What partial pressure of hydrogen gas is required in order for 0.00100 g of the gas to dissolve in 17.6 ml of pure water? the henry's law constant for hydrogen gas is 7.8 × 10–4 m atm–1?

There is a minor typo in this question. The value of 7.8x10^-4 m/atm is nonsense. However, looking up the Henry's constant for hydrogen gas, there is a value of 7.8x10^-4 m/(L*atm) which I will assume is the correct unit and the rest of the problem will be done with that in mind.  
First, determine what the molar concentration would be for 0.00100 g of H2 dissolved in 17.6 ml of water. Start with the atomic weight of hydrogen = 1.00794 g/mol. Molar mass of H2 is twice that, so 2 * 1.00794 = 2.01588 g/mol. So the number of moles of H2 we have is 0.001 g / 2.01588 g/mol = 0.000496061 mol. Finally, the molarity of the solution is 0.000496061 mol / 0.0176 L = 0.0281853 mol/L.  
Now we can use the equation 
Hcp = Ca/p 
where 
Hcp = Henry's constant 
Ca = Concentration in aqueous solution 
p = pressure  
So solve for p, substitute the known values, and calculate: 
Hcp = Ca/p 
p*Hcp = Ca 
p = Ca/Hcp  
p = (0.0281853 mol/L)/(7.8x10^-4 mol/(L*atm)) 
p = 36.135 atm  
Rounding to 3 significant figures gives p = 36.1 atmospheres.

In simplest case mass of reactants is same as mass of products.

Without thinking this question deeper, mass of ZnCl2 would be 49, but..

Explanation: Reaction should be  Zn + 2 HCl ⇒ ZnCl2 + H2

Amount of zinc is  5 g / 65,38 g/mol = 0,076476 mol and amount

of Hydrogen Chloride is 50 g / 36.458 g/mol = 1,371 mol.

Althought HCl is needed 0.152 moles, zinc is an limiting reactant.

So it is possible to produce only 0.076476 mol Hydrogen and its mass

is 0.154 g.  Mass of ZnCl2 would be 0.076476 mol · (65.38 + 2·35.45) =

 10.42 g