Question

Given its notoriety, you might think that uranium is very rare, but it is present at about 2-4 ppm of the earth's crust, which is more abundant than silver or mercury. If the earth's crust is estimated to have a mass of \(8.50 \times 10^{20} \mathrm{~kg}\), what range of mass is thought to be uranium in the crust?

Short answer

The mass of uranium in the earth's crust ranges from \(1.70 \times 10^{15}\) kg to \(3.40 \times 10^{15}\) kg.

Step-by-step solution

Understand the Given Information

We are told that uranium is present at about 2 to 4 parts per million (ppm) in the earth's crust. The total mass of the earth's crust is given as \(8.50 \times 10^{20} \text{ kg}\). We need to find the range of uranium's mass within the crust.

Convert ppm to a Proportion

Parts per million (ppm) refers to out of a million units. Therefore, 2 ppm means 2 parts of uranium per 1,000,000 parts of the earth's crust, or \(2 \times 10^{-6}\) in proportion form. Similarly, 4 ppm is \(4 \times 10^{-6}\) in proportion form.

Calculate Minimum Mass of Uranium

To find the minimum mass of uranium, multiply the total mass of the earth's crust by the lower proportion of uranium (2 ppm): \[(8.50 \times 10^{20} \text{ kg}) \times (2 \times 10^{-6}) = 1.70 \times 10^{15} \text{ kg}\].

Calculate Maximum Mass of Uranium

To find the maximum mass of uranium, multiply the total mass of the earth's crust by the higher proportion of uranium (4 ppm): \[(8.50 \times 10^{20} \text{ kg}) \times (4 \times 10^{-6}) = 3.40 \times 10^{15} \text{ kg}\].

Determine the Range of Uranium Mass

The range of uranium mass in the earth's crust is from \(1.70 \times 10^{15} \text{ kg} \) to \(3.40 \times 10^{15} \text{ kg}\).

Key concepts

Parts Per Million (ppm)

When discussing concentrations, especially in geological contexts, parts per million (ppm) is a common unit used. It helps express very small amounts of substances within larger volumes or masses. Imagine ppm like this: if you have one million grains of sand, and two grains are a different color, those two grains represent 2 ppm.
This unit is particularly handy when describing trace elements in something as massive as the Earth’s crust. In our exercise, uranium's presence is quantified in ppm, meaning it is found in minuscule quantities within the crust's immense total mass.

• 2 ppm equates to 2 units of uranium per million units of the Earth’s crust.
• 4 ppm, consequently, means 4 units of uranium per million units.

This conversion helps to understand how a material that seems scarce can actually be more common than certain metals.

Mass Calculation

Calculating the mass of uranium in Earth's crust involves understanding how to convert ppm into actual mass. We begin by recognizing that ppm is a ratio of the number of uranium parts to one million parts of the Earth's crust. Given the exercise, the Earth's crust has a total estimated mass of \(8.50 \times 10^{20} \text{ kg}\).
To calculate mass:

• For 2 ppm, the calculation is \((8.50 \times 10^{20} \text{ kg}) \times (2 \times 10^{-6})\), equivalent to \(1.70 \times 10^{15} \text{ kg}\).
• For 4 ppm, use \((8.50 \times 10^{20} \text{ kg}) \times (4 \times 10^{-6})\), resulting in \(3.40 \times 10^{15} \text{ kg}\).

This yields a range of uranium mass from \(1.70 \times 10^{15} \text{ kg}\) to \(3.40 \times 10^{15} \text{ kg}\), illustrating how relatively small ppm values can translate into sizeable quantities when applied to large masses like Earth's crust.

Earth's Crust Composition

The Earth's crust is a fascinating yet intricate layer composed of various elements and minerals. It forms the outermost solid shell of our planet, with materials varying from silicates to metals and more exotic minerals. While elements like oxygen, silicon, and aluminum dominate the crust, trace elements like uranium are also present.
In this particular exercise, we are looking at uranium, which is notably more abundant than silver or mercury despite its trace element status.

• Uranium finds its presence in minerals and rocks, contributing to natural radioactivity and heat production within Earth.
• Having a ppm range between 2 and 4 in the Earth’s crust not only signifies its abundance relative to other trace metals but also its significance in geological processes.

Understanding crust composition goes beyond just identifying elements. It involves appreciating how these elements, even in trace amounts, play critical roles in geological and economic processes.