**Question **# a. Derive the expression for the entropy for a three-dimensional monoatomic gas. Show all steps and ignore internal degrees of freedom. b. Consider nitrogen at STP. Calculate its concentration n = N/V from the ideal gas equation. Calculate its quantum concentration no and the associated quantum volume (1/no) and quantum length which is the dimension of a cube whose volume is the quantum volume (Express in correct units!). Determine the ratio non and the resulting entropy. C.A degree of freedom "freezes out" when kT is less than or comparable to the spacing between the lowest energy levels of a system. Estimate the temperature at which the translational degree of freedom of a nitrogen molecule in a one-dimensional box will freeze out.