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It looks like it does not. The specific heat (cp) of oxygen is 0.918 J/(g*K). I do not find this number freely published for ozone, but the gas constant (0.173 J/(g*K)) and cp/cv ratio (1.29) are, and they indicate ozone should have a cp value of 0.770 J/(g*K). So it appears that ozone absorbs about 16% less heat than oxygen to get the same temperature rise, on a mass basis. What ozone does do, is radiate heat in the far infrared... which oxygen cannot do. So heat can be transferred from locally placed oxygen, and the ozone can radiate that heat away.

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