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Delta refers to the difference or change in a quantity. In this case, deltaH is the difference in the enthalpies of vapor and liquid. It seems that so far in this chapter we're given deltaH of vaporization/fusion and we're asked to do calculations from there using those values. For the homework problems I just used Table 8.3 on page 284 when the problems required deltaH of fusion and vaporization.
H is simply enthalpy whereas delta H is the change in enthalpy. Enthalpy itself is the total heat content (energy) of a system: aka the internal energy plus the product of pressure and volume. My guess as to why we use the change in enthalpy is because it is easier to calculate through experiments.
You can't necessarily measure just the enthalpy (H) of a system. Rather, when we calculate the heat that is transferred during an equation, we want to compare the heat before and after the reaction has occurred, thus we would always use in terms of measuring the enthalpy change.
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