1. Maxwell-Boltzmann Distribution for Velocities and Speeds of Particles in Thermal Equilibrium: The normalized Maxwell-Boltzmann distribution of momenta for a particle of mass m in thermal equilibrium has been determined to be f(px,py,pz)dpxdpydpz=(2mkT)3/2e(px2+py2+pz2)/2mkTdpxdpydpz a) Convert this Maxwell-Boltzmann distribution to the distribution of velocities and show that it is properly normalized. b) Sketch the distribution for the x - component of the velocity. c) Convert to spherical coordinates, integrate over the angles to find the expression for the distribution of speeds. Sketch the distribution of speeds. d) Derive an expression for the most probable molecular speed (maximum of the distribution). e) Derive expressions for the average or mean speed and the root mean square speed (v2)1/2=(3kBT/m)1/2. The latter is used to estimate the temperature in MD simulations. f) Evaluate the most probably speed for N2 at room temperature