Question

The price of gold on a certain day in 2004 was \(\$ 315\) per troy ounce. How much did \(1.00 \mathrm{~g}\) of gold cost that day? \((1\) troy ounce \(=31.03 \mathrm{~g} .)\)

Short answer

The price of 1.00 g of gold on that day was approximately \$10.15.

Step-by-step solution

Identify the known values

First, identify the known values from the question. Here, you know the price of a troy ounce of gold is \$315, and 1 troy ounce is equivalent to 31.03 grams.

Calculate the cost per gram

Calculate the cost per gram by dividing the cost of 1 troy ounce by the equivalent grams. This can be expressed as \(\frac{315}{31.03}\).

Round off

The result from Step 2 will most likely be a long decimal. Round this off to the second decimal place to determine the cost per gram in dollars, which gives a more realistic price value.

Key concepts

Stoichiometry

Stoichiometry is the backbone of chemical reactions, measuring the quantitative relationships amongst substances as they participate in various chemical reactions. Think of it as a recipe for a delicious meal—stoichiometry tells you exactly how much of each ingredient you need. When you're given a problem that asks you how much of one substance results from a reaction or how much is needed to produce a certain amount of another substance, you're stepping into the world of stoichiometry.

Our core exercise isn't explicitly about a reaction, but it still calls on basic stoichiometric principles by assessing quantities and conversions, akin to determining how much of one ingredient (gold) you can get for a set amount of money at a given unit price. Even when calculating cost per gram for a precious metal, stoichiometry helps ensure you're not overspending or miscalculating your need.

Unit Conversion

Unit conversion is a fundamental skill in chemistry word problems and involves translating a quantity expressed in one unit of measure into another. This is critical because measurements can be represented in multiple units, and it's important to be able to convert between them to communicate effectively and solve problems accurately.

Returning to the exercise at hand, the need for unit conversion arises from the price being given per troy ounce, but our requirement is to find the cost of 1.00 gram. Knowing that 1 troy ounce is equal to 31.03 grams is our conversion factor. It's like knowing how many slices of bread are in one loaf; once you know this, you can find out various quantities—be it slices, loaves, or even crumbs! In the realm of chemistry, this concept often extends to converting between moles, liters, grams, and atoms or molecules.

Chemical Mathematics

Chemical mathematics or 'chem-math' is what happens when we apply mathematical operations to solve chemical problems. It encompasses the calculations involved in stoichiometry, unit conversions, and much more. It's the numerical side of chemistry, blending the precision of math with the experimental nature of science.

To make these calculations more palatable, let's bring it back to our example. Here, the mathematical operation needed is division, where we divide the cost of gold per troy ounce by the number of grams in one troy ounce. This seemingly simple operation is vital in accurately determining the price per gram of gold. Remember, when it comes to problems involving money and precious substances like gold, getting the math right isn't just academic—it has real-world implications.