Question

TSD.1 In this problem, we will see (in outline) how we can calculate the multiplicity of a monatomic ideal gas This derivation involves concepts presented in chapter 17 Note that the task is to count the number of microstates that are compatible with a given gas macrostate, which we describe by specifying the gas's total energy u (within a tiny range of width dlu), the gas's volume V and the num- ber of molecules N in the gas. We will assume that the gas is confined in a cubical container whose sides have length L. We will also assume that the gas is pure (only one type of molecule). Also note that if it is monatomic, its mol- ecules have no accessible internal energy storage modes. (a) As we saw in chapter T7, the energy of a single mol- ecule in our cubical "box" is 12 (T8.35) where m is the molecule's mass, h is Planck's constant and n,, y and n are independent positive integers. This means that the gas's total thermal energy must be 8mL (T8.36) where N is the number of molecules. Now, imagine a 3N-dimensional space whose perpendicular axes point in the lx, ly, lz, 2x, 2y, 2z, , Nx, Ny, and Nz direc- tions. Each gas microstate corresponds to a complete set of choices of the integers n, through no, and so to a different tiny cubelet of volume 1 1 in such a space The quantity in parentheses is the distance that a given microstate's cubelet is from the origin in that space. In a typical gas, the numbers m through M will be T large enough that we can consider these quantities to be continuous variables instead of integers. The total number of microstates consistent with the gass total energy being U within a tiny range +du, will be the same as the number of cubelets within a shell of radius r = V㎡, + + n = V8nUL/h and thickness dm](V8mlTL/hudu in our space. Look up "area of an n-dimensional sphere" online and use what you find to argue that the number of cublets in such a shell is (T8.37) r(3N/2) u where Tx) is the so-called "gamma function," which is a generalization of the factorial such that x) (x - 1)! when x is an integer, but is a smooth function of x between these values. through nv must be positiu (b) But each value of Argue therefore that the actual number of cubelets the part of the shell that counts is (T8.38) T3N/2) U (c) Because xs10 or larger for most everyda gas samples, the distinction between integer and non- integer values of x is negligible. Stirling's approxima- tion (look it up!) states that V2x (x/e) when 1. Use this approximation to argue that v du (T8.39) where I have ignored the distinction between x-1 and x (for x 102, this is an excellent approximation). (d) However, the molecules in the gas are completely indistinguishable, so microstates that differ only by the rearrangement of values of n, 1n, and n, between dif- ferent molecules don't actually count as distinct micro- states. Therefore, we have overcounted the actual number of microstates by a factor of N! (the number of different ways that we could rearrange the molecules). Use Stirling's approximation again to argue that the correct multiplicity should be N V6TNu (e) Now, ginormous numbers like 2 will not really be changed if we multiply it by even a large number (10") 10-= 10"or, +22) 1010". So we can drop the factor of V6т, the factor of du/u (which will be larger than about 10- in realistic situations), and even the factor of N). Argue that if we drop these factors, we get the Sackur-Tetrode equation (equation T8.8). Indeed, a basic quantum model of the ideal ideal monatomic gas for large U and large N implies (see problem T8D.1) that the gas's entropy is given by the so-called Sackur-Tetrode equation: (T8.8)

Answer 1

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