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# A mixture of $1.65\times {{10}^{21}}$ molecules of X and $1.85\times {{10}^{21}}$ molecules of Y weighs 0.688g. If the molecular mass of Y is 187, what is the molecular mass of X?(A)- 41.35(B)- 42.35(C)- 41.65(D)- 41.75

Hint: Calculate the mass of molecules of Y present in the mixture. Eliminating the mass of molecules of Y from the total mass of the mixture will give us the mass of all the molecules of X in the mixture.

Let us see what we have been given.
Number of molecules of X in the mixture = $1.65\times {{10}^{21}}$
Number of molecules of Y in the mixture = $1.85\times {{10}^{21}}$
Total mass of the mixture = 0.688g
Molecular mass of Y= 187
Molecular mass of Y is equal to the mean mass of one molecule of Y which is 187 (in atomic mass unit).
We know that one mole of a substance contains $6.022\times {{10}^{23}}$ molecules.
Therefore, one mole of Y contains $6.022\times {{10}^{23}}$ molecules and the mass of one mole of Y (in grams) is equal to the average or mean mass of one molecule of Y (in amu), i.e. 187.
Hence, we can write that
Mass of $6.022\times {{10}^{23}}$ molecules of Y = 187g
Then, we can say that the mass of $1.85\times {{10}^{21}}$ molecules of Y = $\dfrac{187g}{6.022\times {{10}^{23}}}\times 1.85\times {{10}^{21}}$
Simplifying the above equation gives the mass of $1.85\times {{10}^{21}}$ molecules of Y = 0.574g.
Since the total mass of the mixture is 0.688g, such that
Mass of $1.65\times {{10}^{21}}$ molecules of X + mass of $1.85\times {{10}^{21}}$ molecules of Y = 0.688g
Then, mass of $1.65\times {{10}^{21}}$ molecules of X = 0.688g - mass of $1.85\times {{10}^{21}}$ molecules of Y
Now, we have calculated the mass of $1.85\times {{10}^{21}}$ molecules of Y to be 0.574g.
Therefore, on subtracting 0.574g from 0.688g, we get
Mass of $1.65\times {{10}^{21}}$ molecules of X is = 0.114g.
Thus, mass of $6.022\times {{10}^{23}}$ molecules of X = $\dfrac{0.114g}{1.65\times {{10}^{21}}}\times 6.022\times {{10}^{23}}$
Simplifying the above equation and calculating we obtain, mass of $6.022\times {{10}^{23}}$ molecules of X = 41.60g.
Since, the mass of one mole of X (in grams), i.e. 41.60g is equal to the average or mean mass of one molecule of X (in amu). Therefore, the molecular mass of X is 41.60g.
The only option close to 41.60 is (C).

Hence, the correct option is (C).

Note: Molecular mass is generally expressed in atomic mass unit (amu) whereas the unit for molar mass is gram/mol. We are likely to make calculation mistakes, so carefully solve the question step by step to avoid any errors.