During collisions, energy is transferred from one molecule to another and as a result the speed of both molecules will change. As a result, not all of the molecules will have the same kinetic energy at a given time, but there will be a distribution of molecular energies and velocities. Maxwell and Boltzmann derived the probability distribution function for molecular velocities (v) in a single direction: p = This can be converted into a distribution function in three dimensions for molecular speeds (c): p = Using E = ½mc2, this can also be converted into a distribution function for molecular energies: p = The first part of this distribution is the Maxwell part and it shows p increasing with this happens √E; because the more energy there is, the greater the number of ways to distribute the energy amongst the molecules and so the more probable the energy (the entropy factor); the second part of this distribution is the Boltzmann part and shows p decreasing rapidly with increasing E; the number of molecules with energy much higher than the mean will be small and will decrease with increasing energy To calculate the probability of a molecule having a or energy within a particular range, it is necessary to integrate the distribution function within those two limits; however this is difficult because the function takes the form y = which is not easily integrated; an approximation can be made at high energies by assuming that the Maxwell part of the equation is approximately constant at high energies; the resulting integral then simplifies to = (; note that this approximately is only valid when the range of energies under consideration is significantly greater than the average energy Increasing the temperature and decreasing the of mr increases the proportion of molecules with higher speeds; the Maxwell-Boltzmann distribution of molecular energies is best demonstrated graphically (the plots of probability against speed and probability against energy are very similar) DIFFERENT MEASURES OF MOLECULAR We know that the kinetic energy of one mole of molecules = ½mrv2 = Therefore v2 = Therefore v (or the root mean square speed) = Analysis of the Maxwell-Boltzmann distribution the most probable speed of the molecules as Further analysis of the Maxwell-Boltzmann distribution gives the of the molecules as average speed The values of root mean square speed, average speed and most probable speed are therefore not the same value. Root mean square speed is important for kinetic energy considerations, but average speed is generally more useful in predicting individual particle behaviour vrms > vav > vmp in the √3:: √2
During collisions, energy is transferred from one molecule to another and as a result the speed of both molecules will change. As a result, not all of the molecules will have the same kinetic energy at a given time, but there will be a distribution of molecular energies and velocities. Maxwell and Boltzmann derived the probability distribution function for molecular velocities (v) in a single direction: p = This can be converted into a distribution function in three dimensions for molecular speeds (c): p = Using E = ½mc2, this can also be converted into a distribution function for molecular energies: p = The first part of this distribution is the Maxwell part and it shows p increasing with √E; this happens because the more energy there is, the greater the number of ways to distribute the energy amongst the molecules and so the more probable the energy (the entropy factor); the second part of this distribution is the Boltzmann part and shows p decreasing rapidly with increasing E; the number of molecules with energy much higher than the mean will be small and will decrease with increasing energy To calculate the probability of a molecule having a or energy within a particular range, it is necessary to integrate the distribution function within those two limits; however this is difficult because the function takes the form y = which is not easily integrated; an approximation can be made at high energies by assuming that the Maxwell part of the equation is approximately constant at high energies; the resulting integral then simplifies to = (; note that this approximately is only valid when the range of energies under consideration is significantly greater than the average energy Increasing the temperature and decreasing the of mr increases the proportion of molecules with higher speeds; the Maxwell-Boltzmann distribution of molecular energies is best demonstrated graphically (the plots of probability against speed and probability against energy are very similar) DIFFERENT MEASURES OF MOLECULAR We know that the kinetic energy of one mole of molecules = ½mrv2 = Therefore v2 = Therefore v (or the root mean square speed) = Analysis of the Maxwell-Boltzmann distribution the most probable speed of the molecules as Further analysis of the Maxwell-Boltzmann distribution gives the average speed of the molecules as The values of root mean square speed, average speed and most probable speed are therefore not the same value. Root mean square speed is important for kinetic energy considerations, but average speed is generally more useful in predicting individual particle behaviour vrms > vav > vmp in the √3:: √2
chem_211.docx