Съдържание

IAO 2007, Теория, Група β, английски — Задача 3. Alcohol in Universe.

Оригинал на условието (изображение)Съобщи за грешка

Всички материали от IAO 2007

Theoretical round. Problems to solve · 29 септември 2007 г.

Бележка

Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may have no real sense. In this case you have to write in your answer (in English or Russian): «situation is impossible – ситуация невозможна». Of course, this answer has to be explained numerically or logically.

Source availability

The numerical table of planetary data is not present in the32-file2007 archive inventory. The file bgtheo6.jpg contains only a translation of column headings, without numerical planetary rows. No table from another year has been substituted. The separately archived airplane/Moon photograph is included in full with question4.

Note for jury and team leaders

The proposed sketches are not full; the team leaders have to give more detailed explanations for students. But the correct solutions in the students’ papers (enough for8pts) may be shorter.

Source photograph damage

The second and fourth official solution photographs lose some initial letters or words at the left edge. Ellipses inside the affected solution text mark those source losses. Legible formulas, results and grading remarks are retained; missing letters have not been silently supplied. The solution for question4 contains no explanatory drawing, although the question requests one.

Printed source annotations

The galaxy key prints the same final logarithm1+HL₂/c in its classical and relativistic final lines, despite different numerical totals22.46 and22.82. The sail key repeats Δp and uses the printed multiplication/division notation. The source results and formulas are reproduced without scientific correction; punctuation and multiplication spacing are normalized only for mathematical rendering.

Условие

Astronomers based at Britain's Jodrell Bank Observatory say they have spotted a cloud of alcohol in deep space that measures 288 billion miles (463 billion kilometres) across, a finding that could shed light on how giant stars are formed from primordial gas. Some nations even decided to send expeditions to this area to taste the cloud but... enthusiasm disappeared after learning that the cloud is so remote that it will take us many millions years to reach it on the modern techniques. The density of molecules in the cloud is large in comparison with the interstellar gas density but very small from our custom point of view, only about 10 atoms/mm³. Estimate the temperature such a cloud should have to be stable and not get dispersed in future (before the international expedition will reach the cloud). Suppose the alcohol is ethyl (C₂H₅OH).

Решение

The problem, of course, estimational. The result may give us only very approximate value. The cloud will be stable and not dissolve in the case the average speed of molecules of alcohol, atomic weight of C₂H₅OH is $m=2\cdot12+6\cdot1+1\cdot16=46$ , so molar weight $\mu=0.046$ kg/mol. $R$ is universal gas constant and $T$ is absolute temperature (i.e. in Kelvins),

V_{\mathrm{av}}=(3RT/\mu)^{1/2}

not exceeds the II cosmic speed (the parabolic speed in other words) for the bodies at the edges of the cloud. $M$ is mass of the cloud and $r$ is its radius (half of mentioned 463 billion kilometres),

V_{\mathrm{II}}=(2GM/r)^{1/2}.

3RT/\mu<2GM/r.

M=\tfrac43\pi r^3\rho=\tfrac43\pi r^3nm=\tfrac43\pi r^3n\mu/N_A,

where n is concentration of molecules (for approximation it may be used value $n=10$ atoms/mm³ $=10^{10}$ m⁻³ or, that is more correct, this value should be deleted by the number of atoms in alcohol molecule, i.e. 9, and $n=1.1\cdot10^9$ m⁻³) and $N_A=6.022\cdot10^{23}$ is Avogadro number.

3RT/\mu<\tfrac83G\pi r^2n\mu/N_A.

T<\tfrac{8\pi}{9}Gr^2n\mu^2/RN_A.

Calculations of $\tfrac{8\pi}{9}Gr^2n\mu^2/RN_A$ give us 42.2 K for n = 10 molecules/mm³ and 4.7 K for 10 atoms/mm³, but since it is only rough approximation one cannot use an accuracy of more than 1 digit, i.e. 40 or 5 Kelvins (or, according to past standards “40 degrees of Kelvin” or “5 degrees of Kelvin”).

Note for jury: It is proposed that the students have to know by hard the fundamental constants as $G$ , $R$ or $N_A$ and also the atomic weights of H, C and O since these constants and these 3 elements are fundamental.

Data

Data from the table of planetary data may be used for solving every problem.

Оригинал в Архива: betatheo.jpg · официални решения: theosol1.jpg · допълнителни данни: theosol2.jpg · допълнителни данни: theosol3.jpg · допълнителни данни: theosol4.jpg · допълнителни данни: photo.JPG