Question

Magnesium occurs in seawater to the extent of \(1.4 \mathrm{g}\) magnesium per kilogram of seawater. What volume of seawater, in cubic meters, would have to be processed to produce \(1.00 \times 10^{5}\) tons of magnesium \((1 \text { ton }=2000 \mathrm{lb}) ?\) Assume a density of \(1.025 \mathrm{g} / \mathrm{mL}\) for seawater.

Short answer

The volume of seawater needed to produce \(1.00 \times 10^{5}\) tons of magnesium is \(1.77 \times 10^{11}\) cubic meters.

Step-by-step solution

Convert tons to grams

In this step, convert the desired amount of magnesium from tons to grams. Note that \(1 \text { ton } = 2000 \text { lb }\) and \(1 \text { lb } = 453.592 \text { g }\). So you multiply \(1.00 \times 10^{5}\) tons by 2000 and then by 453.592 to get the result in grams.

Calculate the amount of seawater needed

After converting the desired amount of magnesium to grams, you can now calculate the required amount of seawater. Given that 1.4g of magnesium is present per kg of seawater, you divide the total amount of magnesium (in grams) by 1.4 to get the required amount of seawater in kg.

Convert kilograms to cubic meters

As the density of seawater is given as \(1.025 \text { g/mL }\) and we know that \(1 \text { kg } = 1000 \text { g }\) and \(1 \text { mL } = 0.001 \text { m$^3$ }\), we can easily convert the required amount of seawater from kg to cubic meters by multiplying by \( \frac{1}{1.025} \) and then by 0.001.

Key concepts

Magnesium Extraction from Seawater

Magnesium is an important element used in a variety of industries, ranging from construction to medicine. One of the prominent sources of magnesium is seawater, which, despite its dilute concentration, offers a vast supply due to the sheer volume of oceans. The concentration of magnesium in seawater is approximately 1.4 grams per kilogram. This means for every kilogram of seawater, only a tiny fraction is magnesium. The process of extracting magnesium from seawater involves several stages, typically starting with the precipitation of magnesium compounds and followed by their reduction. Each step in the extraction requires careful handling and efficient processing to ensure that magnesium is isolated effectively.
To understand the scale of extraction, consider the volume of seawater needed to collect a substantial amount of magnesium, such as 100,000 tons. This reveals the massive nature of industrial-scale extraction processes. Such large-scale extractions must balance economic feasibility with environmental sustainability, ensuring minimal ecological disruption.

Density Calculations and Seawater

Density is a key concept in chemistry and physics, defined as mass per unit volume. It helps us understand how concentrated a substance is, in this case, seawater. Seawater has a density of about 1.025 grams per milliliter, which is slightly denser than pure water due to the salts and minerals dissolved in it. Knowing the density allows us to calculate how much space a given mass of seawater will occupy, which is crucial for industrial applications like magnesium extraction.
For example, to determine the volume of seawater needed to obtain a certain mass of magnesium, we first calculate how many grams of seawater corresponds to the amount we need by utilizing the provided density. We then convert this mass into a volume using the formula:
- Mass = Density × Volume
By rearranging this formula to find volume, we can accurately decide on the capacity needed for processing. In our example with seasalt, it was essential to consider density for converting kilograms to cubic meters, ensuring correct resource allocation for the extraction process.

Unit Conversion in Seawater Processes

Unit conversion is a fundamental skill in chemistry and engineering, especially in scenarios involving large quantities or volumes, such as seawater chemistry. Converting between units ensures compatibility and correct assessments during processes. In the case of magnesium extraction, we often start with mass measurements in tons, which need to be converted into grams for precise calculations. Since 1 ton equals 2,000 pounds and further conversion through multiplying by 453.592 gives the result in grams, these conversions standardize the data for calculations.
Similarly, when calculating the mass and volume of seawater required, conversions between different units like kilograms to cubic meters come into play. One kilogram equals 1,000 grams, while 1 milliliter is equivalent to 0.001 cubic meters. Employing these conversions ensures accurate, seamless calculations. For industrial-scale tasks like these, accuracy in unit conversion is vital, as small errors can lead to significant discrepancies in large-scale resource estimation.