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Enthalpy is defined as $H = U + PV$, but internal energy is given by $U = \frac{3}{2} nk_B T = \frac{3}{2}PV$ for an ideal gas, so why don't we just write

$$H = \frac{5}{2} PV$$

for enthalpy? Is this equation even correct? If so, what significance $H = U + PV$ serve that $H = \frac{5}{2} PV$ does not?

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Enthalpy for any system is H=U+PV; that's its definition. Your statement that H = (5/2)PV is true only for an ideal gas. There are many things one cares about that are not ideal gases!

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