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When formulating a rate a equation two experiments must take place: one in which the reactant concentrations are varied, and one in which the catalyst concentrations are varied.

If I am given a sample of a solid catalyst, such as $MnO_2$, how can I change its concentration? Because I can't dilute a solid and increase the volume, which is how I'd do it for a catalyst in solution.

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You will perform the reaction inside a container or solution of some particular volume. This means that you can change the catalyst concentration inside that volume by adding more or less catalyst in different experiments. So you don't increase the volume of the solid (i.e. change its density), you change the mass of solid inside the reaction volume.

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The rate equation is actually defined for the activity of each reactant and product. In most cases, the activity can be approximated to the concentration.

In the case of solid catalysts, you'll have to use activity. I'm not sure how the actual value of activity is calculated, but the solid catalyst's activity is increased when you increase its surface area. That can be one way to change its effect on the rate equation.

This link has some mention of the activity based rate equation and some examples.

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