{"id":20440,"date":"2019-01-13T14:23:57","date_gmt":"2019-01-13T14:23:57","guid":{"rendered":"https:\/\/www.ch.imperial.ac.uk\/rzepa\/blog\/?p=20440"},"modified":"2019-04-29T17:14:09","modified_gmt":"2019-04-29T16:14:09","slug":"free-energy-relationships-and-their-linearity-a-test-example","status":"publish","type":"post","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=20440","title":{"rendered":"Free energy relationships and their linearity: a test example."},"content":{"rendered":"
\n

Linear free energy relationships<\/a> (LFER) are associated with the dawn of physical organic chemistry in the late 1930s and its objectives in understanding chemical reactivity as measured by reaction rates and equilibria.<\/p>\n

The Hammett equation<\/a> is the best known of the LFERs, albeit derived “intuitively”. It is normally applied to the kinetics of aromatic electrophilic substitution reactions and is expressed as;<\/p>\n

log KR<\/sub>\/K0<\/sub> = \u03c3R<\/sub>\u03c1<\/strong> (for equilibria) and extended to log kR<\/sub>\/k0<\/sub> = \u03c3R<\/sub>\u03c1<\/strong> for rates.<\/p>\n

The equilibrium constants are normally derived from the ionisation of substituted benzoic acids, with K0\u00a0<\/sub>being that for benzoic acid itself and\u00a0KR\u00a0<\/sub>that of a substituted benzoic acid, with\u00a0\u03c3R<\/sub> being known as the substituent constant and\u00a0\u03c1 the reaction constant. The concept involved obtaining the substituent constants by measuring the ionisation equilibria. The value of \u03c3R\u00a0<\/sub>is then assumed to be transferable to the rates of reaction,\u00a0where the values can be used to obtain reaction constants for a given reaction. The latter would then be assumed to give insight into the electronic nature of the transition state for that reaction.<\/p>\n

The term\u00a0log kR<\/sub>\/k0\u00a0<\/sub>(the ratio of rates of reaction) can be related to\u00a0\u0394\u0394G = -RT ln kR<\/sub>\/k0\u00a0<\/sub>and this latter quantity can be readily obtained from quantum calculations, where \u0394\u0394G is the difference in computed reaction activation free energies for two substituents (of which one might be R=H). The most interesting such Hammett plots are the ones where a discontinuity becomes apparent. The plot comprises two separate linear relationships, but with different slopes. This is normally taken to indicate a change of mechanism, on the assumption that the two mechanisms will have different responses to substituents.\u00a0<\/p>\n

A test of this is available via the calculated activations energies for acid catalyzed cyclocondensation to give furanochromanes[1]<\/a><\/span> which is a two-step reaction involving two transition states TS1<\/strong> and TS2<\/strong>, either of which could be rate determining. A change from one to the other would constitute a change in mechanism. In this example, TS1<\/strong> involves creation of a carbocationic centre which can be stabilized by the substituent on the Ar group;\u00a0TS2<\/strong> involves the quenching of the carbocation by a nucleophilic oxygen and hence might be expected to respond differently to the substituents on Ar. As it happens, the reaction coordinate for TS2<\/strong> is not entirely trivial, since it also includes an accompanying proton transfer which might perturb the mechanism.<\/p>\n

\"\"<\/a><\/p>\n

Fortunately for this reaction we have available full FAIR data (DOI: 10.14469\/hpc\/3943<\/a>), which includes not only the computed free energies for both sets of transition states but also the entropy-free enthalpies for comparison. This allows the table below to be generated. For each substituent, the highest energy point is in bold, indicating the rate limiting step. The span of substituents corresponds to a range of rate constants of almost 1010<\/sup>, which in fact is rarely if ever achievable experimentally.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
\n

