The electronic structure of one-dimensional chain of sodium (Na) atoms can be approximately described by the particle-in-a-box model. The energy of each state can be calculated using En=(n^2h^2)/8mL^2, ? = 1, 2, 3, … where L is the length of the 1D chain. Assuming L = a0(N – 1), where N is the number of Na atoms and a0 = 0.360 nm is the internuclear distance. a) Determine the energy gap between the highest occupied energy level and the lowest unoccupied energy level as a function of N. Assume that N is an even number that is large enough (Hint: assume that each Na only contributes 1 electron to the problem and consider the information in Question 1c above.). b) Thermal energy at room temperature is 4.15 × 10–21 J. Calculate the minimum number of Na atoms required so that the energy gap is smaller than the thermal energy.
Part B
The electronic structure of one-dimensional chain of sodium (Na) atoms can be approximately described by the particle-in-a-box model. The energy of each state can be calculated using En=(n^2h^2)/8mL^2, ? = 1, 2, 3, … where L is the length of the 1D chain. Assuming L = a0(N – 1), where N is the number of Na atoms and a0 = 0.360 nm is the internuclear distance.
a) Determine the energy gap between the highest occupied energy level and the lowest unoccupied energy level as a function of N. Assume that N is an even number that is large enough (Hint: assume that each Na only contributes 1 electron to the problem and consider the information in Question 1c above.).
b) Thermal energy at room temperature is 4.15 × 10–21 J. Calculate the minimum number of Na atoms required so that the energy gap is smaller than the thermal energy.
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