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Chapter 10: Problem 8

You and your friend are each drinking cola from separate \(2-\mathrm{L}\) bottles. Both colas are equally carbonated. You are able to drink 1 L of cola, but your friend can drink only about half a liter. You each close the bottles and place them in the refrigerator. The next day when you each go to get the colas, whose will be more carbonated and why?

Short Answer

Expert verified
The second person's cola will be more carbonated the next day because it has less headspace (0.5 L) compared to the first person’s cola (1 L). Less headspace means a higher carbonation level is maintained, as the cola releases less CO2 gas into the headspace. This causes CO2 gas to dissolve back into the cola more efficiently, keeping the second person's cola more carbonated than the first person's cola.

Step by step solution

01

Determine the initial conditions

Both colas started as equally carbonated 2-L bottles. The first person drinks 1 L, leaving 1 L of cola left in their bottle. The second person drinks only 0.5 L, leaving 1.5 L of cola in their bottle.
02

Understand how carbonation is maintained in the closed bottles

Carbonation is regulated by the equilibrium between CO2 gas dissolved in the liquid and the CO2 gas in the air above the liquid trapped in the bottle. When the bottle is closed, and the amount of CO2 gas in the headspace (the empty part of the bottle) is in equilibrium with the amount of carbonation dissolved in the cola, the carbonation levels will stay constant. In each bottle, as the liquid starts losing carbon dioxide, some of the CO2 in the headspace will dissolve back into the liquid to maintain equilibrium.
03

Compare the headspace of the two bottles

The more headspace in a bottle, the greater the amount of CO2 gas that was released when the liquid was consumed. More CO2 gas will be present in the headspace when there is more liquid consumed in a bottle, as the amount of CO2 released will be proportional to the amount of liquid consumed. In this case, the first person's bottle has more headspace. It has 1 L of headspace due to drinking 1 L of cola, while the second person's bottle has 0.5 L of headspace due to drinking 0.5 L of cola.
04

Determine the effect of headspace on carbonation the next day

As the headspace increases, the amount of CO2 gas released from the cola will also increase. It will take longer for equilibrium to be established; however, once the equilibrium is achieved, the carbonation levels in the two colas will be the same. As both colas were initially at the same carbonation level, with the same headspace equilibrium, the cola with less headspace will have more CO2 gas already dissolved in the cola, which will maintain the cola's carbonation to a higher degree. The less headspace will also mean there is less CO2 gas in the air to dissolve back into the liquid.
05

Conclude whose cola will be more carbonated

The second person's cola will be more carbonated the next day because it has less headspace (0.5 L) compared to the first person’s cola (1 L). Less headspace means a higher carbonation level is maintained, as the cola releases less CO2 gas into the headspace. This causes CO2 gas to dissolve back into the cola more efficiently, keeping the second person's cola more carbonated than the first person's cola.

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