What Temperatures Contribute to Air's Heat Capacity Due to Vibration?

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SUMMARY

The discussion focuses on determining the temperatures at which vibrational contributions to the heat capacity of air, primarily composed of diatomic nitrogen (N2), become significant. Using an effective spring constant of 2.3 x 103 N/m and an effective oscillating mass of half the atomic mass, the angular frequency is calculated as ω = 1.67 x 1015 rad/s. The energy associated with this vibration is E = 1.76 x 10-19 J. The key takeaway is that understanding these parameters is essential for accurately modeling the heat capacity of air.

PREREQUISITES
  • Understanding of diatomic gas behavior, specifically nitrogen (N2).
  • Familiarity with the concepts of oscillators in physics.
  • Knowledge of the Boltzmann constant (kB) and its application in thermodynamics.
  • Basic proficiency in quantum mechanics, particularly energy quantization.
NEXT STEPS
  • Research the relationship between temperature and vibrational modes in diatomic gases.
  • Study the derivation of the equipartition theorem in thermodynamics.
  • Learn about the implications of quantum harmonic oscillators in statistical mechanics.
  • Explore the calculation of heat capacity for different states of matter, focusing on gases.
USEFUL FOR

This discussion is beneficial for physics students, thermodynamics researchers, and anyone involved in the study of gas properties and heat capacity modeling.

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Homework Statement



Air is mostly composed of diatomic nitrogen, N2. Assume that we can model the gas as an oscillator with an effective spring constant of 2.3 x 103 N/m and and effective oscillating mass of half the atomic mass. For what temperatures should vibration contribute to the heat capacity of air?

Homework Equations



\omega=\sqrt{\frac{\kappa}{m}}
E=\hbar\omega
K=\frac{3}{2}k_{B}T

The Attempt at a Solution



\omega=\sqrt{\frac{\kappa}{m}}=1.67\times10^{15} rad/s
E=\hbar\omega=1.76\times10^{-19} J

I am not sure what to do next.
 
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