.. _sphx_glr_api_gallery_kinetics_rate_laws:

Rate laws
---------

Zero-, first-, and second-order integrated rate laws and half-lives,
reaction order from initial rates, and Wilhelmy's first-order sucrose
inversion.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="van&#x27;t Hoff (1884) classified reactions by the order n of their rate law -d[A]/dt = k[A]^n. The three textbook cases -- ZeroOrder, FirstOrder, and SecondOrder -- all started from the same initial concentration and with rate constants chosen to give the same half-life, so the different curvature (linear, exponential, hyperbolic) is the only thing distinguishing them. First order is the only one whose half-life doesn&#x27;t depend on the starting concentration; the plot below marks all three half-lives to make that concrete.">

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  .. image:: /api/gallery/kinetics/rate_laws/images/thumb/sphx_glr_plot_01_integrated_rate_laws_thumb.png
    :alt:

  :doc:`/api/gallery/kinetics/rate_laws/plot_01_integrated_rate_laws`

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      <div class="sphx-glr-thumbnail-title">van't Hoff's reaction orders: zero-, first-, and second-order rate laws</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Wilhelmy (1850) followed the acid-catalyzed inversion of cane sugar by polarimetry: the optical rotation \alpha(t) drifts from the dextrorotatory sucrose value towards the (levorotatory) value of the glucose + fructose mixture. Because \alpha is linear in the concentrations, \alpha(t) - \alpha_\infty is proportional to the sucrose still present, and a straight line of \ln[(\alpha - \alpha_\infty)/(\alpha_0 - \alpha_\infty)] against t identifies -d[\text{sugar}]/dt = k[\text{sugar}].">

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  .. image:: /api/gallery/kinetics/rate_laws/images/thumb/sphx_glr_plot_02_wilhelmy_sucrose_inversion_thumb.png
    :alt:

  :doc:`/api/gallery/kinetics/rate_laws/plot_02_wilhelmy_sucrose_inversion`

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      <div class="sphx-glr-thumbnail-title">Wilhelmy's sucrose inversion: the first measured first-order rate law</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/kinetics/rate_laws/plot_01_integrated_rate_laws
   /api/gallery/kinetics/rate_laws/plot_02_wilhelmy_sucrose_inversion