1
$\begingroup$

i'm trying to plot Maxwell distribution and draw a line where average speed varg, most probable speed vm, and rms speed vrms, and label those lines.

my struggle how to plot these lines and combine graphics with plot in Show.

here is the code easy and short!

k = 1.38064852*10^(-23);
m = 28*1.660538782*10^(-27);
t = 300;

max[v_] := ((m/2 Pi k t)^3/2 )*(4 Pi v^2) Exp[-((m*v^2)/(2 k t))]


p1 = Plot[max[v], {v, 0, 1200}]

vm = Sqrt[2 k t/m]
vrms = Sqrt[3 k t/ m]
vavg = Sqrt[8/Pi]*Sqrt[k t /m]

l1 = Graphics[Line[{vm, 0}, {vm, y}]]
l2 = Line[{0, vrms}]
l3 = Line[{0, vavg}]

Show[p1, l1]

enter image description here

$\endgroup$
  • $\begingroup$ Have you defined $y$? With no $y$ yet defined, l1 = Graphics[Line[{vm, 0}, {vm, y}]] will not be well defined. $\endgroup$ – mjw Mar 26 '19 at 22:15
  • $\begingroup$ Note the syntax for Line[{{x1, y1}, {x2, y2}}]: you are missing a set of braces; you should also probably use InfiniteLine here. $\endgroup$ – MarcoB Mar 26 '19 at 22:54
2
$\begingroup$

With your definitions, consider something like this:

l1 = InfiniteLine[{{#, 0}, {#, 1}}] &[vm];
l2 = InfiniteLine[{{#, 0}, {#, 1}}] &[vrms];
l3 = InfiniteLine[{{#, 0}, {#, 1}}] &[vavg];

Show[
 p1,
 Graphics[{Red, l1, Blue, l2, Black, l3}]
]

Mathematica graphics

| improve this answer | |
$\endgroup$
  • $\begingroup$ One might also consider using GridLines. $\endgroup$ – J. M.'s technical difficulties Mar 27 '19 at 1:25
  • $\begingroup$ how do i label them !? $\endgroup$ – Alrubaie Mar 27 '19 at 14:02
2
$\begingroup$

Your definition for max is not a valid PDF

max[v_] := ((m/2 Pi k t)^3/2)*(4 Pi v^2) Exp[-((m*v^2)/(2 k t))]

The total probability should be 1

Assuming[m > 0 && k > 0 && t > 0,
 Integrate[max[v], {v, 0, Infinity}]]

(* (m^(3/2) π^(9/2) (k t)^(9/2))/(4 Sqrt[2]) *)

You probably intended to use

max[v_] := (m/(2 Pi k t))^(3/2)*(4 Pi v^2) Exp[-((m*v^2)/(2 k t))]

This is a valid PDF

Assuming[m > 0 && k > 0 && t > 0,
 Integrate[max[v], {v, 0, Infinity}]]

(* 1 *)

You would be better off using the built-in MaxwellDistribution

dist = MaxwellDistribution[Sqrt[k t/m]];

This has the same PDF as the revised max

Assuming[m > 0 && k > 0 && t > 0 && v > 0,
 PDF[dist, v] == max[v] // Simplify]

(* True *)

vavg = Mean[dist]

(* 2 Sqrt[2/π] Sqrt[(k t)/m] *)

vrms = RootMeanSquare[dist]

(* Sqrt[3] Sqrt[(k t)/m] *)

The mode is

vm = v /. Solve[{D[PDF[dist, v], v] == 0 && 
    m > 0 && k > 0 && t > 0 && v > 0}, v][[1]] // 
  Simplify[#, m > 0 && k > 0 && t > 0] &

(* Sqrt[2] Sqrt[(k t)/m] *)
| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ Additionally, the "magic constants" in the OP are k = QuantityMagnitude[UnitConvert[Quantity[1, "BoltzmannConstant"]]] and m = QuantityMagnitude[UnitConvert[Quantity[28, "AtomicMassUnit"]]]. $\endgroup$ – J. M.'s technical difficulties Mar 27 '19 at 1:22
  • $\begingroup$ Thanks guys so much $\endgroup$ – Alrubaie Mar 27 '19 at 14:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.