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): $$\text { ppmv of } X=\frac{\text { vol of } X \text { at } \mathrm{STP}}{\text { total vol of air at } \mathrm{STP}} \times 10^{6}$$ 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 \(2.479 \times 10^{-4}\) atm. b. The concentration of CO in molecules per cubic meter is \(6.558 \times 10^{18}\) molecules/m³. c. The concentration of CO in molecules per cubic centimeter is \(6.558 \times 10^{12}\) molecules/cm³.

Step-by-step solution

Calculate the partial pressure of CO

We can calculate the partial pressure of CO by first converting the atmospheric pressure (in torr) to atm. 1 atm = 760 torr So, $$ P_\text{total} = \frac{628 \ \text{torr}}{760 \ \text{torr/atm}} = 0.8263 \ \text{atm} $$ Now to find the partial pressure of CO in the air, we use the mixing ratio formula given: $$ \text{ppmv of } X = \frac{\text { vol of } X \text { at STP}}{\text { total vol of air at STP}} \times 10^{6} $$ Rearrange the formula to isolate the partial pressure of CO: $$ P_\text{CO} = \frac{\text{ppmv of CO}}{10^6} \times P_\text{total} $$ Plug in the provided ppmv value and the total pressure: $$ P_\text{CO} = \frac{3.0 \times 10^2}{10^6} \times 0.8263 \ \text{atm} $$ Calculate the partial pressure of CO: $$ P_\text{CO} = 2.479 \times 10^{-4} \ \text{atm} $$ #a. The partial pressure of CO is \(2.479 \times 10^{-4}\) atm.

Convert pressure to pascals and find moles of CO per cubic meter

In order to find the concentration of CO in molecules per cubic meter and per cubic centimeter, we need to convert the pressure to pascals and find the moles of CO per unit volume. 1 atm = 101325 Pa So, $$ P_\text{CO} = 2.479 \times 10^{-4} \ \text{atm} \times \frac{101325 \ \text{Pa}}{1 \ \text{atm}} = 25.11 \ \text{Pa} $$ Use the Ideal Gas Law, \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature. We are given the temperature in Celsius; we need to convert it to Kelvin: $$ T = 0^{\circ} \text{C} + 273.15 = 273.15 \ \text{K} $$ Rearrange the Ideal Gas Law formula to find moles per unit volume, \(n/V\): $$ \frac{n}{V} = \frac{P}{RT} $$ Use the ideal gas constant, \(R = 8.314\) J/(mol K), and plug in the values: $$ \frac{n}{V} = \frac{25.11 \ \text{Pa}}{8.314 \text{J/(mol K)} \times 273.15 \ \text{K}} $$ Calculate the moles per cubic meter: $$ \frac{n}{V} = 1.089 \times 10^{-5} \ \text{mol/m}^3 $$

Calculate molecules of CO per cubic meter and per cubic centimeter

We will now find the concentration of CO in molecules per cubic meter and per cubic centimeter using the moles per cubic meter value and Avogadro's number. Avogadro's number \(N_\text{A} = 6.022 \times 10^{23}\) molecules/mol Multiply moles per cubic meter by Avogadro's number to find molecules per cubic meter: $$ \text{molecules/m}^3 = \left( 1.089 \times 10^{-5} \ \text{mol/m}^3 \right) \times \left( 6.022 \times 10^{23} \ \text{molecules/mol} \right) = 6.558 \times 10^{18} \ \text{molecules/m}^3 $$ Now convert molecules per cubic meter to molecules per cubic centimeter by using: 1 m³ = 1,000,000 cm³ $$ \text{molecules/cm}^3 = \frac{6.558 \times 10^{18} \ \text{molecules/m}^3}{1,000,000 \ \text{cm}^3/\text{m}^3} = 6.558 \times 10^{12} \ \text{molecules/cm}^3 $$ #b. The concentration of CO in molecules per cubic meter is \(6.558 \times 10^{18}\) molecules/m³. #c. The concentration of CO in molecules per cubic centimeter is \(6.558 \times 10^{12}\) molecules/cm³.

Key concepts

Partial Pressure

The concept of partial pressure is an essential aspect of the Ideal Gas Law and pertains to the pressure exerted by a single component in a mixture of gases. In this context, each gas in a mixture behaves independently, as if it were the only gas occupying the entire volume. To find the partial pressure of a specific gas, like carbon monoxide (CO), you need to consider its proportion in the overall mixture.

Partial pressure is calculated using the formula involving the mixing ratio, which is expressed in parts per million volume (ppmv). The key formula used is:

• \(P_\text{CO} = \left(\frac{\text{ppmv of CO}}{10^6}\right) \times P_\text{total}\)

This formula considers both the mixing ratio and the total atmospheric pressure. In our example, Denver's atmospheric pressure was converted from torr to atmospheres to find the partial pressure of CO.
As a result, even in a busy urban environment, the partial pressure of trace gases like CO can be very small, often in the order of micro-atmospheres, indicating the low concentration of these gases in the air.

Mixing Ratios

Mixing ratios are a way to express the concentration of gases within a mixture, typically the atmosphere. They're crucial when dealing with gases like carbon monoxide, which are present in tiny amounts. The key advantage of using mixing ratios, such as ppmv, is their ability to standardize concentrations regardless of temperature and pressure changes.
When we talk about ppmv, or parts per million by volume, we refer to the volume of a gas within one million units of all gases. This measure is particularly helpful for expressing very diluted gases within a mixture, enabling scientists to compare concentrations across different contexts or locations.
To calculate mixing ratios, the general formula is used:

• \(\text{ppmv of } X = \frac{\text{vol of } X \text{ at STP}}{\text{total vol of air at STP}} \times 10^6\)

This formula helps isolate the portion of the gas in the atmospheric mixture, a critical step in identifying its partial pressure and concentration.

Concentration Calculation

Calculating the concentration of a gas, such as carbon monoxide, in terms of molecules per volume involves several steps and conversions. After determining the partial pressure, using the Ideal Gas Law is a practical way to find the number of moles. This step bridges the gap between abstract pressure readings and tangible molecular counts.
The Ideal Gas Law, expressed as \(PV = nRT\), helps compute moles per unit volume. Here, \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant (8.314 J/(mol K)), and \(T\) is temperature in Kelvin. Rearranging the formula, we find moles per cubic meter:

• \(\frac{n}{V} = \frac{P}{RT}\)

With this value, multiplying by Avogadro's number (\(6.022 \times 10^{23}\) molecules/mol) converts moles to molecules, providing a concrete measure of concentration in molecules per cubic meter.
Finally, a further conversion adjusts this concentration to a molecules per cubic centimeter basis by recognizing that 1 m³ equals 1,000,000 cm³.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and physics, connecting macroscopic measurements to molecular scale quantities. Defined as \(6.022 \times 10^{23}\) molecules per mole, it allows us to understand molecular composition in terms of real-world quantities.
In the context of concentration calculations, this constant is invaluable. Once the moles per unit volume are found using the Ideal Gas Law, multiplying them by Avogadro's number converts this into an absolute number of molecules.

• For instance, if you have calculated a concentration in moles per cubic meter, multiplying this value by Avogadro's number gives the number of molecules per cubic meter.

This translation from moles to molecules enables comparison and understanding of gas concentrations in more intuitive, tangible numbers, facilitating us to grasp the actual quantities of different gases in a given volume, such as a cubic meter or a cubic centimeter.