Съдържание

INPhO 2020, Теория — Задача 1. Gas and Carnot cycle

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Всички материали от INPhO 2020

Indian National Physics Olympiad – 2020 · 2 февруари 2020 г. · 13 т.

Внимание: Бележка към темата

The complete retained question booklet includes the summary answer boxes, sixteen blank detailed-answer pages and two blank graph sheets. All matching archived HBCSE key content is retained, including its handwritten graph and comment not for grading purposes. Printed sign/symbol discrepancies and the key cover page-count mismatch remain literal with labelled source notes.

Native cover and instructions

Indian National Physics Olympiad – 2020

Date: 02 February 2020. Time: 09:00–12:00 (3 hours). Maximum Marks: 80.

Roll Number: $2\,0\,\square\,\square-\square\,\square\,\square\,\square-\square\,\square\,\square\,\square$ . Extra sheets attached: [blank box]. INO Centre (e.g. Ranchi): [blank line].

(Do not write below this line)

This booklet consists of 23 pages (excluding this page) and total of 5 questions.
This booklet is divided in two parts: Questions with Summary Answer Sheet and Detailed Answer Sheet. Write roll number at the top wherever asked.
The final answer to each sub-question should be neatly written in the box provided below each sub-question in the Questions & Summary Answer Sheet.
You are also required to show your detailed work for each question in a reasonably neat and coherent way in the Detailed Answer Sheet. You must write the relevant Question Number(s) on each of these pages.
Marks will be awarded on the basis of what you write on both the Summary Answer Sheet and the Detailed Answer Sheet. Simple short answers and plots may be directly entered in the Summary Answer Sheet. Marks may be deducted for absence of detailed work in questions involving longer calculations. Strike out any rough work that you do not want to be considered for evaluation.
Adequate space has been provided in the answersheet for you to write/calculate your answers. In case you need extra space to write, you may request additional blank sheets (maximum two) from the invigilator. Write your roll number on the extra sheets and get them attached to your answersheet and indicate number of extra sheets attached at the top of this page.
Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
Use blue or black pen to write answers. Pencil may be used for diagrams/graphs/sketches.
This entire booklet must be returned at the end of the examination.

Table of Constants

Constant	Symbol	Value
Speed of light in vacuum	$c$	$3.00\times10^8\ \mathrm{m\cdot s^{-1}}$
Planck’s constant	$h$	$6.63\times10^{-34}\ \mathrm{J\cdot s}$
	$\hbar$	$h/2\pi$
Universal constant of Gravitation	$G$	$6.67\times10^{-11}\ \mathrm{N\cdot m^2\cdot kg^{-2}}$
Magnitude of electron charge	$e$	$1.60\times10^{-19}\ \mathrm C$
Rest mass of electron	$m_e$	$9.11\times10^{-31}\ \mathrm{kg}$
Value of $1/4\pi\epsilon_0$		$9.00\times10^9\ \mathrm{N\cdot m^2\cdot C^{-2}}$
Avogadro’s number	$N_A$	$6.022\times10^{23}\ \mathrm{mol^{-1}}$
Acceleration due to gravity	$g$	$9.81\ \mathrm{m\cdot s^{-2}}$
Universal Gas Constant	$R$	$8.31\ \mathrm{J\cdot K^{-1}\cdot mol^{-1}}$
	$R$	$0.0821\ \mathrm{l\cdot atm\cdot mol^{-1}\cdot K^{-1}}$
Permeability constant	$\mu_0$	$4\pi\times10^{-7}\ \mathrm{H\cdot m^{-1}}$

Question	Marks	Score
1	13
2	12
3	15
4	20
5	20
Total	80

HOMI BHABHA CENTRE FOR SCIENCE EDUCATION

Tata Institute of Fundamental Research

V. N. Purav Marg, Mankhurd, Mumbai, 400 088

Native key notice

Please note that alternate/equivalent methods and different way of expressing final solutions may exist.

Source note: The retained key cover says “16 pages (excluding this page)”; the native PDF contains eighteen pages, including its appended handwritten graph. Its cover otherwise repeats the question cover instructions and constants.

Условие

A certain gas obeys the equation of state $U(S,V,N)=aS^7/V^4N^2$ , where $a$ is a dimensioned constant. Here $U$ represents the internal energy of the gas, $S$ the entropy, $V$ the volume and $N$ the fixed number of particles of the system.

Source note: The original writes the equation of state with a solidus as $aS^7/V^4N^2$ ; the key’s displayed derivatives put $V^4N^2$ in the denominator. The original solidus notation remains literal here.

Native answer booklet: PDF pages7–21 and24 are sixteen blank Detailed Answers pages (printed page numbers6–20 and23), with “Que. No.” box and alternating “Last four digits of Roll No.:” field. The complete original remains available; this is its first blank answer-page template.

