Question

Atmospheric scientists often use mixing ratios to express the concentrations of trace compounds in air. Mixing ratios are often expressed as ppmv (parts per million volume): ppmv of \(X=\frac{\text { vol of } X \text { at STP }}{\text { total vol of air at STP }} \times 10^{6}\) On a certain November day, the concentration of carbon monoxide in the air in downtown Denver, Colorado, reached \(3.0 \times 10^{2}\) ppmv. The atmospheric pressure at that time was 628 torr and the temperature was \(0^{\circ} \mathrm{C}\). a. What was the partial pressure of \(\mathrm{CO}\) ? b. What was the concentration of \(\mathrm{CO}\) in molecules per cubic meter? c. What was the concentration of \(\mathrm{CO}\) in molecules per cubic centimeter?

Short answer

a. The partial pressure of CO is \(2.478 × 10^{-4} \text{atm}\) b. The concentration of CO is \(6.68 × 10^{21} \text{molecules/m}^3\) c. The concentration of CO is \(6.68 × 10^{15} \text{molecules/cm}^3\)

Step-by-step solution

Calculate the total pressure in atm

First, we need to convert the total atmospheric pressure given in torr to atmospheres (atm). To do this, we use the conversion factor 1 atm = 760 torr: Total Pressure (atm) = \(\frac{628\,\text{torr}}{760\,\text{torr/atm}} = 0.826\,\text{atm}\)

Calculate the partial pressure of CO

We now use the mixing ratio to find the partial pressure of CO: Partial Pressure of CO = Total Pressure × Mixing Ratio(ppmv) × \(\frac{1}{10^6}\) Partial Pressure of CO = \(0.826\,\text{atm} × 3.0 × 10^{2}\, \text{ppmv} × \frac{1}{10^6} = 2.478 × 10^{-4}\, \text{atm}\) b. Find the concentration of CO in molecules per cubic meter

Use the Ideal Gas Equation

Now, we will use the Ideal Gas Equation and Avogadro's number to find the concentration of CO in the air: \(PV = nRT\) \(n = \frac{PV}{RT}\) Here, P is the partial pressure of CO, V is the volume, n is the number of moles, R is the universal gas constant and T is the temperature in Kelvin. The temperature in Kelvin is 273 K (0°C + 273).

Convert pressure to Pascal

First, we need to convert the partial pressure of CO from atm to Pascal (Pa) by using the conversion factor 1 atm = 101325 Pa: Partial Pressure of CO (Pa) = \(2.478 × 10^{-4}\, \text{atm} × 101325\, \text{Pa/atm} = 25.10\, \text{Pa}\)

Calculate the number of moles of CO per cubic meter

Now, we can calculate the number of moles of CO per cubic meter using the Ideal Gas Equation. We assume a volume of 1 cubic meter (m³) for convenience: \(n = \frac{PV}{RT} = \frac{(25.10\, \text{Pa})(1\, \text{m}^3)}{(8.314\, \text{J/mol·K})(273\, \text{K})} = 0.0111\, \text{mol/m}^3\)

Calculate the concentration of CO in molecules per cubic meter

Now, we will multiply the moles of CO per cubic meter by Avogadro's number (6.022 x 10²³ molecules/mol) to find the concentration of CO in molecules per cubic meter: Concentration of CO (molecules/m³) = \(0.0111\, \text{mol/m}^3 × 6.022 × 10^{23}\, \text{molecules/mol} = 6.68 × 10^{21}\, \text{molecules/m}^3\) c. Find the concentration of CO in molecules per cubic centimeter

Convert the concentration to molecules per cubic centimeter

Finally, we convert the concentration of CO from molecules per cubic meter to molecules per cubic centimeter by using the conversion factor 1 m³ = 10⁶ cm³: Concentration of CO (molecules/cm³) = \(6.68 × 10^{21}\, \text{molecules/m}^3 × \frac{1\, \text{cm}^3}{10^6\, \text{m}^3} = 6.68 × 10^{15}\, \text{molecules/cm}^3\) So, the answers are: a. The partial pressure of CO is \(2.478 × 10^{-4}\, \text{atm}\) b. The concentration of CO is \(6.68 × 10^{21}\, \text{molecules/m}^3\) c. The concentration of CO is \(6.68 × 10^{15}\, \text{molecules/cm}^3\)

Key concepts

Mixing Ratios

Mixing ratios are commonly used in atmospheric chemistry to describe concentrations of gases in the atmosphere. A mixing ratio expresses the ratio of the volume of a trace gas to the volume of air, often denoted as parts per million by volume (ppmv). In simpler terms, if you have a million molecules of air, how many of those are a specific gas? This helps scientists quantify small constituents in the air.

• Calculating Mixing Ratios: The formula is given by \( \\text{ppmv of } X = \frac{\text{volume of } X }{\text{total volume of air}} \times 10^6 \).
• Relevance: Mixing ratios are crucial when calculating partial pressures or converting concentrations between different units. They give a standardized way to describe concentrations regardless of temperature and pressure conditions.

Understanding mixing ratios enables you to easily convert between different atmospheric measurements, helping make sense of global atmospheric changes.

Partial Pressure

Partial pressure is a key concept in understanding how different gases behave in a mixture like the Earth's atmosphere. It refers to the pressure that a gas in a mixture would exert if it alone occupied the entire volume at the same temperature. Knowing the partial pressure allows you to determine how much of the total pressure of the atmosphere is due to a specific gas.To calculate partial pressure:

• Formula: Use the equation \( \text{Partial Pressure} = \text{Total Pressure} \times \text{Mixing Ratio} \times \frac{1}{10^6} \).
• Importance: Partial pressure calculations are critical in chemistry and environmental science to predict reactions and understand atmospheric behavior.

This concept bridges physical chemistry with real-world applications like predicting weather patterns or pollution levels.

Ideal Gas Law

The Ideal Gas Law is a powerful equation that connects pressure, volume, temperature, and number of moles of a gas. It is expressed as \( PV = nRT \), where:

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles of gas,
• \( R \) is the universal gas constant (8.314 J/mol·K),
• \( T \) is the temperature in Kelvin.

Key points to consider:

• Conversions: Remember to convert all values to the SI units before using the equation. Temperature must be in Kelvin, and pressure often in Pascals. This avoids calculation errors.
• Applications: It helps calculate unknown properties of gases in various conditions, useful for determining concentrations or predicting gas behavior.

By applying the Ideal Gas Law effectively, you can solve many real-world problems involving gases.

Concentration Conversion

Converting concentrations from one unit to another is an essential skill in atmospheric chemistry. In this context, converting between molecules per cubic meter and per cubic centimeter is a common task.Here's how you can convert concentrations:

• Conversion Factor: Utilize the factor that 1 cubic meter equals \(10^6\) cubic centimeters.
• Step-by-step: Multiply the concentration in molecules/m³ by \(\frac{1\, \text{cm}^3}{10^6\, \text{m}^3} \) to convert it to molecules/cm³.

These conversions are crucial when comparing measurements that use different units, ensuring consistency and accuracy in calculations. This expertise enhances your ability to analyze scientific data effectively, leading to better understanding and outcomes in your studies or research.