Latent Heat Solid->gas Liquid->gas transitions

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In the discussion on latent heat transitions from solid to gas and liquid to gas at temperatures much lower than the critical temperature, participants focus on demonstrating the relationship L ≈ a + bT, where L is latent heat and a, b are constants. The homework equations provided relate the change in latent heat with temperature to specific heat capacities and volume expansion coefficients. One participant expresses uncertainty about how to begin solving the problem, noting the significant difference in volumes between solid/liquid and gas states. Suggestions include rearranging the provided equations and considering integration or separation of variables as potential methods. The conversation emphasizes the need for a clear approach to tackle the problem effectively.
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Homework Statement


Show that in Solid to gas and Liquid to gas transitions at T \ll {T}_{c} , L\simeq a+bT where {T}_{c}, L are the critic temperature and latent heat respectively and a,b constants.

Homework Equations


\left( \frac{d ( \frac {L} {T})} {dT} \right) = \frac {{c}_{p2}-{c}_{p1}} {T}+ \frac {\alpha_1v_1-\alpha_2v_2} {v_2-v_1} \frac {L} {T}
2 is for gas and 1 is for solid or liquid
cp , \alpha are the calorific coefficient at constant pressure and the cubic expansion coefficient.

3. The Attempt at a Solution
Honestly, I do not have any idea of how to start, I only got to say that v_1 \ll v_2.
 
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Try rearranging the equation given in 2.), Relevant equations, and integrating.
 
Can I separate variables? Or do I have to use another method?
 
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