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INPhO 2023, Теория — Задача 3. Work in progress

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Indian National Physics Olympiad (INPhO)-2023 — Question Paper · 29 януари 2023 г. · 16 т.

Указания

Date: 29 January 2023

Time: 09:00-12:00 (3 hours) Maximum Marks: 60

Instructions Roll No.:

This booklet consists of 6 pages and total of 5 questions. Write roll number at the top wherever asked.
Booklet to write the answers is provided separately. Instructions to write the answers are on the Answer Booklet.
Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
Please submit the Answer Sheet at the end of the examination. You may retain the Question Paper.

Table of Constants

Speed of light in vacuum	$c$	$3.00 \times 10^{8}\ \mathrm{m{\cdot}s^{-1}}$
Planck’s constant	$h$	$6.63 \times 10^{-34}\ \mathrm{J{\cdot}s}$
	$\hbar$	$h/2\pi$
Universal constant of Gravitation	$G$	$6.67 \times 10^{-11}\ \mathrm{N{\cdot}m^2{\cdot}kg^{-2}}$
Magnitude of electron charge	$e$	$1.60 \times 10^{-19}\ \mathrm{C}$
Rest mass of electron	$m_e$	$9.11 \times 10^{-31}\ \mathrm{kg}$
Value of $1/4\pi\epsilon_0$		$9.00 \times 10^{9}\ \mathrm{N{\cdot}m^2{\cdot}C^{-2}}$
Avogadro’s number	$N_A$	$6.022 \times 10^{23}\ \mathrm{mol^{-1}}$
Acceleration due to gravity	$g$	$9.81\ \mathrm{m{\cdot}s^{-2}}$
Universal Gas Constant	$R$	$8.31\ \mathrm{J{\cdot}K^{-1}{\cdot}mol^{-1}}$
	$R$	$0.0821\ \mathrm{l{\cdot}atm{\cdot}mol^{-1}{\cdot}K^{-1}}$
Boltzmann constant	$K_B$	$1.3806 \times 10^{-23}\ \mathrm{J{\cdot}K^{-1}}$
Permeability constant	$\mu_0$	$4\pi \times 10^{-7}\ \mathrm{H{\cdot}m^{-1}}$
1 Angstrom unit	1Å	$1 \times 10^{-10}\ \mathrm{m}$
1 micro unit	$1\mu$	$1 \times 10^{-6}$ units
1 electron volt	1 eV	$1.6 \times 10^{-19}\ \mathrm{J}$

Q No	1	2	3	4	5	Total
Maximum Marks	6	6	16	16	16	60

Указания

Date: 29 January 2023

Time: 09:00-12:00 (3 hours) Maximum Marks: 60

Instructions Roll No.:

This booklet consists of 19 pages and total of 5 questions. Write roll number at the top wherever asked.
Booklet to write the answers is provided separately. Instructions to write the answers are on the Answer Booklet.
Non-programmable scientific calculators are allowed. Mobile phones cannot be used as calculators.
Please submit the Answer Sheet at the end of the examination. You may retain the Question Paper.

Table of Constants

Speed of light in vacuum	$c$	$3.00 \times 10^{8}\ \mathrm{m{\cdot}s^{-1}}$
Planck’s constant	$h$	$6.63 \times 10^{-34}\ \mathrm{J{\cdot}s}$
	$\hbar$	$h/2\pi$
Universal constant of Gravitation	$G$	$6.67 \times 10^{-11}\ \mathrm{N{\cdot}m^2{\cdot}kg^{-2}}$
Magnitude of electron charge	$e$	$1.60 \times 10^{-19}\ \mathrm{C}$
Rest mass of electron	$m_e$	$9.11 \times 10^{-31}\ \mathrm{kg}$
Value of $1/4\pi\epsilon_0$		$9.00 \times 10^{9}\ \mathrm{N{\cdot}m^2{\cdot}C^{-2}}$
Avogadro’s number	$N_A$	$6.022 \times 10^{23}\ \mathrm{mol^{-1}}$
Acceleration due to gravity	$g$	$9.81\ \mathrm{m{\cdot}s^{-2}}$
Universal Gas Constant	$R$	$8.31\ \mathrm{J{\cdot}K^{-1}{\cdot}mol^{-1}}$
	$R$	$0.0821\ \mathrm{l{\cdot}atm{\cdot}mol^{-1}{\cdot}K^{-1}}$
Boltzmann constant	$K_B$	$1.3806 \times 10^{-23}\ \mathrm{J{\cdot}K^{-1}}$
Permeability constant	$\mu_0$	$4\pi \times 10^{-7}\ \mathrm{H{\cdot}m^{-1}}$
1 Angstrom unit	1Å	$1 \times 10^{-10}\ \mathrm{m}$
1 micro unit	$1\mu$	$1 \times 10^{-6}$ units
1 electron volt	1 eV	$1.6 \times 10^{-19}\ \mathrm{J}$

