (3) no forces act on the molecules except during elastic collisions of negligible duration.

The physical properties of a gas is controlled by three variables and these are

Pressure exerted by the gas,
Volume occupied by the gas and
Temperature of the gas.

The behavior of a perfect gas can be studied with the following gas laws that are proved from experimental results.

Boyles Law
Charles’s Law
Gay-Lussac Law

Let’s discuss these laws of gas in detail

Boyle’s Law

Boyle’s law, also called Mariotte’s law, a relation concerning the compression and expansion of a gas at constant temperature. This empirical relation, formulated by the physicist Robert Boyle in 1662,

This Gas Laws states that under constant temperature, the absolute pressure (p) of a given mass of a perfect gas varies inversely with its volume (v).

i.e., in equation form, pv = k, a constant. The relationship was also discovered by the French physicist Edme Mariotte (1676).This law was established by Robert Boyle in 1662.

Mathematically

Mathematically, Boyle’s law can be stated as

$P\alpha\frac{1}{V}$

Pressure is inversely proportional to the volume

or,

$PV= k$ ; Pressure multiplied by volume equals some constant ( $k$ )

The equation states that the product of pressure and volume is a constant for a given mass of confined gas and this holds as long as the temperature is constant. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as

$P_{{1}}V_{{1}}=P_{{2}}V_{{2}}=P_{{3}}V_{{3}}=P_{{n}}V_{{n}}$

Suffixes 1, 2,3 … n represents different sets of conditions.

Explanation:

For understanding this law lets take an experiment.

Boyles Law gif

Suppose we have a cylinder piston assembly which contains a gas. Keeping this system at room temperature (so the temperature remains constant), If we decrease the volume by applying an external force than the pressure of the gas inside the cylinder increases. And if the external load is removed than the volume of the gas starts increasing and pressure will decreases. This proves that when pressure of a perfect gas increases than its volume will decrease and when volume decreases then pressure increases.

Read Also :

Laws of Thermodynamics

Charles’s Law

Charles’s law, a statement that the volume occupied by a fixed amount of gas is directly proportional to its absolute temperature, if the pressure remains constant.

The statement of Charles’s law is as follows: the volume (V) of a given mass of a gas, at constant pressure (P), is directly proportional to its temperature (T). As a mathematical equation, Charles’s law is written as either:

V\propto T\,

V/T=k_{2}

V_{1}/T_{1}=V_{2}/T_{2}

where V is the volume of a gas, T is the absolute temperature and k₂ is a proportionality constant

This empirical relation was first suggested by the French physicist J.-A.-C. Charles about 1787 and was later modified and implemented with sound proof by the chemist Joseph-Louis Gay-Lussac. Charles’s law is a special case of the general gas law and can be derived from the kinetic theory of gases under the assumption of a perfect (ideal) gas. Measurements show that at constant pressure the thermal expansion of real gases, at sufficiently low pressure and high temperature, conforms closely to Charles’s law.

Statement

This law can be stated in two ways ;

Statement 1st : At constant absolute pressure, the volume of a given mass of a perfect gas varies directly to its temperature.

Charles Law Gif

Explanation: Let us again take a piston cylinder assembly which is kept at room temperature. The upper part of the piston is in exposed to the atmospheric pressure (so the system remains at constant absolute pressure). The lower portion of the cylinder is made conductor of heat. Now heat the cylinder form bottom. After some time the piston of the cylinder starts moving upward. Why this has happen? Can you guess? Let me tell you what has actually happened. When the cylinder is heated, temperature of the gas inside the cylinder increases and also increases the pressure which makes the gas expand and exerts a force on the piston and makes it move upward. The piston moves upward to keep the pressure constant (i.e. inside pressure of the gas becomes equal to the atmospheric pressure). The upward movement of the piston increases the volume of the gas

So we can conclude that when a system is at constant absolute pressure, at that time the volume of the gas varies directly to its temperature.

Mathematically

Statement 2nd : At constant pressure, all the prefect gases changes the volume by 1/273th of its original volume at 0 degree Celsius for every 1 degree Celsius change in temperature.

What is absolute zero temperature?

Since the volume of the gas decreases by 1/273 th of its original volume at 0 degree C by every 1 degree decrease in temperature. So if we take the temperature -273 o C, then the volume of the gas reduces to zero.

The temperature at which the volume of a gas becomes equals to zero is called absolute zero temperature.

Gay-Lussac’s Law

Gay-Lussac’s law, Amontons’ law or the pressure law was found by Joseph Louis Gay-Lussac in 1809. It states that, for a given mass and constant volume of an ideal gas, the pressure exerted on the sides of its container is directly proportional to its absolute temperature.

As a mathematical equation, Gay-Lussac’s law is written as either:

P\propto T\,

P/T=k

K=P divided by T

P_{1}/T_{1}=P_{2}/T_{2}

where P is the pressure, T is the absolute temperature, and k is another proportionality constant.

Statement : At constant volume, the absolute pressure of a given mass of a perfect gas varies directly at its absolute temperature.

Explanation:

let’s take a piston cylinder assembly having an ideal gas in it. And the piston is not movable. Since the piston is fixed, so the volume is constant. Now add some heat to the gas present in the cylinder. The temperature of the gas starts increasing and increases the pressure of the gas. The increase in the pressure can be seen on the pressure gauge attached to the system.

The above experiment proves that at constant volume, the pressure of a perfect gas varies directly to its temperature.

Avogadro’s Law

Avogadro law is sometime also called as Avogadro hypothesis or Avogadro’s principle. It is an experimental law which was given in 1811 by Amedeo Avogadro.

Statement: Under constant temperature and pressure, equal volumes of all the gases contains equal no of molecules.

Or,

For a given mass of an ideal gas, the volume of the gas is directly proportional to the amount of substance (moles) of gas, given that the temperature and pressure are constant.

Mathematically

Suffixes 1,2 represents different sets of conditions for a gas.

Where,

V = volume of the gas.
n = the amount of substance of the gas (in moles).
k = constant which is equals to RT/P, where R is the universal gas constant, T is temperature in Kelvin and P is the pressure. Since the temperature and pressure is constant, so RT/P is also constant and represented as k.

ImageSource : Britannica ; Wikipedia ;

ArticleSource : Britannica