Complete first blank native detailed-answer sheet with header, question-number box, roll-number field and entire answer frame.

a) Let such a gas be filled in a box of volume $V$ and the internal energy of the system be $U$ . A partition is placed to divide the box into two equal parts, each having volume $V/2$ . For each part, the internal energy is now $\alpha U$ and the dimensioned constant be $\beta a$ . Obtain $\alpha$ and $\beta$ .

Summary answer boxes: $\alpha=$ [blank]; $\beta=$ [blank]. [3 т.]

b) The temperature $T$ can be expressed in terms of the derivative of internal energy as

T=\left(\frac{dU}{dS}\right)_{V,N}

where the subscripts indicate that the differentiation has been carried out keeping $V$ and $N$ constant. In a similar way, express pressure $P$ in terms of a derivative of the internal energy.

Summary answer box: $P=$ [blank]. [2 т.]

c) Find the equation of state of the given system relating $P$ , $T$ , and $V$ .

Summary answer box: $P=$ [blank]. [1 т.]

d) One mole of this gas executes a Carnot cycle ABCDA between reservoirs at temperatures $T_1$ and $T_2$ ( $T_1>T_2$ ). Obtain the heat change in the process AB ( $Q_{\mathrm{AB}}$ ) and work done by the system in the processes AB and BC ( $W_{\mathrm{AB}},W_{\mathrm{BC}}$ ) of the cycle. Express your answers only in terms of temperatures $T_1,T_2$ , volumes $V_A,V_B$ , and the other constants.

Entire native Carnot cycle ABCDA on P–V axes, all arrows, dashed VA/VB lines and labels.

Summary answer boxes: $Q_{\mathrm{AB}}=$ [blank]; $W_{\mathrm{AB}}=$ [blank]; $W_{\mathrm{BC}}=$ [blank].

Detailed answers can be found on page numbers: [blank]. [7 т.]

Решение

(a)

An extensive parameter of the system gets halved if the size of the system is halved, while an intensive parameter remains unchanged. The internal energy and the entropy, both are extensive parameters. Thus $\alpha=1/2$ , $\beta=1$ .

(b)

P=-\left(\frac{dU}{dV}\right)_{S,N}

(c)

From the definition of temperature,

T=\frac{7aS^6}{N^2V^4}

From part (b)

P=\frac{4aS^7}{V^5N^2}

Eliminating $S$ yields

P=C\frac{T^{7/6}}{V^{1/3}}

where $C=\frac{4N^{1/3}}{7^{7/6}a^{1/6}}$ .

(d)

S=\frac74\frac{PV}{T},\quad U=\frac{PV}{4}

Leg AB: $\mathrm A\to\mathrm B$ (isothermal) $\Rightarrow T_1=\text{constant}$

Q_{\mathrm{AB}}=T_1\int_{\mathrm A}^{\mathrm B}dS\qquad (1.1)

=\frac74(P_BV_B-P_AV_A)\qquad (1.2)

Using equation of state,

\boxed{Q_{\mathrm{AB}}=\frac{7C}{4}\left[V_B^{2/3}-V_A^{2/3}\right]T_1^{7/6}}

W_{\mathrm{AB}}=\int_{\mathrm A}^{\mathrm B}P\,dV=c\int_{\mathrm A}^{\mathrm B}V^{-1/3}T_1^{7/6}\,dV\qquad (1.3)

\boxed{W_{\mathrm{AB}}=\frac{3C}{2}T_1^{7/6}\left[V_B^{2/3}-V_A^{2/3}\right]}

Leg BC is isentropic/adiabatic $\Rightarrow Q=0$ or $S=\text{constant}$ . From first law,

W_{\mathrm{BC}}=-\Delta U=\frac{aS^7}{N^2}\left[\frac1{V_B^4}-\frac1{V_C^4}\right]\qquad (1.4)

=\frac a{N^2}\left[\frac{S^7}{V_B^4}-\frac{S^7}{V_C^4}\right]\qquad (1.5)

Also, $S=\frac{7C}{4}V^{2/3}T^{1/6}$

W_{\mathrm{BC}}=\frac a{N^2}\left(\frac{7C}{4}\right)^7\left[V_B^{2/3}T_1^{7/6}-V_C^{2/3}T_2^{7/6}\right]\qquad (1.6)

\boxed{W_{\mathrm{BC}}=\frac a{N^2}\left[\frac{7C}{4}\right]^7V_B^{2/3}T_1^{1/6}[T_1-T_2]}

Source note: Equation(1.3) prints lowercase $c$ while the surrounding equation-of-state coefficient is uppercase $C$ . This printed symbol remains literal.

Оригинал в Архива: INPHO2020-Questions-en.pdf · решение от архива: INPHO2020-Solutions-20200220.pdf