Material preventing nucleation, why is it not used for soda container?
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Two possible answers so far that I thought about: Either it is too hard to make such a material or polish it "near perfectly", or it would not be worth it. By "worth it" I mean the motivation below would be false.
The motivation to use such materials would be that bubbles of CO2CO2 would not form at all, because if I'm not wrong bubbles are forming on the walls of the bottle or glass thanks to heterogeneous nucleation due to microscopic cracks or "imperfections" (i.e. the crystal isn't plane. In a common glass the atoms aren't ordered like in a crystal and this favors heterogeneous nucleation) while homogeneous nucleation never occurs because the radius of the bubble required for it to occur is too big and so the probability that it's created spontaneously is almost nil. Therefore the bottle or glass would, I believe, only slowly lose CO2CO2 gas through the interface liquid/air which is due to diffusion and occurs because the chemical potential "μμ" of the soda is higher than the chemical potential of the air. So the process will end when there is no more CO2CO2 in the soda, if I assume that there's no CO2CO2 in the air which is a good approximation.
To sum up the motivation: no bubbles formed. The loss of CO2CO2 would be very slow and so we could drink soda with plenty of "gas" even if the bottle has been opened for a long while compared to what we're currently used to.
Now I would like to use some maths to show how much slower the rate of decrease of CO2CO2 would be if we were to use such bottle or glass, compared to a normal bottle or glass.
More precisely: let c(t)c(t) be the concentration of CO2CO2 in function of time. To settle numbers I'll assume that when c(t)=0.1⋅c(0)c(t)=0.1⋅c(0) then there is too few "gas" for the soda to be drank. I want to estimate by calculations via a model how much time it takes until this low concentration threshold is reached in both cases.
Let's assume the bottle or glass is a cylinder of 20 cm height, 3.5 cm radius. This means the area of the interface soda/air is Ainterface soda/air≈38.5 cm2Ainterface soda/air≈38.5 cm2.
Model 1 (no heterogeneous nucleation occurs):
In this case we only have a diffusion equation for c(t):dc(t)dt=−rc(t)c(t):dc(t)dt=−rc(t) where rr is a positive constant proportional to Ainterface soda/airAinterface soda/air, yielding a solution of the form c(t)=c(0)e−rtc(t)=c(0)e−rt. I can now solve for tctc so that c(tc)=0.1⋅c(0)c(tc)=0.1⋅c(0).
Model 2 (heterogeneous nucleation occurs):
In this case I have the same diffusion equation for c(t) except that I have a new term. I am unsure how to write it. I'll assume that there are 2 bubbles per cm2cm2. The total area of the container is Abottle=2πR⋅20 cm+Ainterface soda/air≈478 cm2Abottle=2πR⋅20 cm+Ainterface soda/air≈478 cm2, so there are about 957 bubbles in total.
Again to simplify things I'll assume that a bubble is most of its existing time stuck on the glass rather than going up in the soda. So that nucleation only occurs to bubbles growing on the glass. I'll also assume that all bubbles start with a zero radius and all have the same maximum radius, say 0.1 cm 0.1 cm before they detach from the wall and get replaced instantly by a 0 cm0 cm radius bubble.
Now I believe the bubbles growth rate depend on the current value of c(t)c(t), but I am not sure whether it is the rate of change of the volume or area or radius that depends linearly on c(t)c(t). I'd appreciate a comment here. So that I can set up the differential equation for c(t)c(t), solve it and compare it with the first model.
The motivation to use such materials would be that bubbles of CO2CO2 would not form at all, because if I'm not wrong bubbles are forming on the walls of the bottle or glass thanks to heterogeneous nucleation due to microscopic cracks or "imperfections" (i.e. the crystal isn't plane. In a common glass the atoms aren't ordered like in a crystal and this favors heterogeneous nucleation) while homogeneous nucleation never occurs because the radius of the bubble required for it to occur is too big and so the probability that it's created spontaneously is almost nil. Therefore the bottle or glass would, I believe, only slowly lose CO2CO2 gas through the interface liquid/air which is due to diffusion and occurs because the chemical potential "μμ" of the soda is higher than the chemical potential of the air. So the process will end when there is no more CO2CO2 in the soda, if I assume that there's no CO2CO2 in the air which is a good approximation.
To sum up the motivation: no bubbles formed. The loss of CO2CO2 would be very slow and so we could drink soda with plenty of "gas" even if the bottle has been opened for a long while compared to what we're currently used to.
Now I would like to use some maths to show how much slower the rate of decrease of CO2CO2 would be if we were to use such bottle or glass, compared to a normal bottle or glass.
More precisely: let c(t)c(t) be the concentration of CO2CO2 in function of time. To settle numbers I'll assume that when c(t)=0.1⋅c(0)c(t)=0.1⋅c(0) then there is too few "gas" for the soda to be drank. I want to estimate by calculations via a model how much time it takes until this low concentration threshold is reached in both cases.
Let's assume the bottle or glass is a cylinder of 20 cm height, 3.5 cm radius. This means the area of the interface soda/air is Ainterface soda/air≈38.5 cm2Ainterface soda/air≈38.5 cm2.
Model 1 (no heterogeneous nucleation occurs):
In this case we only have a diffusion equation for c(t):dc(t)dt=−rc(t)c(t):dc(t)dt=−rc(t) where rr is a positive constant proportional to Ainterface soda/airAinterface soda/air, yielding a solution of the form c(t)=c(0)e−rtc(t)=c(0)e−rt. I can now solve for tctc so that c(tc)=0.1⋅c(0)c(tc)=0.1⋅c(0).
Model 2 (heterogeneous nucleation occurs):
In this case I have the same diffusion equation for c(t) except that I have a new term. I am unsure how to write it. I'll assume that there are 2 bubbles per cm2cm2. The total area of the container is Abottle=2πR⋅20 cm+Ainterface soda/air≈478 cm2Abottle=2πR⋅20 cm+Ainterface soda/air≈478 cm2, so there are about 957 bubbles in total.
Again to simplify things I'll assume that a bubble is most of its existing time stuck on the glass rather than going up in the soda. So that nucleation only occurs to bubbles growing on the glass. I'll also assume that all bubbles start with a zero radius and all have the same maximum radius, say 0.1 cm 0.1 cm before they detach from the wall and get replaced instantly by a 0 cm0 cm radius bubble.
Now I believe the bubbles growth rate depend on the current value of c(t)c(t), but I am not sure whether it is the rate of change of the volume or area or radius that depends linearly on c(t)c(t). I'd appreciate a comment here. So that I can set up the differential equation for c(t)c(t), solve it and compare it with the first model.
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Handling Caustic Soda side the boots. DO NOT tuck in.
• Wear chemical resistant clothing for protection of the body. Impregnated vinyl or rubber suits.
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