Question

The Sacagawea gold-colored dollar coin has a mass of \(8.1 \mathrm{~g}\) and contains \(3.5 \%\) manganese. What is its mass in ounces \((1 \mathrm{lb}=16 \mathrm{oz})\), and how many ounces of Mn are in this coin?

Short answer

The mass of the coin is approximately 0.286 ounces, and it contains about 0.010 ounces of manganese.

Step-by-step solution

Convert grams to pounds

To convert the mass of the coin from grams to pounds, use the conversion factor: \(1 \text{ gram} = 0.00220462 \text{ pounds}\). Thus, \[ 8.1 \text{ grams} \times 0.00220462 \text{ pounds/gram} = 0.017857422 \text{ pounds} \]

Convert pounds to ounces

Since there are 16 ounces in a pound, convert the mass from pounds to ounces: \[0.017857422 \text{ pounds} \times 16 \text{ ounces/pound} = 0.285718752 \text{ ounces} \]

Calculate the mass of manganese in grams

The coin contains 3.5% manganese. Calculate the mass of manganese in grams: \[8.1 \text{ grams} \times 0.035 = 0.2835 \text{ grams} \]

Convert the mass of manganese to pounds

Convert the mass of manganese from grams to pounds: \[0.2835 \text{ grams} \times 0.00220462 \text{ pounds/gram} = 0.00062569667 \text{ pounds} \]

Convert the mass of manganese to ounces

Convert the mass of manganese from pounds to ounces: \[0.00062569667 \text{ pounds} \times 16 \text{ ounces/pound} = 0.01001034672 \text{ ounces} \]

Key concepts

mass conversion

Converting mass from one unit to another is important in science and daily life. This involves using conversion factors to change units, like converting grams to pounds and then to ounces. Conversion factors help you switch between units based on a fixed ratio. To convert the Sacagawea coin from grams to ounces, we first convert to pounds, then to ounces. This method ensures accuracy. Understanding how to apply these conversion factors will enable you to tackle various problems more effectively.

percentage composition

Percentage composition tells you the proportion of a specific substance within a mixture or compound. For the Sacagawea coin, 3.5% of its total mass is manganese (Mn). This means that for every 100 grams of the coin, 3.5 grams are Mn. To find this value, multiply the coin’s total mass by the percentage (as a decimal). This way, you can easily find out how much of one component is present in a mixture. It's a valuable skill in chemistry to understand the make-up of different materials.

manganese in coin

When looking at specific elements in a compound or object, like manganese in the Sacagawea coin, knowing the percentage composition helps. For the coin, identifying 3.5% of its total mass as manganese lets us calculate its amount in grams and ounces. This is done through straightforward multiplication and subsequent unit conversions. Knowledge of the element’s percentage allows you to achieve accurate measurements and understand the chemical character of the object. Manganese is often used in coins due to its physical properties, making it important to know its exact amount.

conversion factors

Conversion factors are essential tools in calculations, allowing you to switch from one unit to another easily. They are ratios based on equivalent values of different units, like 1 pound equaling 16 ounces, or 1 gram being 0.00220462 pounds. To apply a conversion factor, multiply your original measurement by the factor. Make sure to set it up so units you want to cancel out are opposite each other. Mastery of conversion factors ensures accurate and consistent measurements across various scientific fields, and helps in everyday problem-solving scenarios.