Question

The volume of seawater on Earth is about \(330,000,000 \mathrm{mi}^{3} .\) If seawater is \(3.5 \%\) sodium chloride by mass and has a density of \(1.03 \mathrm{g} / \mathrm{mL}\), what is the approximate mass of sodium chloride, in tons, dissolved in the seawater on Earth ( 1 ton \(=\) 2000 lb)?

Short answer

To solve this problem, first the total mass of the seawater is calculated using the given volume and density. After that, the mass of the sodium chloride is computed as a percentage of the total seawater mass and finally, this mass is converted from grams to tons using the proper conversion factors.

Step-by-step solution

Find the total mass of seawater

The problem provides the volume of seawater in cubic miles, and the seawater's density in grams per milliliter. Convert the volume of seawater into milliliters \((1 \mathrm{mi}^{3} = 4.16818183 x 10^{15} \mathrm{mL})\), and then multiply by the density to find the total mass of the seawater in grams. Mass = volume x density = \(330,000,000 \mathrm{mi}^{3}\) x \(4.16818183 x 10^{15} \mathrm{mL/mi}^{3}\) x \(1.03 \mathrm{g/mL}\).

Calculate the mass of sodium chloride

We find the mass of sodium chloride in the sea water using the percentage given. According to the problem, seawater is \(3.5%\%\) sodium chloride by mass. That means 3.5g of every 100g of seawater is sodium chloride. So, to find the mass of sodium chloride in the seawater, we only need to find \(3.5\%\) of the total seawater mass calculated in Step 1.

Convert the mass to tons

Lastly, we convert the mass obtained in Step 2 from grams to tons. We know that 1 lb = 453.592g and 2000 lbs = 1 ton. Therefore, using these conversions factors, we transform the mass from grams into tons. Mass (in tons) = Mass (in grams) / \(453.592 \times 2000\) .

Key concepts

Sodium Chloride Mass

Sodium chloride, commonly known as table salt, makes up 3.5% of seawater by mass. This percentage tells us how much salt is present per 100 grams of seawater. To calculate the total mass of sodium chloride in Earth's oceans, you need to first find the entire mass of the seawater using its density and convert it to grams. After determining the total mass of seawater, multiply by 0.035 (which is 3.5% expressed as a decimal) to find the mass of sodium chloride alone. This method uses a direct proportion method where the percentage is converted to a decimal form and applied to the overall mass. By simplifying the calculations this way, you can easily determine the mass of one component within a mixture.

Seawater Density

Density is a measure of how much mass is contained in a given volume. In this problem, the density of seawater is given as 1.03 grams per milliliter (g/mL). This value indicates that every milliliter of seawater weighs 1.03 grams. To use this information, multiply the density by the total volume of seawater (converted into milliliters). This gives the total mass of seawater, which is crucial for further calculations like finding the mass of sodium chloride. Understanding density as a fundamental property helps in converting between mass and volume, allowing for precise calculations in various scientific applications.

Conversion Factors

Conversion factors are constants used to convert one unit of measurement to another. In the context of this problem, they become essential when you need to translate the problem from abstract numbers into concrete, comparable results.

• Cubic miles to milliliters: Seawater volume from cubic miles is converted to milliliters using the conversion factor where 1 cubic mile equals \(4.16818183 \times 10^{15}\) milliliters.
• Grams to tons: Grams are converted to tons using the steps: first convert grams to pounds (1 pound = 453.592 grams), then convert pounds to tons (1 ton = 2000 pounds).

Using correct conversion factors ensure accuracy in calculations and are critical in solving unit-based problems effectively.

Cubic Mile to Milliliter Conversion

To solve problems involving different scales, it's often necessary to convert between units of measurement. In this exercise, the volume of seawater is initially given in cubic miles, a very large unit and not practical for direct use in typical density calculations which use milliliters.One cubic mile equals approximately \(4.16818183 \times 10^{15}\) milliliters. With such a dramatic difference in magnitude, converting cubic miles to milliliters allows for detailed and precise mass calculations using density figures given in grams per milliliter.Converting large-scale geographical measurements like cubic miles into smaller units like milliliters helps integrate vastly different data types into a single computation, providing a clearer picture of the results.