IPhO 2025, theory — Задача 1. Champagne!

Theory — International Physics Olympiad FRANCE 2025 — Q3 — Champagne! (10 points) — English (Official) · 17 юли 2025 г. · 10 т.

Условие

Warning: Excessive alcohol consumption is harmful to health and drinking alcohol below legal age is prohibited.

Champagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide (CO$_2$) in the bottle. The molar concentration of CO$_2$ in the liquid phase $c_\ell$ and the partial pressure $P_{\mathrm{CO_2}}$ in the gas phase are related by $c_\ell = k_{\mathrm{H}} P_{\mathrm{CO_2}}$, known as Henry's law and where $k_{\mathrm{H}}$ is called Henry's constant.

Data

• Surface tension of champagne $\sigma = 47 \times 10^{-3}\ \mathrm{J \cdot m^{-2}}$
• Density of the liquid $\rho_\ell = 1.0 \times 10^{3}\ \mathrm{kg \cdot m^{-3}}$
• Henry's constant at $T_0 = 20\,{}^\circ\mathrm{C}$, $k_{\mathrm{H}}(20\,{}^\circ\mathrm{C}) = 3.3 \times 10^{-4}\ \mathrm{mol \cdot m^{-3} \cdot Pa^{-1}}$
• Henry's constant at $T_0 = 6\,{}^\circ\mathrm{C}$, $k_{\mathrm{H}}(6\,{}^\circ\mathrm{C}) = 5.4 \times 10^{-4}\ \mathrm{mol \cdot m^{-3} \cdot Pa^{-1}}$
• Atmospheric pressure $P_0 = 1\ \mathrm{bar} = 1.0 \times 10^{5}\ \mathrm{Pa}$
• Gases are ideal with an adiabatic coefficient $\gamma = 1.3$

Part A. Nucleation, growth and rise of bubbles

Immediately after opening a bottle of champagne at temperature $T_0 = 20\,{}^\circ\mathrm{C}$, we fill a glass. The pressure in the liquid is $P_0$ and its temperature stays constant at $T_0$. The concentration $c_\ell$ of dissolved CO$_2$ exceeds the equilibrium concentration and we study the nucleation of a CO$_2$ bubble. We note $a$ its radius and $P_\mathrm{b}$ its inner pressure.

In the liquid, the concentration of dissolved CO$_2$ depends on the distance to the bubble. At long distance we recover the value $c_\ell$ and we note $c_\mathrm{b}$ the concentration close to the bubble surface. According to Henry's law, $c_\mathrm{b} = k_{\mathrm{H}} P_\mathrm{b}$. We furthermore assume in all the problem that bubbles contain only CO$_2$.

Since $c_\ell \neq c_\mathrm{b}$, CO$_2$ molecules diffuse from areas of high to low concentration. We assume also that any molecule from the liquid phase reaching the bubble surface is transferred to the vapour.

In practice, bubbles mainly grow from pre-existing gas cavities. Consider then a bubble with initial radius $a_0 \approx 40\ \mathrm{\mu m}$. The number of moles of CO$_2$ transferred at the bubble's surface per unit area and time is noted $j$. Two models are possible for $j$.

• model (1) $j = \dfrac{D}{a}(c_\ell - c_\mathrm{b})$ where $D$ is the diffusion coefficient of CO$_2$ in the liquid.
• model (2) $j = K(c_\ell - c_\mathrm{b})$ where $K$ is a constant here.

Experimentally, the bubble radius $a(t)$ is found to depend on time as shown in Fig. 2. Here $c_\ell \approx 4c_0$, and since bubbles are large enough to be visible, the excess pressure due to surface tension can be neglected and $P_\mathrm{b} \approx P_0$.

Eventually bubbles detach from the bottom of the glass and continue to grow while rising. Fig. 3. shows a train of bubbles. The bubbles of the train have the same initial radius and are emitted at a constant frequency $f_\mathrm{b} = 20\ \mathrm{Hz}$.

For the range of velocities studied here, the drag force $F$ on a bubble of radius $a$ moving at velocity $v$ in a liquid of dynamic viscosity $\eta$ is given by Stokes' law $F = 6\pi\eta a v$. Measurements show that at any moment in time, the bubble can be assumed to be travelling at its terminal velocity.

The quasi-stationary growth of bubbles with rate $q_a = \frac{\mathrm{d}a}{\mathrm{d}t}$ still applies during bubble rise.

