De Broglie wavelength, Maximum kinetic energy, and smoke particles

[Badrakhandama]

I have a few questions:
1. 'small smoke particles in air are seen under a low magnification microscope to move randomly at a speed of 0.10mm/s. The speed of sound in air is 330m/s. Estimate the mass of the smoke particles.

I cannot make the link between speed of particle and speed of sound - I was thinking something to do with relativistic equations but none involve sound

2a) Show that the de Broglie wavelength, L, of a particle of mass m, moving at velocity v, where v<<c is related to the KE of the particle by:

L = h/((2mKE)^.5)

I have done this, however the next part i cannot get:

b) consider the particle in a small rectangular box with sides of length a, b and c. The particle is moving at right angles to the b-c plane. Find an expression for the smallest possible energy. (think about the amplitude of the wave at the wall of the box)


I thought: rearranging gives KE = h^2/(2mL^2)

then differentiate with respect to L, and set it equal to zero but it doesn't seem right to me!

c) the box now contains many particles, and one b-c plane of the box is replaced by a piston. Show that as the length, a, is SLOWLY decreased the resulting change in wavelength ensures that Boyle's Law is obeyed.

My attempt: pV = nRT, where T is constant. From here, I have no clue what to do.

 

[Badrakhandama]

Ok, i have done question 1 now

Any help with the other two would be colosally appreciated

 

[sgd37]

well you have to fit multiples of full wavelengths into the box so the lowest energy is L= a