Highest free energy overall route for HCl catalysed mechanism,<\/p>\n

trans stereochemistry<\/p>\n<\/th>\n<\/tr>\n

Sub<\/th>\n\u0394H\u2021<\/sup>\/\u0394G\u2021<\/sup><\/th>\nReactant<\/th>\n\u0394H\u2021<\/sup>\/\u0394G\u2021<\/sup>, TS1<\/th>\n\u0394H\u2021<\/sup>\/\u0394G\u2021<\/sup>, TS2<\/th>\nRDS<\/th>\n<\/tr>\n
p-NH2<\/sub><\/td>\n0.2\/6.36<\/td>\n0.0\/0.0<\/td>\n0.15\/4.0<\/td>\n0.2<\/b>\/6.4<\/b><\/td>\nTS2\/TS2<\/strong><\/td>\n<\/tr>\n
p-OMe<\/td>\n2.7\/8.48<\/td>\n0.0\/0.0<\/td>\n2.7<\/b>\/8.45<\/td>\n2.1\/8.48<\/b><\/td>\nTS1\/TS2<\/strong><\/td>\n<\/tr>\n
p-Me<\/td>\n5.5\/10.00<\/td>\n0.0\/0.0<\/td>\n5.5<\/b>\/9.9<\/td>\n3.9\/10.00<\/b><\/td>\nTS1\/TS2<\/strong><\/td>\n<\/tr>\n
p-Cl<\/td>\n7.7\/12.28<\/td>\n0.0\/0.0<\/td>\n7.7<\/b>\/12.28<\/b><\/td>\n5.9\/11.84<\/td>\nTS1\/TS1<\/strong><\/td>\n<\/tr>\n
p-H<\/td>\n7.6\/13.01<\/td>\n0.0\/0.0<\/td>\n7.6<\/b>\/13.01<\/b><\/td>\n5.5\/11.51<\/td>\nTS1\/TS1<\/strong><\/td>\n<\/tr>\n
p-CN<\/td>\n10.6\/18.02<\/td>\n0.0\/0.0<\/td>\n10.6<\/b> \/17.61<\/td>\n10.5\/18.02<\/b><\/td>\nTS1\/TS2<\/strong><\/td>\n<\/tr>\n
p-NO2<\/sub><\/td>\n12.4\/19.85<\/td>\n0.0\/0.0<\/td>\n12.4<\/b>\/18.24<\/td>\n12.0\/19.85<\/b><\/td>\nTS1\/TS2<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For the free energies, you can see that TS2<\/strong> is the rate limiting step for the first two electron donating substituents, and the last two electron withdrawing ones, whilst TS1<\/strong> represents the rate limiting step for the middle substituents. This represents two<\/strong> changes of rate limiting step over the entire range of substituents. A different picture emerges if only the enthalpies are used. Now TS1<\/strong> is rate limiting for essentially all the substituents. The difference of course arises because of significant changes to the entropy of the transition states. The Hammett equation, and its use of\u00a0\u00a0\u03c3R\u00a0<\/sub>constants to try to infer the electronic response of a reaction mechanism, does not really factor in entropic responses. Nor is it often if at all applied using a really wide range of substituents. So any linearity or indeed non-linearity in Hammett plots may correspond only very loosely to the underlying mechanisms involved.<\/p>\n

Starting in the 1940s and lasting perhaps 40-50 years, thousands of different reaction mechanisms were subjected to the Hammett treatment during the golden era of physical organic chemistry, but very few have been followed up by exploring the computed free energies, as set out above. One wonders how many of the original interpretations will fully withstand such new scrutiny and in general how influential the role of entropy is.<\/p>\n

References<\/h2>\n
  1. C.D. Nielsen, W.J. Mooij, D. Sale, H.S. Rzepa, J. Bur\u00e9s, and A.C. Spivey, \"Reversibility and reactivity in an acid catalyzed cyclocondensation to give furanochromanes \u2013 a reaction at the \u2018oxonium-Prins\u2019 <i>vs.<\/i> \u2018<i>ortho<\/i>-quinone methide cycloaddition\u2019 mechanistic nexus\", Chemical Science<\/i>, vol. 10, pp. 406-412, 2019. https:\/\/doi.org\/10.1039\/c8sc04302g<\/a>\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

