Question

B.2 The multiplicity of a monatomic ideal gas is given by 2 = f(N)VN U3N/2, where V is the volume occupied by the gas, U its internal energy, N the number of particles in the gas and f(N) a complicated function of N. [2] (i) Show that the entropy S of this system is given by 3 S = Nkg In V + ŽNkg In U + g(N), where g(N) is some function of N. (ii) Define the temperature T of a thermodynamic system, in terms of its entropy. Use the result in part (i) to calculate the temperature of a monatomic ideal gas, as a function of its internal energy U. Then invert this relation to obtain an expression for U. Compare the result with the prediction of U from the equipartition theorem. (iii) State the thermodynamic identity. Explain all the terms you introduce. Use the thermodynamic identity to show that [6] P=T as av UN [5] where P is the pressure of the gas, and the subscripts indicate that U and N are constant. Then use this result and that of part (i) to derive the ideal-gas law. (iv) Consider the gas confined in a container of volume V. What is the probability of finding all N atoms in the leftmost 99% of the container if (a) N = 100; (b) N = 10,000; (c) N = 1023? [2]

Answer 1

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form ear benesy - 36 ln Time 3N NEO (*) In this ). ( tame] NKB Multiply both sides by 2 34 Tame 3N 3 NKB en اله 213 -15 3NKB time en los exp Time 3N 3 NKB E emp 3b2~ temv263 ds 3 NK da a Tds - dut pdv-udn at caust. willene EN. Tds- du (alun Tds= dt

IS 36²N • exp 3NKB 3 NKD simv23 Ex Bro E ZENKBT gas U-2 KBT Intonal energy of a classical ideal Specific heat :- Cu = ( (8) Inka Pressure :- da = de + pdv-udn Tds = dE + opdu- udN po at cunst N and s dEt pou p: -(de luis P 3h²~ exp lonke --)-4.13 sam vole Р 2 3 v pro effe

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