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Cobalt sulfate and lead nitrate combine to make lead sulfate. How and why does this happen?

Answer
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Hint: To predict the direction of the reaction, always check the solubility of different compounds, as it can quickly give us an idea about direction of flow of reaction. Any reaction always moves toward maintaining its Reaction Quotient (Q).So, if $\text{Q K}$reaction moves in backward direction (where K is equilibrium constant).

Complete step-by-step answer: Reaction of cobalt sulphate with lead nitrate is an example of double displacement reaction. 
$CoS{{O}_{4}}+Pb{{(N{{O}_{3}})}_{2}}\xrightarrow{{}}PbS{{O}_{4}}+Co{{(N{{O}_{3}})}_{2}}$ 
Double displacement reactions can be defined as the chemical reaction in which ionic species of both the reacting molecules is exchanged to form the products. During the reaction, cations and anions of two different compounds switch places with each other, producing two entirely different compounds.
To determine in which direction a reaction will proceed, let us check the solubility of the reactant and product.
-$CoS{{O}_{4}}$ : It is soluble in water and forms a dark red color solution.
-$Pb{{(N{{O}_{3}})}_{2}}$ : It is soluble in water as well as has a colorless solution.
-$PbS{{O}_{4}}$ : It is insoluble in water and forms a white color precipitate.
-$Co{{(N{{O}_{3}})}_{2}}$: It is also soluble in water and forms a red color solution.
We conclude that, since all the other compounds other than $PbS{{O}_{4}}$ are soluble in water , so as cobalt sulfate and lead nitrate combine together to form lead sulfate ,the amount of white precipitate increase in the vessel.
Also we know that, Reaction quotient (Q) = $\dfrac{[C][D]}{[A][B]}$ for$A+B\xrightarrow{{}}C+D$. Reaction quotient always remains constant for every particular reaction.
Since concentration of any compound in a reaction is considered if it is in aqueous or gaseous state, whereas it is taken as 1 for solid state. And here we can see that, $PbS{{O}_{4}}$is a precipitate hence being solid, its active mass will always be equal to 1.
Now, we know that:
If $\text{Q < K}$, reaction will always move in forward direction, to attain equilibrium.
Therefore, when the concentration of $PbS{{O}_{4}}$decreases due to precipitation, the value of Q will also decrease and thus the nature of reaction will always be in forward direction.
Thus, $CoS{{O}_{4}}+Pb{{(N{{O}_{3}})}_{2}}\text{ will react to give }PbS{{O}_{4}}+Co{{(N{{O}_{3}})}_{2}}\text{ as product}\text{.}$

Note:  -Make sure you check the solvent given in the question, as different compounds have different solubility in different solvents and thus it may change the direction of reaction flow.
-Equilibrium constant (K) is fixed for every reaction and is dependent only on temperature whereas reaction quotient (Q) changes with respect to concentration of the species.