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Unformatted text preview: Kinetic Theory of Gases Kinetic Topics of Discussion Topics Molecular mass, the Mole and Molecular Avogadro’s Number Avogadro’s Kinetic Theory of Gases Ideal Gas Laws Frick’s Law of Diffusion Molecular Mass, the Mole & Avogadro's Number Avogadro's The atomic mass scale uses the atomic mass unit u to The atomic compare the masses of atom and molecules: 1u = 1.6605 × 10-27 kg. . 10 kg The molecular mass, or molar mass, iis the sum of the s The molecular or molar atomic masses of the molecule. atomic The mole, or one gram-mole, of a substance contains as The mole or one of many particles as 12g of carbon 12. 12 12 The number of particles in 12g of carbon 12 is The 12 12 6.022 × 1023, and is called Avogadro's number NA. 10 and The number of moles n of a substance is given by The n = m / molar mass. molar The Thermodynamic The System for a Gas
Whereas a mechanical system is described in Whereas terms of the position x, the velocity v and the the acceleration a of a body, a thermodynamic system for a gas is described in terms of system pressure P, temperature T and volume V. The temperature The variables P, T and V are called the state variables of the thermodynamic system. variables
S urroundings Q Q Thermodynamic S ystem P, V, T Assumptions for the Kinetic Theory of Gases the The number of molecules is large, and the The distances between molecules is large. distances All molecules are identical. The molecules obey Newton’s Laws of The motion. motion. The molecules interact by short-range The forces during elastic collisions. forces The molecules make elastic collisions with The the walls of the container. the Ideal Gas Law Ideal
A iideal gas held at a pressure P and a deal temperature T in a volume V is given by either PV = NkT or PV = nRT, where n is where PV the number of moles of the ideal gas, 8.31 J/molºK is the universal gas constant, k = J/mol –23 1.38 × 10 –23 J/ºK is Boltzmann’s constant, Boltzmann’s and NA = 6.022 × 1023 is Avogadro's 10 number such that N=nNA. N=nN Derivation of the Ideal Gas Laws Derivation
We derive the ideal gas law. Let us We consider a gas in a container composed of only one type of gas molecule or atom. From a mechanical perspective, the exchange in the x–component of the component momentum with the wall of the container is given by given Δpx = +mvx – ( –mvx ) = 2mvx Derivation of the Ideal Derivation Gas Laws, cont.
Suppose that the molecule interacts with Suppose the wall over some small time interval Δt = 2d / vx , where 2d is the overall distance traveled by the molecule. The Impulse Theorem gives the x–component component of the force Fx exerted on the wall by the molecule, which is given by molecule, Fx = Δpx / Δt = mvx2 / d Derivation of the Ideal Derivation Gas Laws, cont.
The total x–component of the force Fx The component exerted on the wall by N molecules is N given by given 2 Fx = mv xi / d. ∑ i= 1 The average value of the velocity squared The is given by is 1 v= N
2 x ∑ N v i= 1 2 xi . Derivation of the Ideal Derivation Gas Laws, cont.
Thus, the total x–component of the Thus, component force Fx exerted on the wall can now be written as be mN 2 Fx = vx . d Derivation of the Ideal Derivation Gas Laws, cont.
However, each molecule travels in three– dimensional space. So, the velocity v of dimensional each molecule can be written the vector each ˆ ˆ v = vxi + v y ˆ + vz k j
where the magnitude of the velocity v is where given by given v= v = v +v +v
2 x 2 y 2 z Derivation of the Ideal Derivation Gas Laws, cont.
Since the motions of every molecule Since is random, the x, y and z–component and component of each molecules velocity is the same, that is vx = vy = vz . The average velocity squared is given by average v v= . 3
2 x 2 Derivation of the Ideal Derivation Gas Laws, cont.
Thus the average force F exerted on the Thus wall by all the molecules is given by wall N mv . F= 3d
The pressure P experienced by the wall is The given by given 2 F 1N 2 P= = mv . A 3V Derivation of the Ideal Derivation Gas Laws, cont.
We can re-write the pressure P as We 2N1 2 P= mv 3V 2 which states that the pressure is which proportional to the number of particles per unit volume and the kinetic energy of the gas. gas. Derivation of the Ideal Derivation Gas Laws, cont.
By a similar argument, we can show By that the temperature of the gas is proportional to the kinetic energy of the gas, that is the 3 1 2 kT = mv 2 2 where k = 1.38 × 10 –23 J/ºK is called where
Boltzmann’s constant. Boltzmann’s Derivation of the Ideal Derivation Gas Laws, cont.
Using both of the equations and 2N1 2 P= mv 3V 2 3 1 kT = mv 2 2 2 we have the ideal gas law given by PV = NkT = nRT
where R = 8.31 J/molºK is the universal gas where 8.31 constant, N=nNA and R=kNA. N=nN R=kN Fick’s Law of Diffusion Fick’s
The mass m of a solute that diffuses in a time t The through a solvent of length L and cross-sectional area A is given by m = ( DAΔC ) t / L
where ΔC is the concentration difference across where the length L and D is the diffusion constant whose SI units are m/s2. m/s ...
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