Chapter 13, Problem 1FTD

The vibration frequencies of molecules are much higher than those of macroscopic mechanical systems. Why?

To determine

Why the vibration frequencies of molecules are much higher than those of the macroscopic mechanical systems.

Answer to Problem 1FTD

The vibration frequencies of molecules are much higher than those of the macroscopic mechanical systems as the mass of the molecules is really small compared to the mass of the macroscopic materials and the spring constant is large.

Explanation of Solution

The frequency of oscillation is the number of cycle of oscillation that is completed in one second.

The frequency of oscillation of normal simple harmonic oscillation is seen to be inversely proportional to the square root of the mass of the material. Both molecules and macroscopic materials can be taken to be undergoing simple harmonic motion. The mass of the molecules are of the order of few grams per mole or even less. The mass of the macroscopic machines on the other hand have greater mass in the orders of hundreds of kilograms or above. The frequency is inversely proportional to the mass and therefore, the molecules will have much more vibrational frequency than the machines. The amount of the spring constant for molecules which are provided by the bonds existing between the molecules are much greater than the restoring force in the machine. The frequency is being proportional to the spring constant will therefore, also result in the molecules having much more vibrational frequency than the machines.

Conclusion:

Therefore, the vibration frequencies of molecules are much higher than those of the macroscopic mechanical systems as the mass of the molecules is really small compared to the mass of the macroscopic materials and the spring constant is large.