2.6 Problems for Chapter 2

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EXERCISES:

2.1 Variational method of Lagrange: Determine the minimum of the function $f(x,y)=x^{2}+2y^{2}$ under the condition (i.e. along the curve) $g(x,y)=x+y-6=0$, by two different methods:
a) by inserting $y=6-x$ in f(x,y) and differentiating by $x$;
b) using a Lagrange multiplier $\alpha$ and evaluating Lagrange's equations $\partial f / \partial x - \alpha (\partial g / \partial x) = 0$ and $\partial f / \partial y - \alpha (\partial g / \partial y) = 0$. (Shorthand notation: $\delta f - \alpha \delta g = 0$; $\delta f$... ``Variation of $f$'').
For your better understanding sketch the functions $f(x,y)$ and $g(x,y)=0$.

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 $P=(2/3)(N \langle E \rangle/V)$.

2.5 Transport properties: For nitrogen under standard conditions, estimate the mean free path and the transport coefficients $\eta$, $\lambda$ and $D$. ( $\sigma \approx 4 \cdot 10^{-10}m$).



TEST YOUR UNDERSTANDING OF CHAPTER 2:

2.1 Maxwell-Boltzmann distribution: What is the formula for the distribution density $f_{0}(\vec{v})$ of the molecular velocities in equilibrium; what is the respective formula for the speeds (absolute values of the velocity), $f_{0}(\vert\vec{v}\vert)$?

2.2 Pressure in an ideal gas: Derive the pressure equation of state for the ideal gas from simple kinetic theory.

2.3 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 $\sigma$?
- How does the viscosity of a dilute gas depend on the density?