Problem 15 - Chemistry The Central Science

Question

The average concentration of carbon monoxide in air in an Ohio city in 2006 was 3.5 ppm. Calculate the number of CO molecules in 1.0 L of this air at a pressure of 759 torr and a temperature of \(22^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

There are approximately \( 8.64 x 10^{19} \) carbon monoxide (CO) molecules in 1.0 L of this air in an Ohio city in 2006.

Step 1: Convert the temperature to Kelvin

To use the Ideal Gas Law, the temperature must be in Kelvin (K). To convert the temperature from Celsius to Kelvin, use the formula: \[ T(K) = T(^\circ C) + 273.15 \] Plugging in the given temperature, we get: \[ T(K) = 22^\circ C + 273.15 = 295.15 K \]

Step 2: Convert pressure from torr to atm

The Ideal Gas Law requires pressure to be in atmospheres (atm). To convert the pressure from torr to atm, use the conversion factor (1 atm = 760 torr): \[ P(atm) = \frac{P(torr)}{760} \] Given the pressure of 759 torr, we have: \[ P(atm) = \frac{759}{760} = 0.99868 atm \]

Step 3: Calculate moles of air in the 1.0 L volume

Now, we can plug the pressure, volume, and temperature into the Ideal Gas Law to find the moles of air in the 1.0 L volume. The Ideal Gas Law equation is: \[ PV = nRT \] Where: P is the pressure in atm (0.99868 atm), V is the volume in liters (1.0 L), n is the number of moles, R is the ideal gas constant (0.08206 L atm / K mol), and T is the temperature in Kelvin (295.15 K). Rearrange the equation to find n: \[ n = \frac{PV}{RT} \] Now, plug in the values and solve for the number of moles: \[ n = \frac{(0.99868)(1)}{(0.08206)(295.15)} = 0.04097 mol \]

Step 4: Calculate the moles of CO in the 1.0 L volume

Since the concentration of CO is given in ppm (parts per million), we can use this ratio to find the moles of CO in the sample. The ratio is: \[ \frac{moles\ CO}{moles\ total} = \frac{3.5}{10^6} \] Rearrange the equation and plug in the values to find the moles of CO: \[ moles\ CO = \frac{3.5}{10^6} \times 0.04097 = 1.434 x 10^{-4}\ mol\ CO \]

Step 5: Calculate the number of CO molecules

Finally, we can use Avogadro's number (6.022 x 10^23 particles/mol) to convert the moles of CO to the number of CO molecules: \[ molecules\ CO = (1.434 x 10^{-4}\ mol\ CO) (6.022 x 10^{23} particles/mol) \] \[ molecules\ CO = 8.64 x 10^{19} \] So, there are approximately \( 8.64 x 10^{19} \) carbon monoxide (CO) molecules in 1.0 L of this air in an Ohio city in 2006.

Key Concepts

Gas Concentration

Gas concentration refers to the amount of a specific gas present in a mixture of gases. It is typically expressed in parts per million (ppm), which indicates how many parts of the target gas are present per million parts of the overall mixture. In this exercise, the concentration of carbon monoxide (CO) in the air is given as 3.5 ppm. This means that in one million units of air, there are 3.5 units of carbon monoxide.
To calculate the number of CO molecules, the concentration is first used to determine the proportionate amount in moles, which is a fundamental chemical unit for expressing amounts of a substance. By utilizing the ratio of moles CO to moles of total gas, we can work out the actual moles of CO in the specified volume of air.

Pressure Conversion

Pressure conversion is necessary when using the Ideal Gas Law, as it requires the pressure to be expressed in atmospheres (atm). The initial pressure, however, was given in torr, a common unit of pressure in scientific contexts.
To convert from torr to atm, use the conversion factor 1 atm = 760 torr. In the given problem, a pressure of 759 torr must be converted, which is calculated using the formula:

• \( P(atm) = \frac{P(torr)}{760} \)

Plugging in the values, the pressure of 759 torr converts to approximately 0.99868 atm, a value that is crucial for further calculations involving the Ideal Gas Law.

Avogadro's Number

Avogadro's number is an essential constant in chemistry, providing a bridge between the macroscopic and molecular worlds. It is defined as the number of particles (atoms, molecules, ions, etc.) in one mole of a substance, and is approximately \(6.022 \times 10^{23}\) particles/mol.
In this exercise, after determining the moles of carbon monoxide (CO) present in the 1.0 L air sample, Avogadro's number is used to convert these moles into an actual count of CO molecules. By multiplying the calculated moles of CO by Avogadro's number, we find that there are approximately \(8.64 \times 10^{19}\) CO molecules in the sample.

Temperature Conversion

When dealing with gas laws, particularly the Ideal Gas Law, it is essential to express temperature in Kelvin. Kelvin is the SI unit for temperature and relates directly to the energy of molecules within a substance.
To convert temperatures from Celsius to Kelvin, use the equation:

• \( T(K) = T(^{\circ}C) + 273.15 \)

In this problem, the temperature provided is 22°C, which converts to 295.15 K. This converted temperature allows the Ideal Gas Law to be correctly applied, ensuring that calculations involving volume, pressure, and moles conduct accurately.