Hot Ice Melts and Cold Water Freezes

To find the entropy change of the universe when hot ice melts, consider the following data:

As a first approximation, ignore difference in heat capacities:

Figure: If the difference in heat capacities is ignored, $\Delta \ensuremath{\overline{H}}$ independent of temperature

Suppose our ice-system is enclosed in a giant reservoir at $10^\circ$C (the reservoir is so big that its temperature doesn't change, imagine cooling down the ocean with an ice-cube)

$$\input{equations/96B}$$ (14-5)

because $P$ is constant and we suppose that no other heat is added to the system from any other source (constant pressure and adiabatic system).

Suppose the ice melts, then

$$\input{equations/96C}$$ (14-6)

Therefore, $\Delta H_{total} = 0$ However,

$$\input{equations/96D}$$ (14-7)

The entropy of the reservoir decreases.

However the entropy change for our mole of ice that melts is:

$$\input{equations/96E}$$ (14-8)

This corresponds to what we observe, hot ice would melt and the entropy of the universe increases.

Consider the melting of cold ice immersed in $-10^\circ$C reservoir:

$$\input{equations/97A}$$ (14-9)

Entropy of the universe decreases and this is not observed to happen--good!

Hot ice melts and cold water freezes and the entropy of the universe always increases.