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This is the explanation in a graphical sense.
If we visualize just the boiling water (the leftmost point on the red colored line), we see that the heat added which is the heat that transfers into the skin when the skin comes in contact with the water is much less than the right most part of the red line which is steam. It turns out that the energy needed to turn water into steam is ~6 or 7 (if I recall correctly, it was ~6 kj to heat vs 40 kj to turn water into vapor) times as much as it takes to heat water from 30 degrees celsius (the benchmark he uses for your skin temp) to 100 degree celsius (the beginning of boiling water). This energy difference is the cause of the large discrepancy between the damage boiling water can inflict and the damage steam can inflict.
I would also like to add that, when steam comes into contact with skin, it immediately starts to condense. Since condensation entails a phase change from vapor to liquid, essentially hydrogen bonds are being formed, so an exothermic reaction takes place. The heat released in this phase change is transferred to your skin, which causes the burns. The reason why condensation (exothermic) causes more harm than freezing (exothermic) does is because hydrogen bonds are really strong. The energy required to break these bonds through vaporization (endothermic) is large. Thus, when you form these hydrogen bonds through condensation, the energy released is also large. When you melt ice (endothermic), the bonds between molecules or atoms of water are being loosened (from a crystalline structure with atoms or molecules to a liquid in which particles can more easily move around), a process which requires less energy than breaking hydrogen bonds through vaporization does.
In the example in lecture, BOTH water and steam were given at 100 oC, so the reason steam is more painful is NOT because it can get hotter than water. The difference in energy released when 100 oC water comes into contact with your skin is roughly 5 kJ/mol (two thirds of 7.5 kJ/mol, the energy required to raise water from 0 oC to 100 oC. When steam comes into contact with your skin, a phase transition happens which releases ~40.7 kJ/mol, and then the additional 5 kJ/mol to cool the condensed steam to body temperature. This is a much greater exothermic process than for liquid water.
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