Solution:
The Boltzmann equation is:
N2 / N1 = [g2 / g1 ] exp (- E2 – E1) / kT
And from the table, g2 = 8 and g1 = 2
We require the condition that: N2 = N1 so:
1 = [8/ 2 ] exp (- E2 – E1) / kT
But: E2 = - 13.6 eV and E1 = -3.4 eV, therefore:
1 = 4 exp (-10.2 eV)/ kT
Taking natural logs:
ln (4) = (10.2 eV)/ kT
where: k = 8.6174 x 10 -5 eV/K
Solving for T:
T = 10.2 eV/ (ln 4) (8.6174 x 10 -5 eV/K)
T = 10.2 eV/ (1.3862) (8.6174 x 10 -5 eV/K)
T= 85 388 K or T = 8.54 x 10 4 K
2 ) For the Balmer a line (called H- alpha), we know:
E3 – E2 = - 13.6 eV ( 1/ 3 2 - 1/ 2 2 ) = 1.88 eV
a) From this information calculate the ratio N2 / N1
b) Obtain the specific intensity from:
I u = 2h u 3 / c 2 [1/ exp (hc/lkT]
Solution:
N2 / N1 = [g2 / g1 ] exp (- E2 – E1) / kT
From the table in the Dec. 5th post; g2 = 8 and g1 = 2
k = 8.6174 x 10 -5 eV/K
N2 / N1 = [8/ 2 ] exp (- E2 – E1) / kT
N2 / N1 = 4 exp (-1.88 eV)/ (8.61 x 10 -5 eV/K) (10 4 K)
N2 / N1 = 4(0.113) = 0.452
We need to use consistent cgs units. Planck constant h = 6.62 x 10 -27 erg-s
c= 3 x 10 10 cm/s
l = hc/ E = (6.62 x 10 -27 erg-s) (3 x 10 10 cm/s)/ 3.0 x 10 -12 erg
l = 6.62 x 10 -5 cm
k=1.38 x 10 -16 erg/K
[1/ exp (hc/lkT] =
1/ [exp (6.62 x 10 -27 erg-s) (3 x 10 10 cm/s)/ (6.62 x 10 -5 cm) (1.38 x 10 -16 erg/K)( 10 4 K)
= 0.113
I u = 2h u 3 / c 2 [0.113] erg cm -2/s
But u = E/h =
3.0 x 10 -12 erg/ 6.62 x 10 -27 erg-s = 4.53 x 10 14 /s
So:
I u = 0.226(6.62 x 10 -27 erg-s) (4.53 x 10 14 /s) 3 / (3 x 10 10 cm/s) 2
I u = 1.51 x 10 -4 erg cm -2/s
3)Calculate the transition probability you get using the Einstein equation:
A 21= 6.67 x 10 16 [g f/g2 l2 Å]
What possible errors might cause the values to diverge? (Take g = f » 1)
l = 6.62 x 10 -7 m = 6.62 x 10 -5 cm = 6620 Å
With g2 = 8 and g = f = 1
A 21= 1.93 x 10 8 [Å-2]
Compare to standard form (see e.g. Wikipedia, “Einstein coefficients”) given in multiple physics papers as:
A 21= [f g1/g2]{ 2 π u 3 e 2 }/ e o me c3
A 21 = 1.87 x 10 7 or: 0.187 (in defined units of 10 8 s)
Error sources: Imprecise oscillator frequency f
Error in one of the statistical weights.
Error in one of the statistical weights.
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