Question

Table salt contains \(39.33 \mathrm{~g}\) of sodium per \(100 \mathrm{~g}\) of salt. The U.S. Food and Drug Administration (FDA) recommends that adults consume less than \(2.40 \mathrm{~g}\) of sodium per day. A particular snack mix contains \(1.25 \mathrm{~g}\) of salt per \(100 \mathrm{~g}\) of the mix. What mass of the snack mix can an adult consume and still be within the FDA limit? (Assume three significant figures for 100 g.)

Short answer

An adult can consume up to 608 grams of the snack mix to stay within the FDA sodium intake limit.

Step-by-step solution

Calculate the mass of sodium per gram of snack mix

Determine the amount of sodium in the snack mix per gram by using the sodium-to-salt ratio given for table salt. For every gram of snack mix, there is 1.25 grams of salt. Since there are 39.33 grams of sodium per 100 grams of salt, we convert this to sodium per gram of salt by dividing 39.33 by 100.

Calculate the FDA limit of sodium in terms of snack mix

Use the sodium per gram of salt derived in Step 1 to determine the equivalent in snack mix. The FDA recommends less than 2.40 grams of sodium, which we can convert to mass of snack mix by dividing by the sodium content per gram of snack mix.

Determine the maximum allowable mass of snack mix

Having determined how many grams of sodium there are in each gram of snack mix (from Step 1) and knowing the FDA sodium intake limit, we calculate the mass of snack mix that would provide this amount of sodium. This is done by dividing the FDA sodium intake limit by the sodium content per gram of snack mix to obtain the mass of the mix an adult can consume without exceeding the recommended sodium intake.

Key concepts

Chemical Composition

Chemical composition refers to the identification and proportion of the elements that make up any particular substance. In our daily life, this information is crucial, as it affects everything from nutrition to medicine.

For example, to maintain a healthy diet, we need to understand the chemical composition of our food. This is demonstrated in the exercise where we look at the sodium content in a snack mix. To work with such compositions, we typically deal with percentages and ratios, which tell us how much of a particular element or compound is present in a substance relative to its total mass. In the snack mix scenario, we're given that table salt is 39.33% sodium by mass, meaning that 39.33 grams of sodium are present for every 100 grams of salt.

Limiting Reactants

The concept of limiting reactants is pivotal in stoichiometry, which is the branch of chemistry dealing with the quantitative relationships between reactants and products in a chemical reaction. While the exercise at hand doesn't directly deal with a chemical reaction, the principle of limiting factors is still applicable.

Consider that the 'limiting reactant' in our nutritional example is the amount of sodium that the FDA recommends one should consume - the 'reaction' proceeds only as long as the sodium limit is not exceeded. Translating this into our snack mix problem, the FDA recommended sodium intake limit acts as our 'limiting reactant': it is the factor that limits the amount of snack mix an individual can consume while staying within healthy sodium intake levels.

Mole Concept

The mole concept is a central theme in chemistry that relates mass to the number of atoms, molecules, or formula units of a substance. A mole is defined as the amount of substance containing the same number of discrete entities (atoms, molecules, etc.) as there are atoms in 12 grams of carbon-12.

While the mole is not used explicitly in the exercise, understanding the mole concept enriches our comprehension of proportions in chemical composition. For instance, if we were given the number of moles of sodium instead of mass, we would use the mole concept to convert between moles and grams to solve our problem about the snack mix.

Empirical Formula

The empirical formula is the simplest positive integer ratio of atoms present in a compound. It does not provide information about the actual numbers or arrangement of atoms, but it does give us the proportional relationship of different atoms in the substance.

Our snack analysis does not use the empirical formula directly but understanding it helps in determining chemical compositions. For instance, the empirical formula for table salt is NaCl, which gives us a 1:1 ratio of sodium to chloride ions, implying equal numbers of each type of atom. By knowing the percentage composition and the empirical formula, we could theoretically calculate various aspects of a substance's make-up, such as the mass percentage of sodium in sodium chloride.