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latent heat of vaporisation means heat needed to change into gas and latent heat of fusion to change into liquid, in liquid all the molecules are not free of each other that is they still have molecular interactions i.e. all the atoms are not free only some are free so less energy is required for latent heat of fusion while in gases all the atoms are free so all the bonds between molecules must be broken requiring greater energy.
From point of first law of thermodynamics :
∆U is change in internal energy which is equal to sum of work done(W) on system and heat applied to the system(Q).
For real gas ∆U includes both potential and kinetic energy. But changing state means there should be no increase in temperature hence no increase in average kinetic energy of molecules so ∆U should be zero. To change into vapor molecules have to be further apart, that is volume is more than liquid (since molecules have to be further apart potential energy is also more) . ∆U=Q+W . Most substances expand on heating, and thus external work is done by the system on atmospheric pressure to increase in volume. So work done is -W (refer to note below) and since the ∆U is 0 by first law of thermodynamics heat(Q) equal in magnitude to -W should be added to the system. Since W is more incase of vapor hence Q is also more.
Note: Sign convention for the equation. W means work done on system -W is workdone by system, Q is heat added to system and -Q is heat released into environment.
Everything from point of first law I told is for ideal gases and I think the question is referring to ideal gases as well.
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