Cause of change of tea volume during cooling

At work I often drink tea, and i usually fill my cup to the brim with boiling water. When I let it cool though, until it's ready to drink, the liquids surface has sunken by about a fingers width.

To give further information the cup is cylindrical with about 12 cm height and 3,5 cm radius (a pretty big cup). It's made of ceramic and ambient temperature is about 23 °C. I usually check it when it's drinking temperature so, about 55°C I'd say, but could be lower on some days. It's also indoors and we do have air conditioning. The cup was always open on top.

What influences this more, cooling of the liquid, evaporation, expansion of the cup or some other effect?

cheers!

• The coefficient of expansion varies with temperature. Water contracts about 4% when cooling from 99 to 20 centigrade. – Bert Barrois Feb 7 '19 at 12:29
• The volumetric coefficient of expansion of water is about $2 \times10 ^{-4}$ per degree C. The coefficient for the cup is probably at least a factor of 10 times smaller. – alephzero Feb 7 '19 at 12:30
• That's a lot of shrinkage, and there are a lot of relevant factors that you haven't told us about. How long is the delay? How big is the cup, and what's it's shape? I.e., is it tall & narrow, or short & wide? What's it made of? Is it ceramic or metal? What's the ambient temperature? Is this outdoors or indoors, & if indoors, do you have air conditioning? Have you tried covering the cup with an airtight lid to prevent evaporation? – PM 2Ring Feb 7 '19 at 12:41
• @PM2Ring to give further information the cup is cylindrical with about 12 cm height and 3,5 cm radius (a pretty big cup). It's made of ceramic and ambient temperature is about 23 °C. I usually check it when it's drinking temperature so, about 55°C I'd say, but could be lower on some days. It's also indoors and we do have air conditioning. The cup was always open on top. – Radioo Feb 7 '19 at 13:03
• I think evaporation is the predominant factor. You can test it by weighting the cup right after you poured hot water and keep weighting it in regular time intervals. If there's a mass difference you may estimate how much volume was lost. – ErickShock Feb 7 '19 at 13:12

I would say this should almost definitely be caused by the evaporation, not thermal expansion/density changes.

The cup itself should decrease it's internal volume as it cools (raising the tea level), but you would need to consider at what times it has what temperatures. It may be possible that the lagged thermal expansion of the cup when hot water is added actually does increase the volume of the cup as the tea is heating the water. To test if this is significant (which I doubt it is, given that the temperature changes aren't extreme), you could always heat up the cup first, removing the thermal lag. This would allow you to eliminate this as the cause of the height dropping.

The thermal properties of water can be analyzed and we can compare how much volume change that we see to the expected volume change of the temperature difference:

At $$100°C$$, the specific volume is $$~1.0434 \ \frac {cm^3}{g}$$, while at $$55°C$$, the specific volume is $$~ 1.0145 \ \frac{cm_3}{g}$$. This means we should expect approximately 2% change in volume if it is coming only from the change in the volume of the water.

If we look at the volume of the full cup, $$12 \ cm$$ tall with $$3.5 \ cm$$ radius, we get a volume of $$461.81 \ cm^3$$. Assuming that your finger width is 1 cm (fairly conservative estimate, especially considering most cups slope outwards if anything), then we would see a new volume of $$423.33 \ cm^3$$. This represents an 8% change in volume.

That's 4x what we calculated from just the change in water volume. This to me highly suggests that a majority of the tea lost is lost to evaporation, not to the volume changes in the tea. This makes a lot of sense, as you say the cup is open, and your office is air-conditioned. This generally will help get rid of excess humidity in the room, aiding with the evaporation.

To be sure this is what is happening, try pre-heating your cup with warm water. This should remove the expansion of the cup as a variable that reduces the water level (in theory it would actually be counter-acting the lowering tea level if you heated the cup to the starting tea temperature). What I expect is that when you do this, you will find you lose even more tea. That is because the pre-warmed cup isn't a heat sink for the tea, so it would provide a bit more opportunity to use the heat for evaporation instead of just heating up the room temperature cup.

• another factor that came to my mind is, that I always leave my tea infuser in the cup during cooling. So maybe the tea ingredients soaking up water over time might have to be considered aswell? My infuser also has a pretty dense net, so maybe smaller bubbles of air remain inside, that escape once I move the tea a little, or surface tension changes to break through the small holes of the net. – Radioo Feb 8 '19 at 8:00