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104 Chapter 4 4-3. Show that (a, T) = a in one dimension and that in two dimensions. Use these results to show that has a contribution of to its total value for each dimension. Remember that = Then, for one dimension, And for two dimensions, = Now The partition function is proportional to where n is the dimension. So and 4-4. Using the data in Table 1.2, calculate the fraction of sodium atoms in the first excited state at 300 K, 1000 K, and 2000 K. We can use the second line of Equation 4.10 to calculate the fraction of sodium atoms in the first excited state, = = Using the data in Table 1.2, we find that the numerator of this fraction is 2 956.183