    Linear free energy relationships (LFER) are associated with the dawn of physical organic chemistry in the late 1930s and its objectives in understanding chemical reactivity as measured by reaction rates and equilibria. The Hammett equation is the best known of the LFERs, albeit derived “intuitively”. It is normally applied to the kinetics of aromatic electrophilic […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"activitypub_content_warning":"","activitypub_content_visibility":"","activitypub_max_image_attachments":5,"activitypub_interaction_policy_quote":"anyone","activitypub_status":"","footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[2,4,1086],"tags":[2613,2542,984,543,224,1395,1447,2579,2611,1622,2612,2580,1566,2581,2506,1410,1442,2505,2578],"ppma_author":[2661],"class_list":["post-20440","post","type-post","status-publish","format-standard","hentry","category-chemical-it","category-interesting-chemistry","category-reaction-mechanism-2","tag-benzoic-acid","tag-chemical-kinetics","tag-chemical-reaction","tag-chemical-reactivity","tag-chemist","tag-chemistry","tag-electrophilic-aromatic-substitution","tag-energy-point","tag-equations","tag-equilibrium-chemistry","tag-equilibrium-constant","tag-free-energy-overall-route","tag-hammett-equation","tag-linear-free-energy-relationships","tag-natural-sciences","tag-organic-chemistry","tag-physical-organic-chemistry","tag-physical-sciences","tag-reactivity"],"yoast_head":"\nFree energy relationships and their linearity: a test example. - Henry Rzepa's Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=20440\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Free energy relationships and their linearity: a test example. - Henry Rzepa's Blog\" \/>\n<meta property=\"og:description\" content=\"Linear free energy relationships (LFER) are associated with the dawn of physical organic chemistry in the late 1930s and its objectives in understanding chemical reactivity as measured by reaction rates and equilibria. 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The ionisation mechanism and kinetic isotope effects for 1,3-dimethylindolin-2 one","author":"Henry Rzepa","date":"January 7, 2016","format":false,"excerpt":"This is the third and final study deriving from my Ph.D.. The first two topics dealt with the mechanism of heteroaromatic electrophilic attack using either a diazonium cation or a proton as electrophile, followed by either proton abstraction or carbon dioxide loss from the resulting Wheland intermediate. This final study\u2026","rel":"","context":"In "Historical"","block_context":{"text":"Historical","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=565"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":2423,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=2423","url_meta":{"origin":20440,"position":1},"title":"The oldest reaction mechanism: updated!","author":"Henry Rzepa","date":"September 14, 2010","format":false,"excerpt":"Unravelling reaction mechanisms is thought to be a 20th century phenomenon, coincident more or less with the development of electronic theories of chemistry. Hence electronic\u00a0arrow pushing as a term. But here I argue that the true origin of this immensely powerful technique in chemistry goes back to the 19th century.\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2010\/09\/wheland.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":13962,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=13962","url_meta":{"origin":20440,"position":2},"title":"A new way of exploring the directing influence of (electron donating) substituents on benzene.","author":"Henry Rzepa","date":"April 17, 2015","format":false,"excerpt":"The knowledge that substituents on a benzene ring direct an electrophile engaged in a ring substitution reaction according to whether they withdraw or donate electrons is very old. Introductory organic chemistry tells us that electron donating substituents promote the ortho and para positions over the meta. Here I try to\u2026","rel":"","context":"In "Chemical IT"","block_context":{"text":"Chemical IT","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=2"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":9659,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9659","url_meta":{"origin":20440,"position":3},"title":"Understanding the electrophilic aromatic substitution of indole.","author":"Henry Rzepa","date":"March 3, 2013","format":false,"excerpt":"The electrophilic substitution of indoles is a staple of any course on organic chemistry. Indoles also hold a soft-spot for me, since I synthesized not a few as part of my Ph.D. studies., The preference for substitution in the 3-position is normally explained using the arrows shown below (position 3=green,2=blue,1=red).\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"Molecular electrostatic potential. Click for 3D.","src":"https:\/\/i0.wp.com\/www.ch.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2013\/03\/indole-mep.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":12056,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=12056","url_meta":{"origin":20440,"position":4},"title":"The mechanism of diazo coupling: more hidden mechanistic intermediates.","author":"Henry Rzepa","date":"March 8, 2014","format":false,"excerpt":"The diazo-coupling reaction dates back to the 1850s (and a close association with Imperial College via the first professor of chemistry there, August von Hofmann) and its mechanism was much studied in the heyday of physical organic chemistry. Nick Greeves, purveyor of the excellent ChemTube3D site, contacted me about the\u2026","rel":"","context":"In "reaction mechanism"","block_context":{"text":"reaction mechanism","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=1086"},"img":{"alt_text":"cis-diazo","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2014\/03\/cis-diazo.gif?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":9706,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9706","url_meta":{"origin":20440,"position":5},"title":"Kinetic vs Thermodynamic control. Subversive thoughts for electrophilic substitution of Indole.","author":"Henry Rzepa","date":"March 10, 2013","format":false,"excerpt":"I mentioned in the last post that one can try to predict the outcome of electrophilic aromatic substitution by approximating the properties of the transition state from those of either the reactant or the (presumed Wheland) intermediate by invoking Hammond's postulate. A third option is readily available nowadays; calculate the\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"Click for 3D.","src":"https:\/\/i0.wp.com\/www.ch.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2013\/03\/3-NO-indole-ESP.jpg?resize=350%2C200","width":350,"height":200},"classes":[]}],"jetpack_likes_enabled":false,"authors":[{"term_id":2661,"user_id":1,"is_guest":0,"slug":"admin","display_name":"Henry Rzepa","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/897b6740f7f599bca7942cdf7d7914af5988937ae0e3869ab09aebb87f26a731?s=96&d=blank&r=g","author_category":"1","first_name":"Henry","last_name":"Rzepa","user_url":"https:\/\/orcid.org\/0000-0002-8635-8390","job_title":"","description":"Henry Rzepa is Emeritus Professor of Computational Chemistry at Imperial College London."}],"_links":{"self":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/20440","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=20440"}],"version-history":[{"count":10,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/20440\/revisions"}],"predecessor-version":[{"id":20885,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/20440\/revisions\/20885"}],"wp:attachment":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=20440"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=20440"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=20440"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=20440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}