IZhO 2012, Теория — Задача 2. Adventures of a piston

VIII International Zhautykov Olympiad/Theoretical Competition — THEORETICAL COMPETITION, January 17, 2012 · 17 януари 2012 г. · 10 т.

Условие

An open cylindrical vessel of the height $H = 30.0\ \mathrm{cm}$ and cross-sectional area $S = 50.0\ \mathrm{cm^2}$ is filled with the air under the normal conditions, i.e. at the atmospheric pressure $p_0 = 1.01\times 10^5\ \mathrm{Pa}$ and the temperature $T_0 = 273\ \mathrm{K}$. A thin heavy piston of mass $M = 50.0\ \mathrm{kg}$ is gently inserted into the vessel. The vessel wall and the piston are made of the material that conducts heat very poorly. Assume that air is an ideal diatomic gas with an average molar mass $\mu = 29.0\ \mathrm{g/mol}$, the acceleration of gravity is $g = 9.80\ \mathrm{m/s^2}$, and the universal gaseous constant is equal to $R = 8.31\ \mathrm{J/(mol\cdot K)}$. Heat capacities of the piston and the vessel, as well as the friction between the walls and the piston, are to be completely ignored.

The piston is released. The process of transition to final equilibrium consists of two consecutive stages. During the first stage the piston oscillates such that the corresponding gas processes cannot be considered quasistatic. Because of this, the piston oscillations are damped, that is the mechanical energy is dissipated. Assume that a half of the dissipated energy is transferred to the gas under the piston and the other half is given to the atmosphere. At this stage you can also neglect the thermal conductivities of the vessel and the piston. After the oscillations have vanished, the piston stops at some height $H_1$.

The second stage appears to be quite slow, that is during some long period of time the piston achieves its final equilibrium position at some height $H_2$.

2.1 What is the air pressure $p_1$ in the vessel at the end of the first stage? Express your answer in terms of the atmospheric pressure $p_0$, the adiabatic exponent $\gamma$ and the parameter $\alpha = Mg/p_0S$. Calculate the numerical value of $p_1$. [0,5 т.]

2.2 What is the temperature $T_1$ at the end of the first stage? Express your answer in terms of $T_0$, $\gamma$ and $\alpha = Mg/p_0S$. Calculate the numerical value of $T_1$. [1,5 т.]

2.3 Find the height $H_1$. Express your answer in terms of $H$, $\gamma$ and $\alpha = Mg/p_0S$. Calculate the numerical value of $H_1$. [0,5 т.]

2.4 What is the air pressure $p_2$ in the vessel at the end of the second stage? Express your answer in terms of $p_0$ and $\alpha = Mg/p_0S$. Calculate the numerical value of $p_2$. [0,5 т.]

2.5 What is the temperature $T_2$ at the end of the second stage? [0,5 т.]

2.6 Find the height $H_2$. Express your answer in terms of $H$ and $\alpha = Mg/p_0S$. Calculate the numerical value of $H_2$. [0,5 т.]

2.7 Find the frequency $\omega$ of small oscillations around the equilibrium position $H_2$ of the piston, assuming that the process is quasistatic and adiabatic. Express your answer in terms of $g$, $H$, $\gamma$ and $\alpha = Mg/p_0S$. Calculate numerical value of $\omega$. [2 т.]

At the end of the second stage, a great number of small holes are made in the bottom of the vessel whose total cross-sectional area is $S_O = 5.00\times 10^{-4}\ \mathrm{cm^2}$. The size of each hole is much smaller than the mean free flight path of air molecules. After some time the piston starts to move with the constant velocity $u$.

It is known that the average number of molecules $\overline{N}$ hitting the unit of surface area per unit of time is equal to

where $\bar{v} = \sqrt{8RT/\pi\mu}$ is the so-called average thermal velocity of molecules, and $n$ stands for the number density.

The average kinetic energy of translational motion of molecules passing through the holes is

where $k_B$ is the Boltzmann constant.

Assuming that the heat flow through the walls of the vessel and the piston is completely negligible, answer the following questions:

2.8 The steady air pressure under the piston has the form $p_3 = Af(\alpha)$, where $A$ is a constant that depends on $p_0$, while $f(\alpha)$ is a function of $\alpha$. Find $A$ and $f(\alpha)$. Calculate the numerical value of $p_3$. [1 т.]

2.9 The steady velocity of the piston has the from $u = Bg(\alpha)$, where $B$ is a constant depending on $S_O$, $S$, $R$, $T_0$ and $\mu$, while $g(\alpha)$ is a function of $\alpha$. Find $B$ and $g(\alpha)$. Calculate the numerical value of $u$. [2 т.]

2.10 The steady temperature of the gas under the piston has the form $T_3 = Ch(\alpha)$, where $C$ is a constant that depends on $T_0$, while $h(\alpha)$ is a function of $\alpha$. Find $C$ and $h(\alpha)$. Calculate the numerical value of $T_3$. [1 т.]

Отговори

Решение

Оригинал в Архива: IZhO 2012 Theory_eng_f.doc · официални решения: IZhO 2012 Theory_eng_sol_final.doc