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Disclaimer : I'm not sure that the following is the exact/complete answer to the question but maybe these elements could help. Equilibrium of a system under an external field Let say that you have an open macroscopic system $\Sigma$ (composed of identical particles) which is under the influence of an external time-independant but space-dependant ...

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The chemical kinetics of air depend on both how fast you are flying and your altitude. Fortunately, NASA has studied these issues. The figures below are from NASA Report NACA-TN-4359. The predominant chemistry in the stagnation region of an airfoil as a function of flight speed and altitude are shown below: You say $M=7$. If your vehicle is near sea ...

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This problem contains a functional form that is very difficult to work with. A numeric integration is one approach that will give an approximate (but good) answer. To do this, the following procedure can be used: 1) Start at pressure P1, and establish a "small" value for dP 2) Use a trial-and-error method to calculate V 3) Multiply V by dP and keep ...

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The Fermi energy, $\epsilon_F$, is only equal to the chemical potential, $\mu$, when the Fermi gas is at zero temperature. The Fermi energy basically means, "chemical potential at zero temperature". At any other temperature you could find $\mu$ via one of the standard thermodynamic relations (i.e. as the appropriate derivative of a free energy).

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