Okay, let's solve this step by step:
1. We are given a harmonic oscillator with vibrational frequency 421 cm-1. From this we can calculate the excitation energy between vibrational levels:
E = hν = (6.626×10-34 J.s)(421×1017 s-1) = 1.769×10-19 J
2. The temperature is 581.73 K. We can convert this to Joules:
T = 581.73 K x (1.38×10-23 J/K) = 8.018×10-21 J
3. The fraction of oscillators in the zero-point level can be calculated using the Boltzmann distribution:
f0 = e^(-E0/kT)
Where E0 is the zero-point energy and k is the Boltzmann constant.
Since the zero-point energy is E0 = 0.5hν, we have:
f0 = e^(-0.5×1.769×10-19 / 8.018×10-21)
= e^-22.02
= 1.70×10-10
Therefore, the fraction of oscillators in the zero-point level at this temperature is approximately 1.70×10-10.