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# sess4 - 4 Reaction equilibria Reading Shu Vol.I Ch.7 4.1...

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4. Reaction equilibria Reading: Shu, Vol.I, Ch.7 4.1 The Saha equation In Local Thermodynamical Equilibrium the distribution of atoms with ionization state i over the various energy states m is proportional to exp( - χ i,m /kT ), where χ i,m is the excitation energy of state m relative to the ground state. Then N i,m N i, 1 = g i,m g i, 1 exp( - χ i,m kT ) (4 . 1) and further after summing over all m N i,m N i = g i,m u i ( T ) exp( - χ i,m kT ) (4 . 2) where u i ( T ) = X m g i,m exp( - χ i,m kT ) (4 . 3) is called the level partition function . We can extend this treatment to the continuity of states with positive energy by using differen- tials and state densities. We thus derive for the ratio of singly ionized atoms to neutral, both in ground state, dN 1 , 1 N 0 , 1 = 2 g 1 , 1 g 0 , 1 exp( - χ 0 + p 2 2 m e kT ) d 3 x d 3 p h 3 (4 . 4) where the factor 2 takes care of the two independent spin positions of a free electron. Integrating over momentum we derive dN 1 , 1 N 0 , 1 = 2 g 1 , 1 g 0 , 1 (2 π m e kT ) 3 / 2 exp( - χ 0 kT ) d 3 x h 3 (4 . 5) The electron shares the available volume with all electrons, and thus d 3 x = n - 1 e . Hence we finally derive the Saha equation N 1 , 1 N 0 , 1 n e = 2 g 1 , 1 g 0 , 1 (2 π m e kT ) 3 / 2 h 3 exp( - χ 0 kT ) (4 . 6) that, using the level partition sums, can be written as an equation for the ionization equilibrium. N i +1 N i n e = 2 u i +1 u i (2 π m e kT ) 3 / 2 h 3 exp( - χ i kT ) (4 . 7) The density of electrons is determined by the density of ionized atoms, of course.

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sess4 - 4 Reaction equilibria Reading Shu Vol.I Ch.7 4.1...

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