Question

The abundance of iodine in seawater is \(5.0 \times 10^{-8}\) percent by mass. How many kilograms of seawater must be treated to obtain \(1.0 \mathrm{~g}\) of iodine?

Short answer

2.0 million kg of seawater is needed.

Step-by-step solution

Convert Given Percent to Decimal

Convert the percent abundance of iodine to a decimal for easier calculations. Since the abundance is given as \(5.0 \times 10^{-8}\%\), convert this to a fraction of 100: \[ 5.0 \times 10^{-8}\text{\textperthousand} = 5.0 \times 10^{-10} \]

Recognize Desired Iodine Mass

Identify the mass of iodine needed, which is given as \(1.0 \text{ grams}\). Convert this mass into kilograms: \[ 1.0 \text{ grams} = 0.001 \text{ kilograms} \]

Set Up Proportion for Seawater Mass

Establish a proportion to find the mass of seawater required. The proportion is based on the relation that the mass of iodine in the seawater is \(5.0 \times 10^{-10}\) of the total seawater mass. Let \(M\) denote the mass of seawater needed: \[ 0.001 \text{ kg} = M \times 5.0 \times 10^{-10} \]

Solve for M

Solve the equation for \(M\) to find the mass of seawater needed: \[ M = \frac{0.001 \text{ kg}}{5.0 \times 10^{-10}} \] Calculate the quantity: \[ M = 2.0 \times 10^6 \text{ kg} \]

Key concepts

Percent by Mass

Percent by mass means how much of a substance is present in a mixture compared to the total mass. In the context of seawater iodine extraction, the percent by mass tells us the percentage of iodine in seawater. For example, if we say the abundance of iodine is \(5.0 \times 10^{-8}\%\) by mass, it means there are \(5.0 \times 10^{-8}\) grams of iodine in every 100 grams of seawater. This tiny percentage indicates that iodine is present in very small amounts. Converting this percentage to a decimal (\(5.0 \times 10^{-10}\)) makes calculations easier.

Metric Conversions

Metric conversions help in changing units from one metric unit to another, which is necessary in chemical calculations. In the given exercise, we needed to convert the mass of iodine from grams to kilograms. Remember:

• 1 gram \( = 0.001\) kilograms
• 1 kilogram \( = 1000\) grams

In our example, 1 gram of iodine was converted to 0.001 kilograms to match the standard unit used in the chemical equation. Converting units accurately ensures precise calculation and minimizes errors.

Proportional Reasoning

Proportional reasoning involves setting up equations to show how different quantities relate to each other. It's a critical concept in solving the seawater iodine extraction problem. Here's how it works:

• We establish that the mass of iodine in the seawater is \(5.0 \times 10^{-10}\) of the total seawater mass.
• We know the mass of iodine required: 0.001 kg.

So, we set up a proportion:
\(0.001 \text{ kg} = M \times 5.0 \times 10^{-10}\)
By solving for \(M\), we determine the total mass of seawater needed to extract the desired amount of iodine.

Chemical Calculations

Chemical calculations are essential for deriving meaningful results from percentages and proportions. To find the mass of seawater needed, we solve the proportional equation:
\(M = \frac{0.001 \text{ kg}}{5.0 \times 10^{-10}}\)
This gives:
\[ M = 2.0 \times 10^6 \text{ kg} \]
Thus, we need \(2.0 \times 10^6\) kg (or 2 million kilograms) of seawater to obtain 1 gram of iodine. Understanding chemical calculations helps you interpret and solve real-world problems involving mixture and concentration.