(* Start with a simple plot of an exponential All built-in function in Mathematica are capitalized.  The exponential function can use E^(x) or Exp[x]. *)
Plot[Exp[x],{x,0,1}]

(* Lists can be created using {,} and two functions can be plotted using a list. *)

Plot[{E^(2x),Exp[x]},{x,0,1}]

(* Now create a version of the Boltzmann energy distribution *)

Plot[(x^(1/2)) Exp[-x],{x,0,1}]

(* Add in the Boltzmann constant (in eV/K), other constants, and the ability to specify temperature.  Here, the Evaluate[] function is used to replace the T in the formula with 300. *)

Plot[Evaluate[(2/(Pi^(1/2)))((((1/(8.62 (10^(-5))T))))^(3/2))(x^(1/2))Exp[((-x)/(8.62 (10^(-5))T))]/.T->300],{x,0,0.1}]

(* Real-time control of T would be even better, so now use the Manipulate[] function. We copy from above, but don't need the Evaluate[]. Be sure to limit the lower range since the function will blow up at T=0. *)

Manipulate[Plot[(2/(Pi^(1/2)))((((1/(8.62 (10^(-5))T))))^(3/2))(x^(1/2))Exp[((-x)/(8.62 (10^(-5))T))],{x,0,0.1}],{T,10,1000,1}]

(* Here's one way to determine the peak value of the function. Take it's derivative. Note, I switched to U from x for the energy. *)

D[(2/(Sqrt[Pi]))((((1/(8.62 (10^(-5))T))))^(3/2))(U^(1/2))Exp[((-U)/(8.62 (10^(-5))T))],U]

(* Set it equal to 0 and solve for U. *)

Solve[%==0,U]

(* Now put this back in the distribution and simplify the function to find the maximum value at the peak. I copy/pasted the function and the previous output.  Mathematica shows a ` to indicate that there are more digits in the decimal number. The result is very simple. *)

Simplify[(2/(Sqrt[Pi]))((((1/(8.62 (10^(-5))T))))^(3/2))(U^(1/2))Exp[((-U)/(8.62 (10^(-5))T))]/.U->0.0000431` T]

(* Now let's plot 2 distributions at different temperatures and add some labels. I split the lines to make it a little more readable. *)

Manipulate[
Plot[{
Legended[(2/(Sqrt[Pi]))((((1/(8.62 (10^(-5))T1))))^(3/2))(U^(1/2))Exp[((-U)/(8.62 (10^(-5))T1))],T1],Legended[(2/(Sqrt[Pi]))((((1/(8.62 (10^(-5))T2))))^(3/2))(U^(1/2))Exp[((-U)/(8.62 (10^(-5))T2))],T2]},{U,0,0.5},
PlotLabels->{((5614.169942439521` (Sqrt[(1/T1)]))/(Sqrt[T1])),((5614.169942439521` (Sqrt[(1/T2)]))/(Sqrt[T2]))},PlotRange->{0,5}],
{{T1,300,"(Subscript[T, 1])"},1,6000,1},{{T2,600,"(Subscript[T, 1])"},1,6000,1}]