B.5) I think we are to read the plots of Figure T5.2 finding the x=kT/e for
which the y-axis (essentially speciic heat) is 1/2 its high temperature value.
We then convert x to T using the value of e: T=e x/k
1-d oscillator x=.3
1d motion x=.8
diatomic rotation x=.4
a:
* ? 1e-7*.8/kb
.9283650331310271E-03
b:
* ? .00025*.4/kb
1.160456291413784
c:
* ? .29*.3/kb
1009.596973529992
M.3)
* ? hbar*3e14/kb
2291.470533294990
mostly switched off until thousands of K
-----
z1=Sum[(2 j +1) Exp[-e b j(j+1)],{j,0,4}]
avg1=-D[Log[z1], b]
avg1=avg1 /. b->1/(k T)
c1=D[avg1,T]
c1 = c1 /. {e-> 87.6 k}
c1=Simplify[c1/k]
Plot[c1,{T,10,100}]
Export["H2_c1.eps",%]
Howework Questions: Explain what/why of the following bit of code:
avg1=-D[Log[z1], b]
the average energy is (-) the beta derivative of the log of the partition function
c1=D[avg1,T]
the specific heat is how the energy changes with temperature
Plot[c1,{T,1,10}]
fails: "too small to represent as a normalized machine number"
Plot[c1,{T,100,1000}]
shows a falling specific heat due to termination of the infinite sum
at high temperatures the occupation of states equalizes and there is no change with temeprature
z2=Sum[(2 j +1) Exp[-e b j(j+1)],{j,0,8}]
avg2=-D[Log[z2], b]
avg2=avg2 /. b->1/(k T)
c2=D[avg2,T]
c2 = c2 /. {e-> 87.6 k}
c2=Simplify[c2/k]
Plot[c2,{T,100,250}]
Export["H2_c2.eps",%]
Plot[{c1,c2},{T,80,200}]
Export["H2_c1+c2.eps",%]
Plot[c2,{T,250,1000}]
also shows a declining specific heat (but its not that much of a decline)
z3=Integrate[Exp[-e b u],{u,0,Infinity},Assumptions->{e b >0}]
avg3=-D[Log[z3], b]
avg3=avg3 /. b->1/(k T)
c3=D[avg3,T]
c3 = c3 /. {e-> 87.6 k}
In[28]:= c3=Simplify[c3/k]
Out[28]= 1
allc[T_]:=If[T<100,c1, If[T<300,c2,c3]]
Plot[allc[T],{T,10,200},PlotRange->All]
Export["H2_allc.eps",%]