In a canonical ensemble, how does the entropy relate to the thermodynamic probabilities of various energy states?
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Q: The entropy S of a system of N spins, which may align either in the upward or in the downward…
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- According to the energy according to the equipartition theorem of degrees of freedom, what is the internal energy of 5 moles of rigid diatomic ideal gas molecules at equilibrium at temperature T?Using molecular and system partition functions it is possible to derive the Sackur-Tetrode equation, which allows the computation of absolute entropy from molecular parameters: 2лmkвT S = {NkB + NkB In In [ (2TmkyT) 3/² V h² N In this equation, the quantity V/N is sometimes written as 1/p. = (a) Use this expression to compute the entropy due to translation for one mole of CO₂ gas at 400 K, for which mco2 = 44.01 Daltons, and its molar volume (pressure = 1 bar) is V 0.03326 m³. (Note: 1 Dalton = 1.66053 × 10-27 kg). Be careful with the computation- keep track of units and make sure that the argument of the LN function is unitless. Your answer should be between 100 and 200 J/K if you do everything correctly. (b) If the mass of neon were twice as large (88 instead of 44 Daltons), by how much would the extra mass affect its translational entropy? Report the ratio of the entropy with the heavier mass to that found in part (a).Consider a classical ideal gas in three dimensions, with N indistin- guishable atoms confined in a box of volume V = L³. Assume the atoms have zero spin and neglect any internal degrees of freedom. Starting from the energy levels of a single atom in a box, find: (a) The Helmholtz free energy F (b) The entropy o (c) The pressure p (d) The chemical potential u (e) If the effect of the gravitational field of the Earth is taken into account (constant gravitational acceleration g), the chemical potential would also depend on the height h from some reference point which can be taken to be the sea level. Find u(h) in this case
- Problem 1: Consider a classical ideal gas in three dimensions, with N indistinguishable atoms confined in a box of volume N³. Assume the atoms have zero spin and neglect any internal degrees of freedom. Starting from the energy levels of a single atom in a box, find: (a) The Helmholtz free energy F' Hint: ſ. -ax² d.x e Va (b) The entropy o (c) The pressure pConsider a classical ideal gas of N diatomic heterogeneous molecules at temperature T. The charac- teristic rotational energy parameter is € = 1 and the natural frequency of vibrations is wo. Consider the temperature region where T≫er/kB, but T is of the order of ħwo/kB. Ignore contributions from all other internal modes. Calculate the canonical partition function, the average energy, and the heat capacity at constant volume, Cv.Answer in 90 minutes please.