This is my attempt at understanding this (maybe I am wrong but i think i am right lol ) :

First the title say organic chem and this is a Gen chem question

--- Next you wrote the Mole Fraction formula WRONG. A mole fraction is not just : moles solution/ total moles ? That is where you got confused.

The formula for a mole fraction is : Moles Solute / total Moles

**Solution **....................... You cant do mols solute/ (mols of solute + 1 molH2O) Because a mole fraction like I just wrote is just supposed to be over moles of SOLUTION, and you put it as solute + solution so its wrong.

Now to get to Molality....

Solute is what gets dissolved so that is the gas which is in a mole fraction of : 2 x 10^-18 moles solute / 1 mole solution

We have the numerator correct now we need to figure out how to get moles of solution into kg solvent somehow?

2 x 10^-18 moles * 32 grams/ 1 mole = 6.4*10^-17 grams solute

..................................And we know that: grams solute - grams solution = grams solvent

So put it in kg

6.4*10^-17 grams solute * 1 kg/10^3 grams = 6.4*10^-20 kg solute

So now lets get grams of solution:

H20

1 mole H20 * 18 grams/ 1 mole = 18 grams H20

18 grams H20 * 1 kg/10^3 grams = .018 Kg solution

..................................kg solute - kg solution = kg solvent

But when we subtract

6.4*10^-20 grams solute - .018 Kg solution = the number is so small that it remains as .018 Kg

--> so now we have the Kg solvent we needed which is .018 kg solvent

------> (2 x 10^-18) moles solute/ 0.018 kg solvent = 1 x 10^-16 m