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# Kinetic Interpretation of Temperature

## Introduction

There are five states of matter and gas is one of them. Empty spaces are filled by gases. The atmosphere is full of gas molecules. It contains different gas molecules. The oxygen gas molecule which is necessary for breathing is also in our atmosphere along with many other gas molecules. Every matter in the universe is composed of tiny particles called molecules. They attract each other with some attractive force. There exists an intermolecular force of attraction between molecules.

Based on the force of attraction between the molecules, matters are classified into five categories. They are solid, liquid, gas, plasma, and Bose-Einstein. Solids are the matter in which the distance between the molecules is very small and they are tightly bound to each other. Liquids are matter having molecules at a small distance and they bound less tightly than solids. Plasma state molecules have high kinetic energy.

## Assumptions of Kinetic Theory of Gases

The kinetic theory of gases shows the thermodynamic behavior of gases. They have some basic assumptions as,

There are many tiny particles in the gas known as particles.

They make an elastic collision with each other and also with the walls of the container.

The particles are identical to each other and can move in all possible directions.

The time of collision is very small as compared with the time between the collision.

The volume of the molecule is very much smaller when comparing it with the total volume of the gas.

Molecules do not apply any force on the other except during collision.

The separation between the molecules is greater compared to the size of the molecule.

All the molecules obey the laws of motion.

## What do you mean by Kinetic Interpretation of Temperature ?

Let us consider a gas-filled container of volume V. Let the molecules move at all possible velocities. Then the pressure applied by the gas molecules on the walls is given by,

$\mathrm{P=\frac{1}{3}\frac{mnc^2}{v}}$

Where m denotes the mass of one molecule. n denotes the number of molecules. $\mathrm{C^2}$ denotes the root mean square velocity of the molecules. V denotes the volume of the gas.

The 1 mole of gas contains N number of molecules.

$\mathrm{PV=\frac{1}{3}mNC^2}$

By gas law PV=RT

$\mathrm{RT=\frac{1}{3}mNC^2}$

$\mathrm{\frac{RT}{N}=\frac{1}{3}mC^2}$

multiply and divide the right-hand side by 2 we get,

$\mathrm{\frac{RT}{N}=\frac{2}{3} \times \frac{1}{2} mC^2}$

The kinetic energy of the molecule is $\mathrm{K.E=\frac{1}{2}mC^2}$

$\mathrm{\frac{RT}{N}=\frac{2}{3} \times K.E}$

$\mathrm{K.E=\frac{3}{2}\times \frac{RT}{N}}$

$\mathrm{K.E=\frac{3}{2} KT}$

where $\mathrm{K=\frac{R}{N}}$ denotes Boltzmann's constant.

## Relation between Kinetic Energy and Temperature

The work that is done on the body is its energy. The body which possesses the energy due to its motion is called its kinetic energy. It can be transformed from one object to another and also into other forms of energy. This energy depends on its mass and velocity. The S.I unit of kinetic energy is Joule. Thus the kinetic energy of the object is given by,

$\mathrm{K.E=\frac{1}{2} mc^2}$

Degrees of freedom of the molecules are different by the law of equipartition of energy; the total energy of the system is equally allocated for each degree of freedom of the molecule. And the energy allocated equally is 1/2 kT. This is called the law of equipartition of energy. Thus the temperature of the gas is directly related to the kinetic energy of the molecules.

## Gas laws

Many laws were proposed to know about the behavior of gas. Gas laws give the relation between pressure, temperature, and volume. They are Charle’s law, Gay Lussac’s law, Avogadro’s law, Boyle’s law, and Grahm’s law.

**Boyle’s law:**Boyle’s law clearly says that the pressure of the gas is controversial with the volume of the gas. (i.e) when pressure increases volume will decrease. This will be maintained as long as the temperature is constant. Mathematical representation gives,

$\mathrm{P \varpropto \frac{1}{v}}$

$\mathrm{PV = k\:(constant)}$

**Charle’s law:**When temperature increases the gas gets expanded. This was explained by Charle’s law. It shows that the volume of the gas changes directly with temperature change when pressure is maintained constant.

$\mathrm{V \varpropto T}$

$\mathrm{\frac{v}{T} = k(constant)}$

**Gay Lussac’s law:**Pressure of the gas changes directly by the change in temperature when the volume of the gas is kept constant. This is known as Gay Lussac’s law.

$\mathrm{P \varpropto T}$

$\mathrm{\frac{P}{T} = k(constant)}$

**Avogadro’s law:**Avogadro’s law says that in an equal volume of gases the number of molecules will be the same when the same temperature and pressure are the same.

$\mathrm{V \varpropto n}$

$\mathrm{\frac{v}{n} = k}$

**Graham's law:**According to his principle lighter molecules move faster than heavier molecules. I.e. effusion of molecules with low molecular mass is faster than the higher molecular mass at constant pressure and temperature. The molecules diffusing for time are the reciprocal of the square root of their mass density.

$\mathrm{r \varpropto \frac{1}{\sqrt{P}}}$

## Conclusion

In this tutorial, the assumption of the kinetic theory of gases and the kinetic interpretations of temperature are discussed in detail. The relation between the kinetic energy and the temperature and gas laws is also discussed.

## FAQs

**Q1. Explain degrees of freedom.**

Ans: The number coordinates needed for the molecule to mention its position are called degrees of freedom. Generally, monoatomic molecules have three degrees of freedom. Molecules having more than one atom include three more degrees of freedom such as translation motion, rotational motion, and vibrational motion. According to this degrees of freedom for the molecules having more than one atom will vary.

**Q2. State means free path in the gas.**

Ans: In gas molecules make collisions with each other and also with the container’s wall. The mean value of the distance traveled by a molecule between any two consecutive collisions is known as the mean free path. It is denoted by λ.

**Q3. What are the factors affecting the behavior of gases?**

Ans: The factors that affect the behavior of gas are Temperature (T), Pressure (P), Volume, and Quantity. If the temperature of the gas increases the gas expands thereby volume gets increased. Similarly, as temperature decreases volume also decreases due to the compression of gas molecules. An increase in temperature causes the gas to expand and so its pressure increases and vice versa. To convert gas into solid or liquid the temperature must be very low. Quantity and pressure are directly proportional to each other. As pressure is directly proportional to volume and quantity, pressure is increased by making quantity and volume high.

**Q4. List some applications of gas laws?**

Ans: If the environment condition changes then the behavior of gas molecules deviates from their behavior. These changes can be studied well by the gas laws. The syringe used to intake the liquid is under the principle of Boyle’s law. The expansion of cakes and bread while heating comes under Charle's law. It is also used in weather prediction systems. They are also used in thermodynamics and fluid dynamics.

**Q5. How does the gas get compressed?**

Ans: Gas is the state of matter in which molecules are loosely bounded. So there are many gaps between the molecules. When pressure is given to the gas the gaps get reduced and the particles come closer. Thereby the volume is reduced and the gas gets compressed.

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