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Although the integrals (and ) forand cannot be
Use the formula to show that the pressure of a photon gas is 1/3 times the energy density (U/V). Compute the pressure exerted by the radiation inside a kiln at 1500 K, and compare to the ordinary gas pressure exerted by the air. Then compute the pressure of the radiation at the centre of the sun, where the temperature is 15 million K. Compare to the gas pressure of the ionised hydrogen, whose density is approximately 105 kg/m3.
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Near the cells where oxygen is used, its chemical potential is significantly lower than near the lungs. Even though there is no gaseous oxygen near these cells, it is customary to express the abundance of oxygen in terms of the partial pressure of gaseous oxygen that would be in equilibrium with the blood. Using the independent-site model just presented, with only oxygen present, calculate and plot the fraction of occupied heme sites as a function of the partial pressure of oxygen. This curve is called the Langmuir adsorption isotherm ("isotherm" because it's for a fixed temperature). Experiments show that adsorption by myosin follows the shape of this curve quite accurately.
Consider a system of five particles, inside a container where the allowed energy levels are nondegenerate and evenly spaced. For instance, the particles could be trapped in a one-dimensional harmonic oscillator potential. In this problem you will consider the allowed states for this system, depending on whether the particles are identical fermions, identical bosons, or distinguishable particles.
For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is
Consider two single-particle states, A and B, in a system of fermions, where and that is, level A lies below by the same amount that level B lies above . Prove that the probability of level B being occupied is the same as the probability of level A being unoccupied. In other words, the Fermi-Dirac distribution is "symmetrical" about the point where .
For a system of bosons at room temperature, compute the average occupancy of a single-particle state and the probability of the state containing bosons, if the energy of the state is
For a system of particles at room temperature, how large must be before the Fermi-Dirac, Bose-Einstein, and Boltzmann distributions agree within ? Is this condition ever violated for the gases in our atmosphere? Explain.
For a system obeying Boltzmann statistics, we know what is from Chapter 6. Suppose, though, that you knew the distribution function (equation ) but didn't know . You could still determine by requiring that the total number of particles, summed over all single-particle states, equal N. Carry out this calculation, to rederive the formula . (This is normally how is determined in quantum statistics, although the math is usually more difficult.)
Physics Principles with Applications
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