Kinetic Theory Of Gases
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3. Moleular nature of matter
- We know that molecules which are made up of one or more atoms constitute matter.
- In solids these atoms and molecules are rigidly fixed and space between then is very less of the order of few angestrem and hance they can not move.
- In liquids these atoms and molecules can more enabeling liquids to flow.
- In gases atoms are free to travell without colliding for large distances such that if gases were not enclosed in an enclosure they would disappear.
4. Dalton's law of partial pressures :
- Consider a mixture of non-interacting ideal gases with n_{1} moles of gas 1,n_{2} of gas 2 and so on
- Gases are enclosed in an encloser with volume V, temperature T and pressure P.
- Equation of state of mixture
PV = (n_{1}+ n_{2})RT
or P = n_{1}RT/V + n_{2}RT/V + - - - -
= P_{1} + P_{2} + - - --
where,
P_{1} = n_{1}RT/V
is pressure the gas 1 would exert at same V and T if no other gases were present in the enclosure. This is know as law of partial pressure of the gases.
- The total pressure of mixture of ideal gases is sum of partial pressures of individual gases of which mixture is made of.
5. Kinetic Theory of an ideal gas
Following are the fundamental assumptions of kinetic theory of gases.
- Gas is composed of large number of tiny invisible particles know as molecules
- These molecules are always in state of motion with varying velocities in all possible directions.
- Molecules traverse straight line path between any two collisions
- Size of molecule is infinitely small compared to the average distance traverse by the molecules between any two consecutive collisions.
- The time of collision is negligible as compared with the time taken to traverse the path.
- Molecules exert force on each other except when they collide and all of their molecular energy is kinetic.
- Intermolecular distance in gas is much larger than that of solids and liquids and the molecules of gas are free to move in entire space free to them.
6. Pressure of gas
- Consider a cubical vessel with perfectly elastic walls containing large number of molecules say N let l be the dimension of each side of the cubical vessel.
- v_{1x} , v_{1y}, v_{1z} be the x, y, and z componenet of a molecule with velocity v.
- Consider the motion of molecule in the direction perpandicular to the face of cubical vessel.
- Molecule strikes the face A with a velocity v_{1x} and rebounds with the same velocity in the backward direction as the collisions are perfectly elastic.
- If m is the mass of molecule, the change in momentum during collision is
mv_{1x} - (-mv_{1x}) = 2 mv_{1x} (1)
- The distance travelled parallel to x-axis an is between A to A´ and when molecule rebounds from A´and travel towards A is 2L
- Time taken by molecule to go to face A´ and then comeback to A is
Δt = 2l/v_{1x}
- Number of impacts of this molecule with A in unit time is
n = I/Δt = v_{1x}/ 2l (2)
Rate of change of momentum is
ΔF = ΔP/Δt
=nΔP
from (1) and (2)
ΔF = mv_{1x}^{2} / l
this is the force exerted on wall A due to this movecule.
- Force on wall A due to all other molecules
F = Σmv_{1x}^{2}/L (3)
- As all directions are equivalent
Σv_{1x}^{2}=Σv_{1y}^{2}=Σv_{1z}^{2}
Σv_{1x}^{2}= 1/3Σ((v_{1x})^{2} + (v_{1y})^{2} +( v_{1z})^{2} )
= 1/3 Σv_{1}^{2}
Thus F = (m/3L) Σv_{1}^{2}
- N is total no. of molecules in the container so
F = (mN/3L) (Σ(v_{1})^{2}/N)
- Pressure is force per unit area so
P = F/L^{2}
=(M/3L^{3})(Σ(v_{1})^{2}/N)
where ,M is the total mass of the gas and if ρ is the density of gas then
P=ρΣ(v_{1})^{2}/3N
since Σ(v_{1})^{2}/N is the average of squared speeds and is written as v_{mq}^{2} known as mean square speed
Thus, v_{rms}=√(Σ(v_{1})^{2}/N) is known as roon mean squared speed rms-speed and v_{mq}^{2} = (v_{rms})^{2}
- Pressure thus becomes
P = (1/3)ρv_{mq}^{2} (4)
or PV = (1/3) Nmv_{mq}^{2} (5)
from equation (4) rms speed is given as
v_{rms} = √(3P/ρ)
= √(3PV/M) (6)
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