There are $N_\mathrm{b}$ nucleation sites of bubbles. Assume that the bubbles are nucleated at a constant frequency $f_\mathrm{b}$ at the bottom of a glass of champagne (height $H_\ell$ for a volume $V_\ell$), with $a_0$ still negligible. Neglect diffusion of CO$_2$ at the free surface.

Part B. Acoustic emission of a bursting bubble

Small bubbles are nearly spherical as they reach the free surface. Once the liquid film separating the bubble from the air thins out sufficiently, a circular hole of radius $r$ forms in the film and, driven by surface tension, opens very quickly (Fig. 4. left). The hole opens at constant speed $v_\mathrm{f}$ (Fig. 4. right). The film outside the rim remains still, with constant thickness $h$.

Due to dissipative processes, only half of the difference of the surface energy between $t$ and $t + \mathrm{d}t$ of the rim and the accumulated liquid is transformed into kinetic energy. We further assume that the variation of the surface of the rim is negligible compared to that of the film.

When the film bursts, it releases internal pressure and emits a sound. We model this acoustic emission by a Helmholtz resonator: a cavity open to the atmosphere at $P_0$ through a bottleneck aperture of area $S$ (Fig. 5. left). In the neck, a mass $m_\mathrm{p}$ makes small amplitude position oscillations due to the pressure forces it experiences as the gas in the cavity expands or compresses adiabatically. The gravity force on $m_\mathrm{p}$ is negligible compared to pressure forces. Let $V_0$ be the volume of gas under the mass $m_\mathrm{p}$ for $P = P_0$ as $z = 0$.

The Helmholtz model may be used for a bubble of radius $a$. $V_0$ is the volume of the closed bubble. From litterature, the mass of the equivalent of the piston is $m_p = 8\rho_\mathrm{g} r^3/3$ where $r$ is the radius of the circular aperture and $\rho_\mathrm{g} = 1.8\ \mathrm{kg \cdot m^{-3}}$ is the density of the gas (Fig. 5. right). During the bursting process, $r$ goes from 0 to $r_\mathrm{c}$, given by $r_\mathrm{c} = \dfrac{2}{\sqrt{3}} a^2 \sqrt{\dfrac{\rho_\ell g_0}{\sigma}}$. At the same time, the frequency of emitted sound increases until a maximum value of $40\ \mathrm{kHz}$ and the bursting time is $t_b = 3 \times 10^{-2}\ \mathrm{ms}$.

Part C. Popping champagne

In a bottle, the total quantity of CO$_2$ is $n_\mathrm{T} = 0.2\ \mathrm{mol}$, either dissolved in the volume $V_\mathrm{L} = 750\ \mathrm{mL}$ of liquid champagne, or as a gas in the volume $V_\mathrm{G} = 25\ \mathrm{mL}$ under the cork (Fig. 6. left). $V_\mathrm{G}$ contains only CO$_2$. The equilibrium between both CO$_2$ phases follows Henry's Law. We suppose that the fast gaseous CO$_2$ expansion when the bottle is opened, is adiabatic and reversible. Ambient temperature $T_0$ and pressure $P_0 = 1\ \mathrm{bar}$ are constant.

Another step of champagne production (not described here) leads to the following values of $P_\mathrm{i}$ that we will use for the next questions: $P_\mathrm{i} = 4.69\ \mathrm{bar}$ at $T_0 = 6\,{}^\circ\mathrm{C}$ and $P_\mathrm{i} = 7.45\ \mathrm{bar}$ at $T_0 = 20\,{}^\circ\mathrm{C}$.

During bottle opening, two different phenomena can be observed, depending on $T_0$ (Fig. 6. right).

• either a blue fog appears, due to the formation of solid CO$_2$ crystals (but water condensation is inhibited);
• or a grey-white fog appears, due to water vapor condensation in the air surrounding the bottleneck. In this latter case, there is no formation of CO$_2$ solid crystals.

The saturated vapor pressure $P_\mathrm{sat}^{\mathrm{CO_2}}$ for the CO$_2$ solid/gas transition follows : $\log_{10}\left(\dfrac{P_\mathrm{sat}^{\mathrm{CO_2}}}{P_0}\right) = A - \dfrac{B}{T + C}$ with $T$ in K, $A = 6.81$, $B = 1.30 \times 10^{3}\ \mathrm{K}$ and $C = -3.49\ \mathrm{K}$.

