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$$\ce{CaCO3 + 2H2O + Pb^{2+} -> Pb(CO3) + Ca(OH)2 + H2}$$

Does this equation balance? On the right you have lead with 2+ charge, whereas on the right its completely neutral. So my question is do you need to balance the charges in order to make the equation correct?

If that equation is wrong, what is the right equation?

I need to show that hydrogen gas is formed (or some other gas but I don't know what else it can be), that the solution will become more basic (proven with $\ce{Ca(OH)2}$), and that the lead has to replace the calcium.

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Unfortunately your equation does not add up, let me briefly explain why.

Calcium carbonate is a not very soluble salt. There is ionic binding between the calcium cation and the molecular carbonate anion. If you dissolve it in water you, the following equilibrium will establish: $$\ce{CaCO3~(s) <=>[\ce{H2O}] Ca^{2+}~(aq) + CO3^{2-}~(aq)}$$

Lead carbonate is also a salt, it is even less soluble. Only a little bit will dissolve according to the following equilibrium:$$\ce{PbCO3~(s) <=>[\ce{H2O}] Pb^{2+}~(aq) + CO3^{2-}~(aq)}$$

Calcium hydroxide is also a salt. It is a little bit more soluble than calcium carbonate. The following equilibrium is established when put in water:$$\ce{Ca(OH)2~(s) <=>[\ce{H2O}] Ca^{2+}~(aq) + 2{}^{-}OH~(aq)}$$

You write the following:

$$\ce{CaCO3 + 2H2O + Pb^{2+} -> Pb(CO3) + Ca(OH)2 + H2}$$

So in principle you have $\ce{Ca^{2+}}$, $\ce{Pb^{2+}}$ and $\ce{CO3^{2-}}$ on both sides of the equation. What is left is not a balanced equation, since there are two electrons missing: $$\ce{2H2O ~\nrightarrow 2{}^{-}OH + H2}$$

You need a source for the electrons, one way of balancing the equation is $$\ce{CaCO3 + 2H2O + Pb -> Pb(CO3) + Ca(OH)2 + H2}$$

Given the insolubilities of all the species involved, it might be hard to see any reaction at all. And I have not checked if the redox reaction is favourable, with potentials.

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The reaction would make more sense, and be balanced for charge, if the $\ce{H2}$ were replaced by $\ce{2H+}$:

$$\ce{CaCO3 + 2H2O + Pb^{+2} -> Pb(CO3) + Ca(OH)2 + 2 H+}$$

Even that is a bit weird because it suggests that the water would be acidified by this (non-redox) process. You could also write it as:

$$\ce{CaCO3 + 2OH- + Pb^{+2} -> Pb(CO3) + Ca(OH)2}$$

Lead hydroxide is apparently a compound that has never been isolated: the hydrated lead oxides are more stable than a true hydroxide salt.

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Yup, your equation falls apart like dominoes.

Nothing balances the charge in lead ion, so you need to eliminate that reaction.

Nothing is left to balance the lead in lead carbonate, so that has to go away.

Nothing is left to balance the carbon in calcium carbonate, so good-bye to that reactant.

Nothing is left to balance the calcium in calcium hydroxide, so there can't be that product either.

Nothing is left to balance the oxygen in water, so -- no water.

And nothing is left to balance the hydrogen in elemental hydrogen gas, so that goes away.

Everything that's left, meaning zero, works.

You could have avoided this disaster by rendering the lead as the element not lead ion. But Curt's answer is more likely as an actual chemical reaction.

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