This problem is referring to an anhydrous salt that is derive from a hydrated Epson salt (magnesium sulfate hydrate) and the atom economy is required. However, not enough information is provided, yet we can use the attached file as a reference to provide a methodology of calculation.
To begin with, we need to calculate the [tex]x[/tex] which refers to the degree of hydration of the salt, so that we use the mass before and after heating the sample to figure out the grams and subsequently moles of water that are in there:
[tex]m_w=3.648g-1.782g=1.866g\\\\n_w=0.103molH_2O[/tex]
Next, since 1.782g corresponds to the anhydrous salt, we calculate the moles in such a amount:
[tex]1.782gMgSO_4*\frac{1molMgSO_4}{120.4gMgSO_4}=0.0148mol[/tex]
Then, we divide the moles of water by those of the anhydrous salt to calculte [tex]x[/tex]:
[tex]x=\frac{0.103mol}{0.0148mol} =6.96=7[/tex]
This means the formula of the hydrate is:
[tex]MgSO_4\ 7H_2O[/tex]
Finally, for the atom economy we just divide the molar mass of the anhydrous salt by that of the hydrate:
[tex]A.E=\frac{120.4g/mol}{246.47g/mol} *100\%\\\\A.E=48.8\%[/tex]
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