One plot with two corresponding x axes in the FrameTicks

Question

I am having difficulty plotting the following ListPlot. Below I simplified my code. I would like the top x axis to show the corresponding w values that resulted the dots in the plot. For example for the first dot, b=9 and w=1.9. How can I add the w=1.9 value corresponding to this dot on the top x-axis? Thanks

a[b_, w_] := b/(w);
k[b_, w_] := (1 - Exp[-a[b, w]])^2*Exp[-2/w];


ListPlot[{{9, k[9, 1.9]}, {19, k[19, 3]}, {29, k[29, 4]}, {39, 
   k[39, 5]}}, Frame -> True, 
 FrameTicks -> {{Automatic, None}, {Automatic, {1.9, 3, 4, 5}}}, 
 FrameLabel -> {{k, None}, {"b", "w"}}]

Answer 1

$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global`*"]

a[b_, w_] := b/(w);
k[b_, w_] := (1 - Exp[-a[b, w]])^2*Exp[-2/w];

wValues = {5, 4, 3, 2.5, 1.9};

Use ParametricPlot

EDIT 2: Your points are on the associated w curve, the value of w is not related to the x-axis (i.e., b) in any straightforward manner.

Show[ParametricPlot[{b, k[b, w]}, {b, 9, 39}, {w, 1.9, 5}, 
  PlotStyle -> Opacity[0.15], AspectRatio -> 1, BoundaryStyle -> None], 
 ParametricPlot[
  Evaluate[Tooltip[{b, k[b, #]}, StringForm["w = ``", #]] & /@ wValues], {b, 
   9, 39}, AspectRatio -> 1, 
  PlotLegends -> LineLegend[wValues, LegendLabel -> Style[w, 14]]],
 ListPlot[{
   Callout[{9, k[9, 1.9]}, 1.9], 
   Callout[{19, k[19, 3]}, 3],
   Callout[{29, k[29, 4]}, 4],
   Callout[{39, k[39, 5]}, 5, Below]}], 
 FrameLabel -> (Style[#, 14] & /@ {b, k})]

EDIT: The curves for w asymptoticly approach

Asymptotic[k[b, w], b -> Infinity, Assumptions -> w > 0]

(* E^(-2/w) *)

or equivalently,

Limit[k[b, w], b -> Infinity, Assumptions -> w > 0]

(* E^(-2/w) *)

For the wValues

E^(-2./#) & /@ wValues

(* {0.67032, 0.606531, 0.513417, 0.449329, 0.349018} *)