Showing posts with label Kinetic Theory. Show all posts
Showing posts with label Kinetic Theory. Show all posts

Work done against Gravitational Force and Potential Energy

Work has to be done in moving a body against the gravitational force. Once if the work is done, the job is over and body gets displaced. This work done can not disappear all of a sudden. It is against the law of conservation of energy. As per law of conservation of energy, energy is neither created not destroyed. It just converts from one form to other. Work done and energy are just two forms of same expression. If we have energy, we can do work and if we are doing work means, we are using our energy for that.

Here the work done in displacing the body will convert into gravitational potential energy in the body and gets stored in the body. It is also called potential energy.

Let us consider a body of mass m on the surface of the earth. Here surface of  the earth is take like a reference to measure work done and energy. By default, gravitational force is acting on it and it is in the downward direction and towards the centre of the earth always.

To lift the body by body to a certain height, we shall apply some force and that shall be at least equal to the weight of the body acting in the down ward direction so that equilibrium can be disturbed and body can be moved.

Let the displacement is along the same direction of the body and hence the work done is maximum. We can measure the work done as the dot product of force and displacement and here, being they are in the same direction, work done is maximum.

After work is done against the gravitational force, the body acquires a new position and it is at a certain height from the reference point. At that height, the work done is stored in the format of potential energy and that energy is also called gravitational potential energy.

Work done is independent of path in which the body is moved and the important issue is only final point and destination. As it is the dot product of force and displacement. Displacement is a vector and it is the shortest distance between initial and final positions and is independent of path. Hence work done and potential energy are also independent of path.

To calculate the work done by gravitational force, the scenario is little bit different. Here the gravitational force is in the downward direction and the displacement is in the opposite direction and hence work done here is negative and of course the magnitude is same.

 This concept  is further explained with a video as show below.





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Molar Specific Heats of Gas and Relation between Them

Molar specific heat of a gas at constant volume

The amount of the heat energy required to rise the temperature of unit mole of gas by 1°C at constant volume.

During this process the volume of the gas is Constant. As the volume is constant no external work is done in this process. According to first law thermodynamics all the heat energy supplied in this process is used only to increase the internal energy.

Molar specific heat of a gas at constant pressure

The amount of heat energy required to rise the temperature of unit mole of a gas by 1°C at constant pressure.

In this process the pressure of the system is kept constant. Here the heat energy supplied is used not only to increase the internal energy but also to do some external work.

It is obvious that Molar specific heat of a gas at constant pressure is greater than that of the Molar specific heat of a gas at constant volume.




Relation between two different specific heats of the gas

Basing on the very definition of the specific heats with can find the relation as shown below.

We also use first law Thermodynamics to derive the conclusion. It is proved below that the difference in the specific heats is equal to universal gas constant. It is constant for all the gases at all the conditions and the entire universe.

The same can be proved for the unit mass of the gas. But in this case you will be getting the difference between the specific heats as only gas constant which varies from one gas to another gas. That is the reason why mole is more standard in the nature while we are referring the gases than that of the mass in grams.




The ratio of specific heats depends on the nature of the gas.

For a Mono atomic gas its value is 5/3, for a diatomic gas its value is 7/ 5 and for a trial atomic gas its value is the 8 /6.

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Heat and Thermodynamics Complete

Heat is a form of energy. Heat can be converted to other forms of energy’s and other forms of energy’s can also be converted into heat. Heat can be measured with a physical quantity called temperature. When heat energy is given to your body it expands. The expansion has to be studied separately for solids, liquids and gases.

The conversion of the heat energy into other forms and its applications are studied in the chapters called calorimetry and thermodynamics. The reason behind the temperature is nothing but the internal collisions of molecules of a gas and it is studied in the kinetic theory of gases.

This post is a collection of all the topics that are relevant to the heat and Thermo dynamics.

Posts available in the blog are 


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Degree of freedom and Law of the Equipartisien energy

Degree of  Freedom

It is the ability of a gas molecule which can move freely with respect to the available circumstances.The degree of the freedom of a gas molecule depends on the nature of the gas molecule.

The molecule of a mono atomic gas can have only translatory motion and hence it has 3 degree  of the freedom along its x-axis y-axis and z-axis.

