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Lesson 45 notes – Boltzmann’s Constant Objectives Be able to select and solve problems using the ideal gas equation expressed as pV = NkT and pV = nRT, where N is the number of atoms and n is the number of moles; Be able to explain that the mean translational kinetic energy of an atom of an ideal gas is directly proportional to the temperature of the gas in kelvin; Be able to select and apply the equation E = 3/2kT for the mean translational kinetic energy of atoms. Outcomes Understand the difference between R and k and when to apply them. Be able to state that the Boltzmann constant is the gas constant for individual molecules whilst the molar gas constant is the constant used when dealing with quantities in moles. Be able to explain that the mean translational kinetic energy of an atom of an ideal gas is directly proportional to the temperature of the gas in Kelvin. Be able to select and apply the equation E = 3/2kT for the mean translational kinetic energy of atoms correctly in different situations. Be able to derive equation for the translational KE of an atom in an ideal gas E = 3/2kT. The Boltzmann Constant The Boltzmann constant is used when we are looking at amounts of molecules rather than amounts of moles. From lesson 44 we know that pV = nRT We can now write that pV = NkT where N is the number of atoms and n is the number of moles So we can see that nR=Nk Also N=nNA Therefore k=R/NA Check that you understand this from the definitions of these symbols. Translational kinetic energy of an atom of an ideal gas From lesson 33 we saw that from the kinetic theory, gas pressure is given by: P = ⅓ρ<c>2 Letting <c> = the mean square speed which is the same as saying: P Nmv2 / 3V . From lesson 44 we saw that the Ideal Gas equation is: PV = nRT Therefore: nRT = 1/3 Nm<c>2 Now, KE = 1/2m v2 Rearranging 3nRT / N = m<c>2 So, 1/2 m<c>2= (3/2) nRT/N We know that NA = N/n so n/N = 1/NA (N is the number of molecules in a volume of gas, n is the amount of gas in moles, NA is Avagadro’s number). Therefore: E = 1/2 m<c>2= (3/2) RT / NA = (3/2)kT This tells us that the mean kinetic energy of a molecule of an ideal gas is proportional to the thermodynamic temperature. (lesson 35).