If you double the average speed of the molecules in a gas, by what factor does the pressure change?12/24/2022 Relate particle flux to the velocity distribution. The steps in this process are as follows: Relate particle velocity to height. We have a description of an ideal gas system which can be used to help develop a plausibility argument for the Maxwell velocity distribution. We have an experimentally tested expression for molecular kinetic energy. In addition, we know that conservation of energy in this case involves just the balancing of kinetic energy and gravitational potential energy so long as we treat the atmosphere as an ideal gas.įrom the expression for kinetic temperature In this approach we make use of the fact that the average kinetic energy of the molecules can be expressed in terms of the kinetic temperature. The following treatment follows the development by Rohlf. One way to approach the solution in a more intuitive way is to deal with a physical example which we know - namely the physics of an atmosphere under the influence of gravity as reflected in the barometric formula. Maximizing the probability distribution subject to those constraints in general is a formidable mathematical task (see Richtmyer, et al., for example). ![]() But this most probable distribution (the Maxwell-Boltzmann distribution) is subject to constraints, namely that the number of particles is constant and that the total energy is constant (conservation of energy). So we expect that the description of the velocities of molecules in a gas will in fact be the most probable distribution, since we are dealing with particle numbers in the range of Avogadro's number. Statistical methods become a more precise way to study nature when the number of particles is large. Most probable speed = v p = m/s = km/hr = mi/hrĭevelopment of the Boltzmann Distribution The nominal average molecular mass for dry air is 29 amu. The three characteristic speeds may be calculated. The calculation of molecular speed depends upon the molecular mass and the temperature. Molecular Speed Calculation The speed distribution for the molecules of an ideal gas is given by Some comments about developing the relationship Why does the probability peak at some finite value, when the average velocity is zero?ĭevelopment of Maxwell speed distribution from Boltzmann distribution If the mass m of an individual molecule were used instead, the expression would be the same except that Boltzmann's constant k would be used instead of the molar gas constant R. Note that M is the molar mass and that the gas constant R is used in the expression. ![]() It is used in calculating the rates of many phenomena. Maxwell Speed Distribution The speed distribution for the molecules of an ideal gas is given byįrom this function can be calculated several characteristic molecular speeds, plus such things as the fraction of the molecules with speeds over a certain value at a given temperature. Substitution gives the root mean square (rms) molecular velocity:įrom the Maxwell speed distribution this speed as well as the average and most probable speeds can be calculated. Molecular Speeds From the expression for kinetic temperature ![]() HyperPhysics***** Heat and Thermodynamics The kinetic temperature is the variable needed for subjects like heat transfer, because it is the translational kinetic energy which leads to energy transfer from a hot area (larger kinetic temperature, higher molecular speeds) to a cold area (lower molecular speeds) in direct collisional transfer. When you try to assess specific heat, you must account for all the energy possessed by the molecules, and the temperature as ordinarily measured does not account for molecular rotation and vibration. This distinction becomes quite important when you deal with subjects like the specific heats of gases. That is, they are treated as point masses and no account is made of internal degrees of freedom such as molecular rotation and vibration. It is important to note that the average kinetic energy used here is limited to the translational kinetic energy of the molecules. The more familiar form expresses the average molecular kinetic energy: Where N is the number of molecules, n the number of moles, R the gas constant, and k the Boltzmann constant. Comparisonwith the ideal gas law leads to an expression for temperature sometimesreferred to as the kinetic temperature. The expression for gas pressure developed from kinetic theory relatespressure and volume to the average molecular kinetic energy. Kinetic Temperature, Thermal Energy Kinetic Temperature
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |