Calculating noise of a transducer from molecular vibration

[curiously new]

TL;DR

Searching for a noise floor figure to compare to op-amp noise floor (nV per square root Hertz).

I'd like approximate the noise floor of an ultrasonic air transducer starting from molecular vibrations.

Simply put, if I treat atmospheric air as an ideal gas and I confine each air molecule to exist in a volumetric cube with a square face $A_\mathrm{face}$, then I approximate the number of air molecules that are abutted against a transducer's element with area $A_\mathrm{T}$ by $$n=\frac{A_\mathrm{T}}{A_\mathrm{face}} .$$ Now, if I assume the rough estimate that an air molecule travels on average at a velocity $v$, then I can approximate the frequency of total collisions against the transducer's element by $$f=\frac{1}{6}\frac{1}{t}n=\frac{n}{6}\frac{v}{\sqrt{A_\mathrm{face}}}=\frac{1}{6}\frac{vA_\mathrm{T}}{A_\mathrm{face}^{3/2}} ,$$ where we divide by six because only a single side of each molecule's volumetric cube is abutted against the element.

Then, I can assume that this frequency varies as Gaussian whose standard deviation $\sigma$ is the square root of the frequency, which is the variation in the number of collisions. Then, I could say that my noise $N$ is given by $$N=\sqrt{f}$$ in units square root Hertz ($\sqrt{\text{Hz}}$).

So, I would like to compare this noise floor against the noise floor approximation of an op-amp, which are usually given in units $\text{nV}/\sqrt{\text{Hz}}$. My task lies in understanding how to translate this frequency of collisions (in square root Hertz) into a comparable unit to the op-amp's noise floor. Should I consider the momentum energy transferred by each molecule upon the element to get a pressure that I can then use a transducer's conversion factor (usually $\text{V}/\mu\text{bar}=\text{dB}$) to convert?

 

[Fred Wright]

I am puzzeled by why you think that the random collisions of air molecules are the source of noise for your sensor. Afik the sources of noise in integrated pressure sensors are shot noise and 1/f noise. Shot noise is the result of non-uniform flow of carriers across a junction and is independent of temperature. The second, 1/f, results from crystal defects and also due to wafer processing. This noise is proportional to the inverse of frequency and is more dominant at lower frequencies. However, you can test your hypothieses if you have access to a vacuum chamber. You could measure the noise floor of your sensor as a function of increasing pressure. If the noise floor rises with increasing pressure, then maybe your on to something.

 

[curiously new]

That is interesting, and I will look into that experiment. I'm working in the ultrasonic range around 40 kHz with a a bandwidth of about 2 kHz. At these frequencies, I have just assumed that 1/f noise was negligible. However, I hadn't given much thought to the shot noise. Thank you for your comments.

My larger concern, however, is trying to tie my calculation for the noise back into a comparable with the op-amp's noise floor.