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if you look at the heating curve of water, going from solid to liquid to vapor, you will see the increase in temperature along with the increase of heat being applied to the water. As the water goes from solid to liquid, the enthalpy does not change by a significant amount. As water completes a phase change from liquid to vapor, the enthalpy is significantly greater. Therefore, the more enthalpy being applied to the skin, the worse the burn. Because vapor has a higher enthalpy than liquid, it creates a worse burn! Hope this was helpful!
As said above, when looking at the phase change graph it is evident that it requires a larger input of heat to turn water into vapor than it does to turn solid ice into liquid. This is because the horizontal line that represents a phase change from liquid to vapor is also much lengthier than that of solid to liquid. This indicates that the state of vapor can hold a higher amount of heat and energy than that of boiling liquid, therefore the burn present with vapor is significantly worse than that of boiling water.
Liquid water just needs to cool down when it touches your skin--the resulting release of energy is actually quite low. Steam, however, needs to go through the process of changing phases on top of cooling down. The release of energy is almost ten times as much in this case, which explains why steam burns so much worse than boiling water.
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