Question

The total volume of seawater is \(1.5 \times 10^{21} \mathrm{~L}\). Assume that seawater contains 3.1 percent sodium chloride by mass and that its density is \(1.03 \mathrm{~g} / \mathrm{mL}\). Calculate the total mass of sodium chloride in kilograms and in tons \((1\) ton \(=2000 \mathrm{lb} ; 1 \mathrm{lb}=453.6 \mathrm{~g})\).

Short answer

The total mass of sodium chloride is \(4.79 \times 10^{19}\) kg or \(5.28 \times 10^{18}\) tons.

Step-by-step solution

Convert Volume to Mass

First, convert the total volume of seawater from liters to milliliters since the density is given in grams per milliliter. Since 1 liter equals 1000 milliliters, the volume is \(1.5 \times 10^{21} \text{ L} = 1.5 \times 10^{24} \text{ mL}\). Now, use the density \(1.03 \text{ g/mL}\) to find the total mass: \(\text{mass} = \text{volume} \times \text{density} = 1.5 \times 10^{24} \text{ mL} \times 1.03 \text{ g/mL} = 1.545 \times 10^{24} \text{ g}\).

Calculate Mass of Sodium Chloride

Determine the mass of sodium chloride (NaCl) by using the percentage given. If seawater contains 3.1% sodium chloride by mass, then the mass of NaCl is \(0.031 \times 1.545 \times 10^{24} \text{ g} = 4.7895 \times 10^{22} \text{ g} \).

Convert Mass to Kilograms

Convert the mass of NaCl from grams to kilograms. Since 1 kilogram equals 1000 grams, the mass in kilograms is given by \(4.7895 \times 10^{22} \text{ g} = 4.7895 \times 10^{19} \text{ kg}\).

Convert Mass from Kilograms to Tons

Using the conversion factors provided, first convert kilograms to pounds: \(4.7895 \times 10^{19} \text{ kg} \times \frac{2.20462 \text{ lb}}{1 \text{ kg}} = 1.0565 \times 10^{22} \text{ lb}\). Then, convert pounds to tons: \(\frac{1.0565 \times 10^{22} \text{ lb}}{2000 \text{ lb/ton}} = 5.2825 \times 10^{18} \text{ tons}\).

Key concepts

Density

Density is a fundamental concept in chemistry and physics. It is defined as mass per unit volume. The formula for density is given by \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). In this specific exercise, we use the density of seawater, which means how much mass (in grams) is contained in a certain volume (in milliliters) of seawater. Here, the density is given as 1.03 grams per milliliter.
Understanding density is crucial as it helps to calculate the total mass of a substance when its volume and the density are known. This is particularly useful in solving real-life problems involving large bodies of substances, such as seawater in this exercise. Moreover, density can help in determining whether a substance will float or sink in another, aiding in many industrial applications.

Sodium Chloride

Sodium chloride, commonly known as table salt, is a chemical compound with the formula NaCl. It is one of the most common salts and is essential in the lives of plants and animals. In the exercise, we are considering seawater, which naturally contains sodium chloride. This is important for understanding how mineral and salt concentrations work in different natural bodies of water, such as oceans and seas.
Sodium chloride has a significant presence in seawater, affecting both its chemical and physical properties. Knowing its concentration allows scientists and engineers to predict and utilize these properties, for instance, in desalination or chemical extraction processes. It also has a range of uses, from culinary purposes to de-icing roads, showcasing its importance in everyday life.

Unit Conversion

Unit conversion is the process of changing the measurement of a quantity from one unit to another while maintaining its value. This skill is critical in chemistry as different parameters are often given in various units that need to be aligned for effective calculation.
In this exercise, it was necessary to convert volumes and masses into compatible units. For example, converting liters to milliliters is essential when density is in grams per milliliter. Similarly, converting grams to kilograms or pounds and ultimately to tons helps in understanding massive quantities better. Such conversions ensure calculations are coherent and make communication of results easier for a global audience that might use different measurement systems.

Percentage Calculation

Percentage calculation is widely applied in scientific and everyday problems to determine the ratio of a part to the whole expressed as a percentage. For instance, the exercise states that seawater contains 3.1% sodium chloride by mass.
To calculate this, the total mass of seawater is multiplied by the percentage expressed as a decimal (0.031 in this case) to find the mass of sodium chloride alone. This is a typical procedure in chemistry when dealing with solutions or mixtures, helping to isolate and measure specific components accurately. Such calculations are vital in pharmaceuticals, chemical reactions, and food industries for ensuring the correct proportions of various substances.