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If I have an opening on the side of my electrochemical cell, I'm wondering if I can use this to insert a reference electrode. It is normally used to insert an L-shaped tube where you place the reference electrode, and is connected to luggin capillaries. I'm wondering if we can take out the entire connection with the capillaries and replace it with only the reference electrode (sans capillaries). This is in a 3-electrode system, so if I could insert the counter electrode in this way, that would work too. In my setup, I cannot do this with the working electrode.

From what I've read, it's good practice to not have the reference electrode vertical because of bubbles on the bottom of the frit (so most tilt it slightly), but I haven't read anything about a sideways electrode.

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As long as there's nothing about the design of the reference electrode (RE) that will make it malfunction when oriented horizontally, it should be fine. Obviously if there's free liquid inside the RE that will slosh away from the frit or the electrode internals (or spill out onto the lab bench!) when you tip it, then that won't work well at all. The concern about bubbles collecting on the bottom surface is very real, especially if you're not stirring very vigorously.

One practical downside to inserting the CE directly like this is that the frit of the RE will be liable to clogging/degrading much more rapidly than if you use a Luggin. Also, the composition of the internal electrolyte of the RE will change more rapidly if directly exposed to your working solution without a Luggin.

Of course, depending on your setup, this will allow you to get your reference much closer to your working electrode, reducing the $iR$ drop in your measurement.

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