The total mass of Earth's atmosphere is \(5 \times 10^{18} \mathrm{kg}\). Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) makes up about 0.06 percent of Earth's atmospheric mass. a. What is the mass of \(\mathrm{CO}_{2}\) (in kilograms) in Earth's atmosphere? b. The annual global production of \(\mathrm{CO}_{2}\), is now estimated to be 3 \(\times 10^{13} \mathrm{kg} .\) What annual fractional increase does this represent? c. The mass of a molecule of \(\mathrm{CO}_{2}\) is \(7.31 \times 10^{-26} \mathrm{kg}\). How many molecules of \(\mathrm{CO}_{2}\) are added to the atmosphere each year? d. Why does an increase in \(\mathrm{CO}_{2}\) have such a big effect, even though it represents a small fraction of the atmosphere?

Step by step solution

01

Calculate the mass of CO2 in Earth's atmosphere

To calculate the mass of CO2 in Earth's atmosphere, use the formula: \[ \text{Total mass of CO2} = \text{Total mass of atmosphere} \times \text{Percentage of CO2} \]Given that the total mass of Earth's atmosphere is \(5 \times 10^{18} \text{kg}\) and CO2 makes up 0.06% of this mass, we get:\[ \text{Total mass of CO2} = 5 \times 10^{18} \text{kg} \times \frac{0.06}{100} = 3 \times 10^{15} \text{kg} \]

02

Calculate the annual fractional increase in CO2

First, find the annual CO2 production, which is \(3 \times 10^{13} \text{kg}\). The fractional increase is given by:\[ \text{Fractional increase} = \frac{\text{Annual CO2 production}}{\text{Total CO2 mass in atmosphere}}\]Using the values: \[ \text{Fractional increase} = \frac{3 \times 10^{13} \text{kg}}{3 \times 10^{15} \text{kg}} = 0.01 \]Thus, the annual fractional increase is 0.01 or 1%.

03

Calculate the number of CO2 molecules added each year

To find the number of CO2 molecules added each year, use the formula: \[ \text{Number of molecules} = \frac{\text{Annual CO2 production}}{\text{Mass of one CO2 molecule}}\]Given that the mass of a CO2 molecule is \(7.31 \times 10^{-26} \text{kg}\) and the annual CO2 production is \(3 \times 10^{13} \text{kg}\), we get:\[ \text{Number of molecules added} = \frac{3 \times 10^{13} \text{kg}}{7.31 \times 10^{-26} \text{kg}} \]Using division, we find:\[ \text{Number of molecules added} eq 4.1 \times 10^{38} \]

04

Explain the impact of increased CO2 levels

CO2, while a small fraction of the atmosphere, is very efficient at trapping heat through the greenhouse effect. This means that even small increases in CO2 concentrations can significantly impact global temperatures and climate patterns, leading to various environmental issues.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

carbon dioxide concentration

Carbon dioxide (CO2) concentration in the atmosphere is a crucial aspect of understanding the greenhouse effect and climate change. While CO2 makes up a relatively small fraction of the Earth's atmospheric mass, its concentration has a significant impact. CO2 concentration is typically expressed as a percentage of the total atmospheric mass. For example, in this exercise, CO2 accounts for 0.06% of the Earth's atmosphere. This small percentage might seem insignificant, but it represents a substantial mass of CO2: Using the given values, we calculate the mass of CO2 in the atmosphere to be 3 x 10^{15} kg. Just because its concentration is small does not mean its effects are negligible.

fractional increase

The fractional increase helps us understand how much CO2 is added to the atmosphere annually compared to the current mass. This concept is essential because it shows how rapidly CO2 levels are rising, given the annual production of CO2 is approximately 3 x 10^{13} kg. Calculating the fractional increase involves dividing the annual CO2 production by the total CO2 mass in the atmosphere. In this case, the fractional increase is 0.01 or 1%. This means every year, there is a 1% increase in atmospheric CO2 mass. Even small fractional increases can accumulate over time, leading to significant environmental impacts.

molecular mass

Every molecule of CO2 has a specific mass, contributing to the overall atmospheric mass of CO2. The molecular mass of CO2 is calculated by considering the mass of the carbon atom and the two oxygen atoms it contains. A CO2 molecule weighs approximately 7.31 x 10^{-26} kg. To find out how many individual CO2 molecules are added to the atmosphere each year, we divide the annual CO2 production by the molecular mass of CO2. Using the values from the problem, we find that approximately 4.1 x 10^{38} CO2 molecules are added annually. Understanding the molecular mass of CO2 is vital for grasping how such a small mass per molecule can still lead to significant increases in atmospheric CO2 concentrations.

greenhouse effect

The greenhouse effect is a natural process where certain gases, like CO2, trap heat within the Earth’s atmosphere, keeping the planet warm enough to support life. However, increases in greenhouse gases amplify this effect, leading to higher global temperatures. CO2 plays a critical role in this process. Despite its small concentration, CO2 is very efficient at absorbing and reradiating infrared radiation. Therefore, even small increases in CO2 levels can significantly affect the Earth's climate by enhancing the greenhouse effect. This explains why monitoring CO2 levels and mitigating its increase is crucial to managing global climate change.