2.1 Variational method of Lagrange:
Determine the minimum of the function
under the condition (i.e. along the curve)
, by two different methods:
a) by inserting in f(x,y) and differentiating by ;
b) using a Lagrange multiplier and evaluating
``Variation of '').
For your better understanding sketch the functions
2.2 Method of the most probable distribution:
Having understood the principle of Lagrange variation,
reproduce the derivation of the Boltzmann distribution of energies
(see text). Compare your result with the bottom right histogram
in Applet LBRoulette.
2.3 Moments of the Maxwell-Boltzmann distribution:
Verify the expressions for the most probable velocity,
the average and the r.m.s. velocity.
2.4 Pressure in a dilute gas:
Verify the formula
2.5 Transport properties:
For nitrogen under standard conditions, estimate the mean free path and the
, and . (
2.1 Maxwell-Boltzmann distribution:
What is the formula for the distribution density
of the molecular velocities in equilibrium;
what is the respective formula for the speeds (absolute values
of the velocity),
Pressure in an ideal gas:
Derive the pressure equation of state for the ideal
gas from simple kinetic theory.
Transport coefficients: - Write down the defining equation for one of the three transport coefficients.
- What is the mean free path in a gas of spheres with diameter ?
- How does the viscosity of a dilute gas depend on the density?