Question

A concentration of \(8.00 \times 10^{2}\) ppm by volume of \(\mathrm{CO}\) is considered lethal to humans. Calculate the minimum mass of \(\mathrm{CO}\) (in grams) that would become a lethal concentration in a closed room \(17.6 \mathrm{~m}\) long, \(8.80 \mathrm{~m}\) wide, and \(2.64 \mathrm{~m}\) high. The temperature and pressure are \(20.0^{\circ} \mathrm{C}\) and \(756 \mathrm{mmHg}\), respectively.

Short answer

Approximately 381.78 grams of CO is lethal in the given room.

Step-by-step solution

Calculate Room Volume

First, find the volume of the room in cubic meters by multiplying the length, width, and height: \[\text{Volume} = 17.6 \times 8.80 \times 2.64 = 409.15 \, \text{m}^3\]

Convert Volume to Liters

Since the issue involves ppm, convert the volume from cubic meters to liters, knowing that 1 cubic meter equals 1000 liters:\[\text{Volume in liters} = 409.15 \times 1000 = 409150 \, \text{L}\]

Use PPM by Volume

The ppm (parts per million) by volume indicates that for every 1 million parts of air, there are 800 parts of CO. Calculate the volume of CO:\[\text{Volume of CO} = \frac{800}{10^6} \times 409150 = 327.32 \, \text{L}\]

Convert Lethal Volume to Moles

Use the ideal gas law to convert the volume to moles of CO. PV = nRT, solve for n:\[ n = \frac{PV}{RT} \]Ensure units are consistent (R = 0.0821 L atm/mol K, 20°C = 293 K, 756 mmHg = 0.995 atm):\[ n = \frac{0.995 \times 327.32}{0.0821 \times 293} \approx 13.62 \, \text{moles}\]

Convert Moles to Grams

Multiply the moles of CO by its molar mass (28.01 g/mol) to find the mass:\[\text{Mass of CO} = 13.62 \times 28.01 = 381.78 \, \text{grams}\]

Conclusion

The minimum mass of CO that would become a lethal concentration in the room is approximately 381.78 grams.

Key concepts

PPM Concentration

PPM stands for "parts per million," and it's a unit of measurement used to express small concentrations of substances within larger mixtures. When we talk about ppm concentration in terms of gas laws, it refers to how many parts of a gas, such as carbon monoxide (CO), are present in one million parts of air. For example, 800 ppm of CO means there are 800 parts of CO for every million parts of air.
To apply this in a practical scenario, consider a room filled with air. If the air contains CO at a lethal concentration of 800 ppm, this means that if you took a sample of one million air molecules, 800 of them would be CO molecules. The ppm measurement helps in understanding hazardous levels of substances like CO, which can be essential for ensuring safety in closed environments like the room in our exercise.

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry that relates the pressure, volume, temperature, and amount of gas. It's typically expressed mathematically as:\[ PV = nRT \]where:

• P is the pressure
• V is the volume
• n is the number of moles of the gas
• R is the ideal gas constant
• T is the temperature in Kelvin

Using this formula, you can solve for any one of the variables if you know the others. This is particularly useful when converting the volume of a gas to moles, as in our example where we needed to determine the amount of CO within a certain volume of the room under specific temperature and pressure conditions. By rearranging the ideal gas law to solve for n, we can find out how many moles of a gas are present, which is crucial for further calculations such as determining mass.

Molar Calculations

Molar calculations involve determining the amount of a substance in terms of moles, which is a standard scientific unit for measuring large quantities of very small entities like atoms or molecules. One mole corresponds to Avogadro's number, approximately \(6.022 imes 10^{23}\) of these entities, and provides a bridge between the atomic scale and real-world quantities.
In our problem, after finding the number of moles of CO gas using the ideal gas law, the next step was to convert those moles into grams. This requires knowing the molar mass of the substance, which is the mass of one mole of that substance. For CO, the molar mass is approximately 28.01 grams per mole. By multiplying the moles of CO found with the ideal gas law by its molar mass, we transitioned from moles to a more tangible measure of quantity—grams, completing the picture of how much CO would reach a lethal concentration.

Room Volume

Understanding room volume is essential when dealing with problems involving gases and their concentrations. Room volume is simply the amount of space within a room, often calculated using the dimensions of length, width, and height. In our problem, the room has dimensions of 17.6 m long, 8.80 m wide, and 2.64 m high.
To find the volume, you multiply these three dimensions together:\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} = 17.6 \times 8.80 \times 2.64 \]This results in a volume of 409.15 cubic meters. Since gas laws often use liters as a unit, it's necessary to convert this volume to liters by recognizing that 1 cubic meter equals 1000 liters. Therefore, 409.15 cubic meters is equivalent to 409150 liters, which allows us to perform further calculations related to gas concentrations, such as determining ppm. Understanding and accurately calculating room volume is crucial for addressing safety concerns related to air quality and potential exposure to gas hazards.