Question

Rivers add salt (sodium chloride, \(\mathrm{NaCl}\) ) to the oceans of the world at a rate of approximately \(2 \times 10^{16} \mathrm{~g} / \mathrm{yr}\). Assume that Earth is a sphere with a diameter of \(8,000 \mathrm{mi}\), \(67 \%\) of which is covered by oceans to a depth of \(1 \mathrm{mi} .\) The average sodium chloride concentration of the oceans is \(3 \% \mathrm{NaCl}\) by mass and the average density of seawater is \(1.03 \mathrm{~g} / \mathrm{cm}^{3}\). If the rate at which the \(\mathrm{NaCl}\) addition to the oceans has been constant, calculate the approximate age of the oceans. Comment on the reasonableness of your answer given that the age of Earth is \(4.5 \times 10^{9}\) yr.

Short answer

The age of the oceans calculated is much less than Earth's age, suggesting that not all salt comes solely from river input.

Step-by-step solution

Calculate Ocean Volume

First, calculate the volume of Earth's oceans. Earth is a sphere with a diameter of 8000 miles, so its radius is 4000 miles. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Convert 1 mile to centimeters (1 mile \( \approx 1.60934 \times 10^5 \) cm). Only \(67\%\) of Earth is covered by oceans of 1 mile depth. The total volume \( V_{Earth} \) in cm³ = \( \frac{4}{3} \pi (4000 \times 1.60934 \times 10^5)^3 \). Multiply this by 0.67 and 1 mile in cm for the depth to find the ocean volume \( V_{oceans} \).

Calculate Ocean Mass

With the volume \( V_{oceans} \) known, multiply it by the density of seawater, \(1.03 \, \text{g/cm}^3\), to find the total mass of the oceans. Mass of oceans \( M_{oceans} = V_{oceans} \times 1.03\, \text{g/cm}^3 \) in grams.

Calculate Total NaCl in Oceans

Since sodium chloride concentration is \(3\%\) by mass in the oceans, the total mass of \(\text{NaCl}\) in the oceans is \(M_{NaCl} = 0.03 \times M_{oceans}\).

Calculate Age of Oceans

The oceans gain \(\text{NaCl}\) at a rate of \(2 \times 10^{16}\, \text{g/yr}\). Therefore, divide the total mass of \(\text{NaCl}\) in oceans by this rate to find the age of the oceans. Age of oceans \( = \frac{M_{NaCl}}{2 \times 10^{16} \text{ g/yr}}\).

Compare with Earth's Age and Comment

Calculate the age from Step 4 and compare it against Earth's age \(4.5 \times 10^9 \) yr. Comment whether the calculated ocean age seems reasonable or not.

Key concepts

Ocean Volume Calculation

Understanding the volume of Earth's oceans is essential to grasp how much space is available for substances like sodium chloride. The Earth is approximated as a sphere with an 8,000-mile diameter, meaning its radius is 4,000 miles. To calculate the ocean's volume, we use the formula for the volume of a sphere:

• \( V = \frac{4}{3} \pi r^3 \) where \( r \) is the radius in centimeters

Since 1 mile converts to about \( 1.60934 \times 10^5 \) cm, we need to use this conversion for accurate calculation. Furthermore, not all of Earth is ocean-covered. Only 67% of it is, and it's estimated that this coverage averages a depth of 1 mile.
Hence, the ocean volume \( V_{oceans} \) is a fraction of the total sphere volume, multiplied by the conversion factor for the depth in centimeters. This step sets the foundation for later calculations, providing the space in which substances like sodium chloride are dissolved.

Sodium Chloride Addition

In the context of ocean chemistry, discovering how much sodium chloride (NaCl) is added to the oceans annually helps understand salinity levels. As rivers flow into oceans, they contribute approximately \( 2 \times 10^{16} \text{ g/year} \) of NaCl. Over time, this consistent addition builds up the salinity seen in seawater today.
While this calculation is theoretical, it assumes a constant rate over geological timescales. Thus, the total mass of sodium chloride in the oceans can be obtained by accounting for its percent concentration, given as 3%.

• Total NaCl in oceans: \( M_{NaCl} = 0.03 \times M_{oceans} \)

This tells us how much sodium chloride is present at any given time, derived from our volume and density calculations.

Density of Seawater

The density of seawater is crucial to assign mass to the calculated ocean volume. With an average density of \(1.03 \text{ g/cm}^3\), we can determine the ocean's total mass. This step involves multiplying the ocean volume by seawater's density to obtain precise results.

• Mass of oceans: \( M_{oceans} = V_{oceans} \times 1.03\, \text{g/cm}^3 \)

This conversion from volume to mass reflects the real weight of seawater, integrating both its aqueous content and dissolved salts. Accordingly, these calculations lead directly into the assessment of total dissolved sodium chloride.
Mas of the oceans is a stepping stone to calculating how much salt has accumulated over time and aids in determining the potential age of the oceans.

Earth's Age Comparison

Comparing the calculated age of oceans to that of Earth offers insight into geological processes and constraints of our models. According to the solution, dividing the total sodium chloride mass by its annual addition rate gives the ocean's age.

• Ocean Age: \( \text{Age} = \frac{M_{NaCl}}{2 \times 10^{16} \text{ g/yr}} \)

This theoretical age can then be contrasted with Earth's widely accepted age of \(4.5 \times 10^9\) years. If the ocean's calculated age is significantly less, it suggests the assumptions (constant NaCl addition, initial conditions, etc.) might not fully capture Earth's complex history.
Such exercises highlight the dynamic nature of Earth’s geological and hydrological processes and the necessity of continually refining our models and assumptions.