Serial of year 19

• Three cards are randomly selected from 36 cards. Calculate probability of possibility that (i) just one ace was selected, (ii) at least one ace is selected and (iii) no ace is selected.
• $N$ identical particles is in a container. Calculate probability of case, that in the left part of container is $m$ particles more than in the right half. Draw a graph of dependence for $N=10^{10}$. Range of $m$ select so that the probability on the sides will be one tenth of the probability in the middle. How the width depends on $N?$ (Width is difference $m_{2}-m_{1}$, where $m_{2}>0$ a $m_{1}<0$ are values of $m$, for which the probability is half compared to the maximum).
• Estimate size of ln$n!$ (without using Stirling formula).

• What is the relation between the number of microstates Ω($E)$ of thermostat with energy ≤ $E$ and quantity defined as $η(E)$ (e.g. the number of microstates with energy in interval $E±Δ)$ for small Δ?
• Assume a system of $N$ independent harmonic oscillators, where energy of each oscillator can be one of following values:$nhω$, where $n=0,1,2$,…

(neglecting energy of zero oscillations). What will be the quantity $η(E)$ and $β(E)$ for big $N$ and $E?$

• Find the same quantity as in the previous case for system of $N$ non-interacting free electrons trapped on line or in square or in cube, respectively.

Hint: Use de Broglie relation between momentum and wavelength of de Broglie wave. The integer multiple of half-wavelengths must fit on the line. De Broglie wavelength in square can be imagined as product of waves in direction of axis $x$ and $y$, quantum condition is similar to the condition for line.

• Using similar idea as in the case in theoretical text (sorry, only available in czech:-( ) calculate the form of Gultberg-Waage law for more complicated reactions (e.g. 2$A+B->A_{2}B)$. Try to find, if (and how well) this law follows reality.
• From Maxwell-Boltzmann distribution derive which power of temperature determine mean kinetic energy of particles of gas. Check, that you are able, using same method, find out dependence of any power of velocity on power of temperature.
• Lets have a system of independent spins, discussed in text, at temperature $T_{1}$, which is located in magnetic field $B_{1}$. Then the system is adiabatically isolated (i.e. is closed in to vacuum flask (thermos) to avoid any temperature exchange with environment) and the magnetic field is slowly reduced to value $B_{2}$. Explain, why the temperature of the system will decrease. Calculate final temperature $T_{2}$.

Hint: The work done on the system with magnetic moment $M$ at small change of the field $B$ of d$B$ is given by the equation d$W=-MdB$.

• What is the heat capacity of 3-atomic gas predicted by classical physics? The atoms are arranged in a triangular shape. To what capacity will it fall at 100K?
• Find out behaviour of the equations for internal crystal energy and energetic spectrum of black body radiation for low temperatures. Derive so called Wien's displacement law. It says that the frequency

ω_{m}, at which the spectral radiation of black body has its maximum is directly proportional to temperature.

• Build a better theory of heat capacity of crystals, to include collective vibrations of atoms. Do not calculate resulting difficult integrals.

Hint: There are sounds waves (longitudinal and transverzal at different speeds) propagating through crystal. Number of modes cannot be bigger than the degree of freedom 3N (N is number of atoms).

• Find a density of states $g(E)$ for free electrons and using this equation find relation between number of electrons and Fermi energy at zero temperature. Find out, what must be the Fermi energy dependence on temperature (for low temperatures) for constant number of electrons. Estimate number of excited electrons at room temperature

Hint: Take inspiration from last parts of our series and the problems accompanying them.

• Calculate dependence of μ versus temperature at low temperatures and constant number of particles in the system o identical bosons. Find the temperature dependence of number of excited bosons at low temperatures.

• Describe qualitatively how the behavior of heat capacity of Ising model with zero external magnetic fields around critical temperature.
• Using similar approach, as we calculated behavior of magnetization $m$ in vicinity of critical point, calculate behavior of susceptibility $\chi (lim_{B\rightarrow0}\partial m&#8260;\partial$ B)$ and dependence of magnetization on magnetic field around critical temperature.
• Show, that the model of lattice gas gives condensation and find critical temperature.
• Investigate the model of binary alloy.