Hello dear students, in previous article we have studied the derivation for pressure exerted by the gas. According to assumptions of kinetic theory of gas, the formula for the pressure exerted by the gas is given as,
Sir J.C.Maxwell, Sir Boltzmann have studied the structure of gas molecules in more details and expressed the kinetic energy of gas in form of parameters like pressure, temperature and volume.
The parameters like pressure, temperature and volume are very important to discuss the details about the gas and its behaviour. The detailed about such parameters can be studied with help of the following gas laws,
1.) Boyle’s law: At constant temperature pressure exerted by fixed mass of gas is inversely proportional to its volume.
∴ PV = constant
2.) Charles’s law:-At constant pressure the volume of a gas is proportional to its absolute temperature.
3.) Dalton’s law: The total pressure of a mixture of ideal gases is the sum of partial pressures. i.e. if the number of gases at pressures P1, P2,—– etc is combined in a single container then the pressure of the mixture is given by:
P = P1 + P2 + – – – – – —
Let’s discuss the kinetic energy of gas in terms of temperature……………!
Consider a perfect gas is filled in cubical vessel of side ‘l’. By assumptions of kinetic theory of gases, the gas molecules are constantly moving in all possible direction with all possible velocities. Therefore they possess momentum thus exerts pressure on wall of container which is give as,
Where,
N= number of molecules of gas.
m = mass of each molecule of gas.
V= Volume of cube.
Adjusting equation (1) by multiplying and dividing RHS of it by 2 we get,
Using equation (3) in equation (1) we get,
∴ PV = 2/3 Kinetic energy of gas
Or
∴Kinetic energy of gas, E = 3PV/2…………………….(4)
According to ideal gas equation, we have,
PV=nRT
∴E = 3nRT/2
Kinetic energy per mole of gas is then given as,
∴E/n =3RT/2
We know that in one of gas, number of molecules present is equal to Avogadro’s number.
The above equation indicates that the energy of gas is directly proportional to absolute temperature of gas.
Some important derivation…..!
By kinetic theory, the pressure exerted by gas molecules is,
But according to Ideal gas equation, PV=nRT
For one of gas, PV=RT, then equation (1) becomes,
Hence the rms speed of gas is directly proportional to the square root of its absolute temperature.
Let’s solve some numerical on the same article…..!
Ex:1) Compare the rms speed of gas at 1270C and 3270C.
Here, T1 = 1270 C = 400 K, T1 = 3270 C = 600 K