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In the rate equation, consider k:

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I know that $k$ is fixed unless temperature changes, but what else besides temperature is $k$ dependent on?

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The rate constant $k$ includes all the factors other than the dependence on concentration. A significant factor in the rate is collision frequency and energy, so temperature has a significant influence on the value of $k$. Other factors like the range of angles of collision necessary for a successful reaction, the strength of dipoles, solvent effects, quantum effects, and whatever factor corrects the units of the equation are not determined.

However the rate constant can be expressed by the Arrhenius equation $$k=Ae^{-{E_a/RT}}$$ Doing this decomposes the constant into an exponential relationship including the activation energy, $E_a$, the molar gas constant, $R$, and of course temperature, $T$, which represents the fraction of total molecules that have sufficient energy to overcome the activation energy. This equation comes with it's own new constant, the pre-exponential factor, $A$, which now includes all the factors in the rate constant that aren't the exponential relationship with $E_a$, $R$, or $T$. There may still also have some temperature dependence hidden within $A$, but usually it can be assumed constant over small temperature ranges. Factors like collision frequency and angle, or action of a catalyst, are still obscured within this constant.

Even if these obscured effects were determined there would still be some new constant, $C$, which would include subtle effects such as interactions with surfaces in a container or non-reacting particles, temperature and concentration gradients, ionic strength, entropy changes in the solution due to changing composition, and any other factor that influences the reaction between the particles that hasn't been explicitly considered.

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    $\begingroup$ Nice answer! It might be worth explicitly mentioning that rate constants can depend on pressure, ionic strength (if the reaction is in liquid), and the concentration of non-reacting species (a.k.a. catalysts) in addition to all the factors that you listed. $\endgroup$ – Curt F. Feb 16 '16 at 17:04
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k is a characteristic of the reaction being considered. It depends on the stability of the reactants with respect to intermediate and intermediate with respect to products and the mechanism of the reaction.

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