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While writing the cell notation for a cell, doesn't one have to write the two states in which a species exists in a sequence appropriate to the reaction in which they participate?

For example, in the Daniell cell reaction, shouldn't it always be written $\ce{Zn}|\ce{Zn^2+}$, and never, $\ce{Zn^2+}|\ce{Zn}$?

I didn't see anything in the definitions, and I saw a question asking for the cell reactions for this;

$$\ce{Pt,Cl2}|\ce{Cl^-(aq)}||\ce{Ag^+(aq)}|\ce{Ag}$$

Obviouly, chlorine doesn't react in that direction, but gets oxidised.

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Good question. You can remember the cell notation with the mnemonic "reduction on the right."

So, on the right hand side, you should reduction process and on the left hand side you should show the oxidation process.

For example, for a Daniel cell, one can write

$$\ce{Zn|Zn^{2+}, SO_4^{2−}}||\ce{SO_4^{2−},Cu^{2+}}|\ce{Cu}$$

Now coming to your cell, $$\ce{Pt,Cl_2}|\ce{Cl^-(aq)}||\ce{Ag^+(aq)}|\ce{Ag}$$ is indeed written incorrectly because the writer seems to have the misconception that even in the cell notation, half cells must be written as a reduction.

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