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Elementary gas kinetic theory. Today we will give kinetic theory definitions to main macroscopic properties of a gas and discuss molecular magnitude. Main gas macroparameters: Molecular magnitudes: ● P - pressure ● m - molecular mass ● T - temperature ● v - mean (average, flow) velocity ● Ru - universal gas constant - thermal velocity ● R - gas constant ● ● ● v v v rms - - molecular velocity Root-mean-square thermal speed● k - Boltzmann constant We will also derive the equation of state of ideal gas and introduce the absolute temperature scale. 1 Gas Pressure ● Pressure=Force/Area ● Force=Rate of change of momentum. ● Let's first consider a gas at rest in a container. v strikes a If a molecule with velocity specularly reflecting (ideally smooth) surface, the change of momentum of the molecule is 2mvn , where vn is velocity component normal to the surface and m is mass. ● During a time interval dt all such molecules with velocity v that are within the distance vndt from the surface will strike it. 2 Gas Pressure ● Net rate of change of momentum normal to dS for n v dt dS 2 m v v = molecules with velocity dt where nv – number density of molecules with velocity v. v ● n n Pressure is total net rate of change of momentum normal to dS for all molecules per unit area: p=∑ n v 2m v =n m v 2 n 2 n v , v n0 where n is the total number density, overline denotes average. 3 Gas Pressure ● In equilibrium gas all directions of velocity are equally probable: 2 2 2 2 2 vn =v x =v y =v z=v / 3. where v 2=v 2x v 2y v 2z module. - square of velocity ● Then ● Note that pressure is thus proportional to mean kinetic energy of molecules. n m v2 p= 3 4 Mean and thermal velocity ● is the mean (average) velocity for a system of molecules, i.e. v0 N v0= v x , v y , v z , v x = ∑i=1 v x , i N . Gas at rest is the gas with zero mean velocity. ● v − v is v= 0 thermal (or peculiar) velocity of a molecule in a system.Note that average value of thermal velocity is always zero: v x =v y =v z=0. 5 Pressure of moving gas ● The pressure in the moving gas is best defined as the rate of transfer of normal momentum across a surface that is moving “with the gas”, i.e. with the mean velocity of gas: 2 n mv − v0 n m v2 p= = . 3 3 ● Note that pressure in moving gas is thus proportional to mean kinetic energy of thermal motion. 6 Derivation of Avogadro's Law ● Molecular hypothesis states that gas is a system of molecules in constant random motion. Such systems of randomly moving particles are described by statistical mechanics. ● Equipartition Theorem of statistical mechanics states that the mean kinetic energy associated with each degree of freedom of a mechanical system in statistical equilibrium has the same value. Thus for a gas with two kinds of molecules 1 and 2: If p1=p2: 1 1 m1 v 21= m 2 v 22 21 12 2 2 n1 m 1 v 1 = n 2 m 2 v 2 . 2 2 Therefore, n1=n2 i.e. ideal gases at the same pressure and temperature contain the same number of molecules per unit volume. 7 Avogadro's Law ● Number of molecules in a volume of gas is a quantity of particles in a given amount of matter. A unit of quantity of matter in SI system is mole. ● 1 Mole=number of atoms in 12 grams of C12=6.022x1023 atoms. NA=6.022x1023 – Avogadro number. ● ● The volume V0 occupied by a mole of ideal gas under standard conditions (p=1 atm, melting ice temperature) is thus a universal constant. The accepted value of V0 =22.4 l. 8