Single Displacement Reactions: A + AX =?
Nothing. All you would have is just a mixture of white powder and lead metal.

In general, lead (II) nitrate is synthesized by:

$$\ce{PbO (s) + 2 HNO3 (aq) -&gt; Pb(NO3)2 (aq) + H2O (l)}$$

Lead (II) nitrate is soluble in water.

Alternatively, you could do:

$$\ce{PbCO3(s) + 2HNO3(aq) -&gt; Pb(NO3)2(aq) + H2O(l) + CO2(g)} $$

Single displacement reactions are a kind of redox reaction. For single displacement reactions to occur:

1.) A and B must either be different metals OR

$$ \ce{2AgNO3(aq) + Zn(s) -&gt; 2Ag(s) + Zn(NO3)2(aq)}$$

2.) A and B must be a halogen

$$\ce{Mg(s) + 2 HCl(aq) → MgCl2(aq) + H2(g)}$$

This is necessary as for the redox reaction to occur, you need two differing metals on the reactivity series in order for the single replacement to work, or alternatively, use compounds that contain halogens.

Even so, in combining two compounds with metals or halogens, if the element is less reactive than the element in the compound, the single replacement reaction will not proceed.

Some examples are:

$$\ce{Ag (s) + Cu(NO3)2 (aq) -&gt; } \text{No reaction}$$

$$ \ce{I2 + 2KBr -&gt;} \text{No reaction} $$