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USAPhO 2022, Полуфинал — Задача 4. Question B1

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Всички материали от USAPhO 2022

USA Physics Olympiad Exam — Part A / Part B · 25 т.

Внимание: Бележка към темата

Both original question and worked-key packets have been read in full. They refer to a separate answer-sheet packet, including a blank plotting grid, which is not present in the retained question/key sources or the matching catalogue entry. All native question data and the actual key graph are retained. Printed variable/sign/wording defects and grading allowances remain source faithful and are documented.

Important Instructions for the Exam Supervisor — DO NOT DISTRIBUTE THIS PAGE

This examination has two parts. Part A has three questions and lasts for 90 minutes. Part B also has three questions and lasts for 90 minutes.
For each student, print out one copy of the exam and one copy of the answer sheets. Print everything single-sided, and do not staple anything. Divide the exam into the instructions (pages 2–3), Part A questions (pages 4–7), and Part B questions (pages 8–10).
Begin by giving students the instructions and all of the answer sheets. Let the students read the instructions and fill out their information on the answer sheets. They can keep the instructions for both parts of the exam. Also give students blank sheets of paper to use as scratch paper throughout the exam.
Students may bring calculators, but they may not use symbolic math, programming, or graphing features of these calculators. Calculators may not be shared, and their memory must be cleared of data and programs. Cell phones or other electronics may not be used during the exam or while the exam papers are present. Students may not use books or other references.
To start the exam, give students the Part A questions, and allow 90 minutes to complete Part A. Do not give students Part B during this time, even if they finish with time remaining. At the end of the 90 minutes, collect the Part A questions and answer sheets.
Then give students a 5 to 10 minute break. Then give them the Part B questions, and allow 90 minutes to complete Part B. Do not let students go back to Part A.
At the end of the exam, collect everything, including the questions, the instructions, the answer sheets, and the scratch paper. Students may not keep the exam questions, but they may be returned to the students after April 19th, 2022.
After the exam, sort each student’s answer sheets by page number. Scan every answer sheet, including blank ones.

We acknowledge the following people for their contributions to this year’s exam (in alphabetical order):

Tengiz Bibilashvilli, Abijith Krishnan, Brian Skinner, and Kevin Zhou.

Instructions for the Student

You should receive these instructions, the table on the next page, answer sheets, and blank paper for scratch work. Read this page carefully before the exam begins.
You may use a calculator, but its memory must be cleared of data and programs, and you may not use symbolic math, programming, or graphing features. Calculators may not be shared. Cell phones or other electronics may not be used during the exam or while the exam papers are present. You may not use books or other outside references.
When the exam begins, your proctor will give you the questions for Part A. You will have 90 minutes to complete three problems. Each question is worth 25 points, but they are not necessarily of the same difficulty. If you finish all of the questions, you may check your work, but you may not look at Part B during this time.
After 90 minutes, your proctor will collect the questions and answer sheets for Part A. You may then take a short break.
Then you will work on Part B. You have 90 minutes to complete three problems. Each question is worth 25 points. Do not look at Part A during this time. When the exam ends, you must return all papers to the proctor, including the exam questions.
Do not discuss the questions of this exam, or their solutions, until after April 19th, 2022. Violations of this rule may result in disqualification. Below are instructions for writing your solutions.
This year, all of your solutions will be written on official answer sheets. Nothing outside these answer sheets will be graded. Before the exam begins, write your name, student AAPT number, and proctor AAPT number as directed on the answer sheets.
There are several answer sheets per problem. If you run out of space for a problem, you may use the extra answer sheets, which are at the end of the answer sheet packet. To ensure this work is graded, you must indicate, at the bottom of your last answer sheet for that problem, that you are using these extra answer sheets.
Only write within the frame of each answer sheet. To simplify grading, we recommend drawing a box around your final answer for each subpart. You should organize your work linearly and briefly explain your reasoning, which will help you earn partial credit. You may use either pencil or pen, but sure to write clearly so your work will be legible after scanning.
You may have to fit a line to data, which you can plot on the answer sheet with grid lines. You may use a ruler, pencil, pen, or piece of paper as a straightedge.

Reference table of possibly useful information

$g=9.8\ \mathrm{N/kg}$	$G=6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
$k=1/4\pi\epsilon_0=8.99\times10^9\ \mathrm{N\cdot m^2/C^2}$	$k_m=\mu_0/4\pi=10^{-7}\ \mathrm{T\cdot m/A}$
$c=3.00\times10^8\ \mathrm{m/s}$	$k_B=1.38\times10^{-23}\ \mathrm{J/K}$
$N_A=6.02\times10^{23}\ (\mathrm{mol})^{-1}$	$R=N_Ak_B=8.31\ \mathrm{J/(mol\cdot K)}$
$\sigma=5.67\times10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}$	$e=1.602\times10^{-19}\ \mathrm C$
$1\ \mathrm{eV}=1.602\times10^{-19}\ \mathrm J$	$h=6.63\times10^{-34}\ \mathrm{J\cdot s}=4.14\times10^{-15}\ \mathrm{eV\cdot s}$
$m_e=9.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2$	$(1+x)^n\approx1+nx$ for $\lvert x\rvert\ll1$
$\sin\theta\approx\theta-\theta^3/6$ for $\lvert\theta\rvert\ll1$	$\cos\theta\approx1-\theta^2/2$ for $\lvert\theta\rvert\ll1$

You may use this sheet for both parts of the exam.