During bottle opening, the cork stopper pops out. We now determine the maximum height $H_\mathrm{c}$ it reaches. Assume that the friction force $F$ due to the bottleneck on the cork stopper is $F = \alpha A$ where $A$ is the area of contact and $\alpha$ is a constant to determine. Initially, the pressure force slightly overcomes the friction force. The cork's mass is $m = 10\ \mathrm{g}$, its diameter $d = 1.8\ \mathrm{cm}$ and the length of the cylindrical part initially stuck in the bottleneck is $\ell_0 = 2.5\ \mathrm{cm}$. Once the cork has left the bottleneck, you can neglect the net pressure force.

A.1 Express the pressure $P_\mathrm{b}$ in terms of $P_0$, $a$ and $\sigma$. [0,2 т.]

A.2 Express the critical radius $a_\mathrm{c}$ above which a bubble is expected to grow in terms of $P_0, \sigma, c_\ell$ and $c_0$ where $c_0 = k_\mathrm{H} P_0$. Calculate numerically $a_\mathrm{c}$ for $c_\ell = 4c_0$. [0,5 т.]

A.3 Express the number of CO$_2$ moles in the bubble $n_\mathrm{c}$ in terms of $a$, $P_0$, $T_0$ and ideal gas constant $R$. Find $a(t)$ for both models. Indicate which model explains the experimental results in Fig. 2. Depending on your answer, calculate numerically $K$ or $D$. [1,2 т.]

A.4 Give the expression of the main forces exerted on a vertically rising bubble. Obtain the expression of $v(a)$. Give a numerical estimate of $\eta$ using $\rho_\ell$, $g_0$ and quantities measured on Fig. 3. [0,8 т.]

A.5 Express the radius $a_{H_\ell}$ of a bubble reaching the free surface in terms of height travelled $H_\ell$, growth rate $q_a = \frac{\mathrm{d}a}{\mathrm{d}t}$, and any constants you may need. Assume $a_{H_\ell} \gg a_0$ and $q_a$ constant, and give the numerical value of $a_{H_\ell}$ with $H_\ell = 10\ \mathrm{cm}$ and $q_a$ corresponding to Fig. 2. [0,5 т.]

A.6 Write the differential equation for $c_\ell(t)$. Obtain from this equation the characteristic time $\tau$ for the decay of the concentration of dissolved CO$_2$ in the liquid. [1,1 т.]

B.1 Express $v_\mathrm{f}$ in terms of $\rho_\ell, \sigma$ and $h$. [1,1 т.]

B.2 Express the frequency of oscillation $f_0$ of $m_\mathrm{p}$. Hint: for $\epsilon \ll 1$, $(1+\epsilon)^{\alpha} \approx 1 + \alpha\epsilon$. [1,1 т.]

B.3 Find the radius $a$ and the thickness $h$ of the champagne film separating the bubble from the atmosphere. [1,1 т.]

C.1 Give the numerical value of the pressure $P_\mathrm{i}$ of gaseous CO$_2$ in the bottle for $T_0 = 6\,{}^\circ\mathrm{C}$ and $T_0 = 20\,{}^\circ\mathrm{C}$. [0,4 т.]

C.2 Give the numerical value $T_\mathrm{f}$ of the CO$_2$ gas at the end of the expansion, after opening a bottle, if $T_0 = 6\,{}^\circ\mathrm{C}$ and if $T_0 = 20\,{}^\circ\mathrm{C}$, if no phase transition occured. Choose which statements are true (several statements possible):

1. At $T_0 = 6\,{}^\circ\mathrm{C}$ a grey-white fog appears while opening the bottle.
2. At $T_0 = 6\,{}^\circ\mathrm{C}$ a blue fog appears while opening the bottle.
3. At $T_0 = 20\,{}^\circ\mathrm{C}$ a grey-white fog appears while opening the bottle.
4. At $T_0 = 20\,{}^\circ\mathrm{C}$ a blue fog appears while opening the bottle. [0,7 т.]

C.3 Give the numerical value of $H_\mathrm{c}$ if the external temperature is $T_0 = 6\,{}^\circ\mathrm{C}$. [1,3 т.]

Решение

Оригинал в Архива: IPhO_2025_Q3.pdf · официални решения: IPhO_2025_S3.pdf