All the fertilizers listed below contribute nitrogen to the soil. If all these fertilizers are sold for the same price per gram of nitrogen, which will cost the least per 50 -lb bag? urea, \(\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}\) ammonia, \(\mathrm{NH}_{3}\) ammonium nitrate, \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) guanidine, \(\mathrm{HNC}\left(\mathrm{NH}_{2}\right)_{2}\)

Use the periodic table to find the atomic weights of the elements within each fertilizer, and then add these weights together to obtain the molecular weight of the compound. Molecular weight of urea: $$(\mathrm{NH}_{2})_{2}\mathrm{CO} = 2(\mathrm{N} + 2\mathrm{H}) + \mathrm{C} + \mathrm{O} = 2(14 + 2) + 12 + 16 = 60$$ Molecular weight of ammonia: $$\mathrm{NH}_{3} = \mathrm{N} + 3\mathrm{H} = 14 + 3 = 17$$ Molecular weight of ammonium nitrate: $$\mathrm{NH}_{4}\mathrm{NO}_{3} = \mathrm{N} + 4\mathrm{H} + \mathrm{N} + \mathrm{O} + 2\mathrm{O} = 2\mathrm{N} + 4\mathrm{H} + 3\mathrm{O} = 2(14) + 4 + 3(16) = 80$$ Molecular weight of guanidine: $$\mathrm{HNC}\left(\mathrm{NH}_{2}\right)_{2} = \mathrm{H} + \mathrm{N} + \mathrm{C} + 2(\mathrm{N} + 2\mathrm{H}) = 1 + 14 + 12 + 2(14 + 2) = 59$$

Count the nitrogen atoms in each fertilizer molecule: Urea: 2 nitrogen atoms Ammonia: 1 nitrogen atom Ammonium nitrate: 2 nitrogen atoms Guanidine: 3 nitrogen atoms

Multiply the number of nitrogen atoms by the molecular weight of nitrogen (14) and divide by the molecular weight of the fertilizer compound to find the weight of nitrogen content in each fertilizer: Urea: \(2(14) / 60 = 0.467\) Ammonia: \(1(14) / 17 = 0.824\) Ammonium nitrate: \(2(14) / 80 = 0.350\) Guanidine: \(3(14) / 59 = 0.712\)

Multiply the weight of nitrogen content in each fertilizer (in lb) by the price per gram of nitrogen: Cost of urea per 50-lb bag: \(50(0.467) = 23.35\) Cost of ammonia per 50-lb bag: \(50(0.824) = 41.20\) Cost of ammonium nitrate per 50-lb bag: \(50(0.350) = 17.50\) Cost of guanidine per 50-lb bag: \(50(0.712) = 35.60\) The ammonium nitrate fertilizer will cost the least per 50-lb bag, as its cost is \(17.50\).