End of Instructions for the Student

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Important Instructions for the Exam Supervisor — Key Packet Pagination

For each student, print out one copy of the exam and one copy of the answer sheets. Print everything single-sided, and do not staple anything. Divide the exam into the instructions (pages 2–3), Part A questions (pages 4–13), and Part B questions (pages 14–22).

Условие

Part B

Virial Reality

The ideal gas law states that $PV_m=RT$ , where $V_m=V/n$ is the volume per mole of gas. However, any real gas will exhibit deviations from the ideal gas law, described by the virial expansion,

PV_m=RT\left(1+\frac{B(T)}{V_m}+\frac{C(T)}{V_m^2}+\cdots\right).

For gases with low density, the higher-order terms are negligible, so in this problem we will neglect all of the temperature-dependent terms in parentheses except for $B(T)/V_m$ . The table below shows measurements of $B$ for nitrogen gas ( $N_2$ ) at atmospheric pressure, $P=1.01\times10^5\ \mathrm{Pa}$ .

$T$ (K)	$B$ ( $\mathrm{cm^3/mol}$ )
100	$-160$
200	$-35$
300	$-4.2$
400	9.0
500	16.9
600	21.3

a. According to the ideal gas law, what is the value of $V_m$ at temperatures 100 K, 300 K, and 600 K? Give your answers in SI units.

b. What is the percentage change in $V_m$ at these temperatures if one accounts for $B(T)$ ?

c. In 1910, van der Waals was awarded the Nobel Prize for formulating the equation

\left(P+\frac a{V_m^2}\right)(V_m-b)=RT

which accurately describes many real gases. According to this equation, what is the form of $B(T)$ ? You may assume that $b\ll V_m$ .

d. Using the data above, extract the values of $a$ and $b$ . Give your answers in SI units.

e. In this problem, we have neglected terms in the virial expansion beyond $B(T)$ , which is a good approximation as long as the volume correction due to $B(T)$ itself is small. Assuming the van der Waals equation holds, numerically estimate the temperature range within which the volume correction due to $B(T)$ is at most 10%, for nitrogen gas at atmospheric pressure.

Решение

a.

Plugging in the values and converting to $\mathrm{m^3/mol}$ gives

V_m=\frac{RT}P=\begin{cases}8.23\times10^{-3}\ \mathrm{m^3/mol}&T=100\ \mathrm K\\2.47\times10^{-2}\ \mathrm{m^3/mol}&T=300\ \mathrm K\\4.94\times10^{-2}\ \mathrm{m^3/mol}&T=600\ \mathrm K\end{cases}

b.

The fractional change in volume is

\frac{\Delta V_m}{V_m}\approx\frac{B(T)}{V_m}\approx\begin{cases}-1.9\%&T=100\ \mathrm K\\-0.02\%&T=300\ \mathrm K\\0.04\%&T=600\ \mathrm K.\end{cases}

c.

We have

P+\frac a{V_m^2}=\frac{RT}{V_m-b}\approx\frac{RT}{V_m}\left(1+\frac b{V_m}\right).

Rearranging yields

PV_m\approx RT\left(1+\frac b{V_m}-\frac a{V_mRT}\right)

from which we conclude

B(T)=b-\frac a{RT}.

d.

First, we convert the data points to SI units, giving

$T$ (K)	$B$ ( $\mathrm{m^3/mol}$ )
100	$-1.6\times10^{-4}$
200	$-3.5\times10^{-5}$
300	$-4.2\times10^{-6}$
400	$9.0\times10^{-6}$
500	$1.69\times10^{-5}$
600	$2.13\times10^{-5}$

Now, plotting $B$ versus $1/T$ , the slope is $-a/R$ and the $y$ -intercept is $b$ .

Complete original blue scatter/linear-fit graph B vs1/T: all six points, dotted blue fit, axes/ticks/grid, native equation y=-0.0219x+6E-05 and R²=0.9913; original axes have no separate unit labels.

The final result is

a=0.18\ \mathrm{J\,m^3/mol^2},\qquad b=6\times10^{-5}\ \mathrm{m^3/mol}.

Any answer within 20% of this is acceptable.

e.

We are looking for the temperature range where

|B(T)|\lesssim\frac{V_m}{10}\approx\frac{RT}{10P}.

At high temperatures, $B(T)$ always yields a small correction for nitrogen at atmospheric pressure. At low temperatures, $B(T)$ is dominated by the $-a/RT$ term, so we require

T\gtrsim\frac{\sqrt{10Pa}}R=50\ \mathrm K.

Any answer within 25% of this is acceptable.

Оригинал в Архива: 2022-USAPhO-Exam.pdf · официални решения: 2022-USAPhO-solutions.pdf