Diatomic gas molecule can have not only a translatory motion but also rotatory motion along two other possible axis. Thus it can have five degrees of freedom.

Tri/poly atomic gas molecule can have translatory motion, vibratory motion as well as a rotatory motion.



Law of the Equipartisien energy:

As per this law the average kinetic energy per each degree of the freedom per molecule is fixed.



Basing on this concept we can find the specific heat of a gas at constant volume, at constant pressure and the ratio of their specific heats as shown below.It can be also proved easily that in all the supplied heat energy of hundred percent,around 60% is used to increase the internal energy in the case of mono atomic gas.



Basing on the law of  Equipartisien energy we can also find the ratio of the specific heats of the gases at constant pressure and volume and it can be proved that about the total heat energy supplied around 71% is used to increase its internal energy and the remaining 29% is used to do the additional work.



As per the law of  Equipartisien energy we can find the ratio of the specific heat of the gases for tri atomic gas at constant pressure and volume as shown below. It can we also proved that out of the total supplied heat energy 75% of the heat energy is used to increase the internal energy and only the remaining 25% is used to do the external work.



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RMS Velocity of Gas Molecule and Applications

RMS velocity is useful in calculating and estimating effective velocity of a gas molecule.we can find the velocity of the sound in a gas using Newton’s Laplace formula. It can be further modified in terms of the absolute temperature and the relation can be found is shown below.



we can try a small problem on this concept is shown below.

Compare the RMS velocity of gas molecule of the same nature at two different temperatures.We shall always covert the temperature into kelvin  before we solve the problem as shown.




Another Problem

If the temperature of a gas molecule is increased by 44% calculate by what percentage it’s RMS velocity is affected?




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Kinetic theory of gases and Expression for Pressure

The purpose of the Kinetic theory of the gases is to link to the macroscopic properties of the gases like pressure volume,temperature with the macroscopic properties of the gas molecules like displacement,velocity, momentum, force and kinetic energy.

To apply the connect theory of the gases to the molecules we shall make some assumptions.

1. We shall assume that the size of the molecule is very small and when compared with the volume of the gas occupied the volume of the gas molecules occupied is negligible.
2. The gas molecules are tiny in size , spherical in shape , neutral with respect to charge and all are identical.
3. The molecules moves  in all directions randomly with all possible speeds.
4. The collision of the gas molecules is an elastic. That means during the collision both momentum and kinetic energy are conserved. There is no wastage of energy in the format of sound light and heat.

Mean free path:

The average distance molecule can travel without colliding the neighboring molecules is called as mean free path.When the molecule starts its journey, its motion is so random under regular that the we cannot predict what is going to be its path is. 

 In between any of the two collisions, the molecule travel some distance and by measuring this total distances and by dividing it with the number of the collisions we can calculate the value of the mean free path.

The gas which obeys all gas laws at all temperatures and pressures is called an ideal gas. In reality no gas is actually Ideal and all the existing gases are called real gases.While explaining kinetic theory of the gases, anyway we assume that the gas is Ideal.

Practically real gas obeys all gas laws only at high temperatures and low pressures.

Expression of the pressure of an ideal gas:

Being the gas molecules are having collisions among themselves and with the walls of the container, they are going to  exert some pressure and here we are going to calculate that pressure. Let us  consider a container who is having cube shape. Let the side of the cube is  L and a gas molecule of mass m is moving in parallel to YZ plane along the x-axis.

As the collision is elastic the molecule will come back with the same velocity.Thus we can calculate the change of the momentem with respect to time which leads the calculation of the force. The derivation for the pressure is made are shown below.


This can be further continued by writing all the forces and then further writing equation for the pressure as defined as the force per unit area as shown further.



The expression for the pressure of a gas molecule can be expressed in terms of kinetic energy of the gas molecule also. Here RMS velocity of the gas molecule can be expressed  as the root mean square velocity of the gas molecule. Basing the ideal gas equation even we can measure the impact of the temperature here. The temperature at which the RMS velocity of  gas molecule becomes zero is called a absolute zero temperature and that is taken as a reference to define Kelvin scale.


We can further find a relation between temperature of the gas and the RMS velocity of the gas molecule as shown below.We can also define a particular temparatue at which RMS velocity of gas molecule become zero called absolute temperature.




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