I\'m wanting to know how others would solve part b, for me I took the molecular

Question

I'm wanting to know how others would solve part b, for me I took the molecular weight of H20 (18.016g/mol) converted it to a volume using the density and then took the cubic root of it, but that answer seems way too large to be correct (I got 0.026215 m). Am I correct, or is there a different way to do this that I am missing?

Some other constants/things you may need: Heat diffusivity of water D=1.4 x 10^(-7) m^2/s, Density of water 1000 kg/m^3, and speed of sound is 1480 m/s

1. Consider an ultrasonic wave propagating in water having an acoustic pressure amplitude of 2 kPa, and a frequency of 10 MHz. a. What is the wavelength of this wave? b. What is the mean distance between water molecules (@STP)? c. Calculate its acoustic velocity amplitude and compare it to the mean thermal velocity of water molecules. d. Calculate its acoustic displacement and compare it to the mean distance between water molecules e. Consider a cubic volume of water of a linear size equal to the wavelength. What is the heat diffusion time across the volume? acoustic f. Compare the diffusion time to a period time of a 10 MHz wave. Use your results to argue whether this wave is predominantly adiabatic or isothermal process. g. What is the amplitude of temperature oscillation associated with this wave? Note: The diffusion length, d, is given by d D r, where r is the diffusion time and D is the coefficient of heat diffusivity. This relation can be obtained from the 1-d solution of the diffusion equation.

Explanation / Answer

(a)

Wavelength

=c/f

Where c= speed of sound in water =1480 m/s

f = frequency =10 Mhz=10*106 Hz

So

=c/f=1480/107 = 1.480*10-4 m

(b)

At STP

P=1atm

R=0.082 lit/atm/mol K

T=298 K

So V=nRT/P

V=0.0245 m3

Thus each molecule occupies volume V/NA

NA =Avogadro number = 6.023*1023

So 0.0245/ 6.023*1023 =4.0677*10-25 m3

So distance =(4.0677*10-25)1/3 =7.40*10-9 m =7.40 nm

(c)

(d)

Acoustic displacement = P/2Z

Z=Characterisic impedance =density*speed =1000*1480=1.48*106 kg.m-2.s-1

So displacement =2*103/(2*3.14*1.48*106)=2.15*10-4 m

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