One ice cube, five times the chill

A thirty-gram ice cube was dropped into a glass holding two hundred and fifty grams of juice at twenty degrees. A few minutes later the drink had cooled to about nine degrees.

A $30$ g ice cube in $250$ g of juice at $20{}^{\circ }\text{C}$: a few minutes later the drink was at about $9{}^{\circ }\text{C}$.

Juice $m_j=250\text{g}$, ice $m_i=30\text{g}$, start $T_0=20{}^{\circ }\text{C}$, end $T\approx 9{}^{\circ }\text{C}$.

The reason is that melting ice takes a lot of energy: about three hundred and thirty-four kilojoules for every kilogram, as much as it takes to warm the same water by eighty degrees. The thirty-gram cube took about ten kilojoules from the juice just to melt, while its own temperature stayed at zero.

Why: melting takes $L\approx 334$ kJ/kg, as much as warming water by $334/4.2\approx 80{}^{\circ }\text{C}$. The cube takes $0.03\times 334\approx 10$ kJ from the juice just to melt, at $0{}^{\circ }\text{C}$.

Heat of melting $L\approx 334$; $334/4.2\approx 80$; $Q=m_{i}L=0.03\times 334\approx 10$.

Those ten kilojoules alone would cool the juice by about nine and a half degrees. The melted water then warms a little from zero and takes a small share more, so the final temperature is nine point three degrees: a drop of ten point seven.

That alone cools the juice by $10.02/(0.25\times 4.2)\approx 9.5{}^{\circ }\text{C}$. The meltwater then warms from $0{}^{\circ }\text{C}$: $1.05(20-T)=10.02+0.126T\Rightarrow T\approx 9.3$, a drop of $\Delta T\approx 10.7$.

$1.05(20-T)=10.02+0.126T\Rightarrow T\approx 9.3$; $\Delta T\approx 10.7$.

The surprise: had thirty grams of water at zero degrees been poured in instead of ice, the drink would have cooled by only about two point one degrees. The ice chills about five times more.

Surprise: $30$ g of water at $0{}^{\circ }\text{C}$ instead of ice gives $T=21/1.176\approx 17.9$, a drop of $\Delta T\approx 2.1$. Ice chills $10.7/2.1\approx 5$ times more.

Water instead: $T\approx 17.9$, $\Delta T\approx 2.1$; $10.7/2.1\approx 5$.