Q No	1	2	3	4	5	Total
Maximum Marks	6	6	16	16	16	60

Please note that alternate/equivalent methods and different way of expressing final solutions may exist. A correct method will be suitably awarded.

Условие

One mole of an ideal monoatomic gas goes through a linear process from A to B as shown in the pressure-volume ( $P$ - $V$ ) diagram. The temperature at A is $T_{\mathrm{A}} = 227^{\circ}\mathrm{C}$ . The process is such that, the temperature decreases and the heat is continuously supplied to the gas. The ratio of the specific heat at the constant pressure to that at the constant volume is 5/3. Obtain the expression for the maximum work ( $W_{\max}$ ) the gas can perform in such a process. Calculate $W_{\max}$ .

P–V diagram: vertical axis P and horizontal axis V with arrows; a straight segment descends from point A (upper left) to point B (lower right).

Отговори

Решение

Solution: The variation of $P$ is linear with respect to $V$ , hence it can be written as

P = -aV + b \qquad (3.1)

where $a$ and $b$ are positive constants. At A,

P_A V_A = R T_A \qquad (3.2)

(-aV_A + b)V_A = R \times 500 \qquad (3.3)

where $T_A = 500\,\mathrm{K}$ . Using the ideal gas equation $PV = RT$

T = \frac{PV}{R} = \frac{-aV^2 + bV}{R} \qquad (3.4)

\frac{dT}{dV} = \frac{-2aV + b}{R} \qquad (3.5)

In this process, $V$ is increasing but the temperature is decreasing, hence

\frac{dT}{dV} \leq 0 \qquad (3.6)

Using Eq. (3.5)

\frac{-2aV + b}{R} \leq 0 \qquad (3.7)

\Longrightarrow V \geq \frac{b}{2a} \qquad (3.8)

This is the lower bound on the volume. This means if we want work done to be maximum

V_{\min} = V_A = \frac{b}{2a} \qquad (3.9)

Using the first law of thermodynamics $dQ = dU + PdV$ ,

dQ = \frac{R}{\gamma - 1} dT + PdV \qquad (3.10)

where we use $dU = C_V dT$ and Eq.(3.2), Eq. (3.5), and $\gamma = 5/3$ in the above equation yields

dQ = \frac{3R(-2aV + b)dV}{2R} + (-aV + b)dV \qquad (3.11)

\frac{dQ}{dV} = \left(-4aV + \frac{5b}{2}\right) \qquad (3.12)

In the process, heat is taken and volume is also increasing. hence

\frac{dQ}{dV} \geq 0 \qquad (3.13)

-4aV + \frac{5b}{2} \geq 0 \qquad (3.14)

\Longrightarrow V \leq \frac{5b}{8a} \qquad (3.15)

This is the upper bound on the volume. This means if we want work done to be maximum

V_{\max} = V_{\mathrm{B}} = \frac{5b}{8a} \qquad (3.16)

To get maximum work, The gas must expand from $V_A$ to $V_B$

W_{\max} = \int_{V_A}^{V_B} PdV = \int_{V_{\mathrm{A}}}^{V_{\mathrm{B}}} (-aV + b)dV \qquad (3.17)

= \left[\frac{-aV^2}{2} + bV\right]_{b/2a}^{5b/8a} \qquad (3.18)

Substituting the limits, we get,

W_{\max} = \frac{7}{128} \frac{b^2}{a} \qquad (3.19)

Solving $(-aV_A + b)V_A = 500R$ , we get $\frac{b^2}{a} = 500 \times 4R$ . Substituting this in the above equation, we get,

W_{\max} \approx 909\,\mathrm{J} \qquad (3.20)

Оригинал в Архива: INPhO2023-Question.pdf · официални решения: INPhO2023-Solution-20230215.pdf