Problem 2.35

At what temperature would the entropy of an ideal monatomic gas become negative?  Consider helium originally at room temperature and atmospheric pressure.  Cool it at constant density.  At what temperature would the entropy go to zero?  (Below that, it would become negative.)

Clear["Global`*"]

Sackur-Tetrode Equation

S[n_, V_, T_] := N k (Log[((V/n)(((((4 π m U[n, T])/(3 n (h^2)))))^(3/2)))]+ (5/2))

U[n_, T] := (3/2)n k T (* For a monatomic ideal gas *)

S[n, V, T]

Solve[S[n, V, T] == 0, T]

Tz = T 1 /. %[[1]]

V = n k Ti/pi; (* Ideal gas law applied to initial conditions *)

Tz

Tz = FullSimplify[Tz, Assumptions->n > 0]

Ti = Quantity[300, "Kelvins"]

pi = Quantity[1, "Atmospheres"]

k = Quantity[1, "BoltzmannConstant"]

h = Quantity[1, "PlanckConstant"]

m = ElementData["Helium", "AtomicMass"]

Tz

UnitConvert[Tz, "Kelvins"]