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The reason for more reactivity of more substituted alkyl halides for E2 Reaction is said to be due to the more stability of its more substituted alkene character having transition state but here (https://home.iitk.ac.in/~madhavr/CHM102/Lec13.pdf) at page 6 it is mentioned that:

The reaction rate increases with the increase in number of alkyl substituents on the carbon with the leaving group ($\alpha$-Carbon)

However, it further says that the reason is the relatively more stable transition state (which has partial double bond) due to the more alkyl substituents (probably Hyperconjugation effect). But my question is that Will not the stability of the same transition state increase if the $\beta$-carbon is also more substituted? Why it specifically mentions for the $\alpha$-carbon to be more substituted?

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You are on to something. Check out the product ratios here: https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Supplemental_Modules_(Organic_Chemistry)/Alkyl_Halides/Properties_of_Alkyl_Halides/Introduction_to_Alkyl_Halides/Elimination_by_the_E2_mechanism. The halide with 2 beta methyls and only one proton gives the same product ratios as the halide with one despite one less proton albeit not equivalent. The original kinetic data should be consulted. Substitution at the beta carbon changes steric and entropic effects. [An error in the first example in the link, cyanide is a weaker base than carbonate but a stronger base than bicarbonate.]

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  • $\begingroup$ From your provided page, I can only understand the same thing that the transition state has double bond character and that's why "alkene stability differences will be reflected in the transition states of elimination reactions, and therefore in the activation energy of the rate-determining steps". It doesn't answers the question about the effect of beta carbon substitution. But confuses more on the point that if there are e- withdrawing groups at beta then it should help C-H breaking but donating grps stabilize TS. And "Substitution at[...]entropic effects" should only matter for bulky bases. $\endgroup$
    – D13G
    Sep 23 at 10:45
  • $\begingroup$ Also the page is hard to follow due to poor readability and styling:') $\endgroup$
    – D13G
    Sep 23 at 10:53
  • $\begingroup$ Electron withdrawing on the beta carbon increase E2 rates On the alpha carbon increase rates by about half of the beta [My PhD thesis unfortunately not published]. Read between the lines a bit. With one beta the Hs are not equivalent one gives cis the other trans with different rates adding a second group removes one of the rates but the product ratios ae the same suggesting the rate for one H has increased. research similar situations $\endgroup$
    – jimchmst
    Sep 23 at 19:26

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