Question

Carbon monoxide poisons humans by irreversibly binding to hemoglobin in the bloodstream. Although not dangerous in minute concentrations over long exposure times, a CO concentration of \(0.4 \%\) by volume is quickly lethal. If a defective automobile in a sealed garage having a volume of \(4.1 \times 10^{4}\) liters produces \(0.60\) mole of \(\mathrm{CO}\) per minute, how long will it take to reach this lethal concentration of CO? Assume that the volume remains constant at \(27^{\circ} \mathrm{C}\) and the pressure at 1 atm.

Short answer

To find the time it takes to reach the lethal CO concentration, follow these steps: 1. Convert the temperature to Kelvin: \(T = 27 + 273.15 = 300.15 \: K\) 2. Find initial moles of air with Ideal Gas Law: \(n = \frac{PV}{RT} \) 3. Calculate moles needed to reach 0.4% concentration: \(n_{CO} = n \times 0.004\) 4. Calculate the time required: \(time = \frac{n_{CO}}{0.60}\) After completing these calculations, you will obtain the time in minutes.

Step-by-step solution

Convert the temperature to Kelvin

In the exercise, we are told that the temperature remains constant at 27°C. To use the Ideal Gas Law, we need the temperature in Kelvin. We can get this by adding 273.15 to the Celsius temperature: \( T = 27 + 273.15 = 300.15 \: K \)

Find initial moles using given pressure, volume, and temperature

We will use the Ideal Gas Law to find out the initial number of moles of air in the garage: \( PV = nRT \) Rearranging the equation for n: \( n = \frac{PV}{RT} \) We have P = 1 atm, V = \(4.1 \times 10^4 L\), R = 0.0821 \( \frac{L \cdot atm}{mol \cdot K} \) (Ideal Gas Constant), and T = 300.15 K. Plug in these values: \( n = \frac{(1)(4.1 \times 10^4)}{(0.0821)(300.15)} \)

Calculate the moles needed to reach 0.4% concentration

Once we have the initial number of moles, we need to find out how many moles are required to reach a 0.4% concentration of CO by volume. To do this, we multiply the initial number of moles by the percentage concentration: \(n_{CO} = n \times 0.004\)

Calculate the time required to produce the necessary moles of CO

We are given that the defective automobile produces 0.60 mole of CO per minute. To find out how long it will take to reach the lethal concentration of CO, we calculate the time as follows: \(time = \frac{n_{CO}}{production\:rate}\) where the production rate is 0.60 mole/minute. Plug in the values: \(time = \frac{n_{CO}}{0.60}\) Calculate the values from Steps 2 and 3, then plug them into the equation to obtain the time in minutes.

Key concepts

Carbon Monoxide Poisoning

Carbon monoxide (CO) is an invisible, odorless gas that can be incredibly dangerous due to its ability to poison humans. This type of poisoning occurs when someone breathes in CO gas, which can enter the bloodstream quickly. Even though small amounts of CO might not be harmful if exposed over a long period, concentrations that are too high can be lethal very quickly. Here's why CO is so dangerous:

- It can accumulate in spaces where there isn't enough ventilation. - CO prevents the blood from carrying oxygen, which is crucial for our body to function. - Symptoms of poisoning can include dizziness, headaches, weakness, and eventually loss of consciousness or even death if not addressed immediately.

Understanding how CO operates and its effects on the human body is critical for preventing dangerous situations in enclosed spaces like garages.

Hemoglobin Binding

At the core of carbon monoxide's lethality is its ability to bind with hemoglobin, the protein in our blood that carries oxygen. Normally, hemoglobin binds with oxygen molecules, transporting them from our lungs throughout our body to sustain our organs and tissues.

However, CO binds to hemoglobin much more effectively than oxygen, about 200 times more strongly. When CO enters the bloodstream, it attaches to hemoglobin, forming a compound known as carboxyhemoglobin.

Here are the impacts of this binding:

• Oxygen transport is inhibited, leading to oxygen deprivation of critical tissues.
• This strong bond prevents hemoglobin from picking up fresh oxygen from the lungs.
• Once CO attaches, it takes a long time for hemoglobin to release it, compounding the effects of poisoning.

Understanding this mechanism illustrates why even small concentrations of CO can be so hazardous.

Lethal Concentration

In enclosed spaces like a garage, carbon monoxide can reach lethal concentrations quite rapidly. A concentration of CO that is just 0.4% by volume is enough to be fatal. This percentage means that out of every 1000 parts of air, 4 parts are carbon monoxide.

Determining what qualifies as a lethal concentration is vital in assessing the risk in scenarios like being in a garage with a running automobile. Safety guidelines help define these limits to prevent harmful exposure. In this specific case, understanding how quickly a defective car can increase CO levels in an enclosed environment helps us comprehend the importance of ventilation and monitoring. - High concentrations in spaces without airflow can lead to rapid poisoning. - Safety measures should always be in place where combustion engines are used indoors.

Gas Volume Calculation

The Ideal Gas Law is instrumental in calculating the gas volume required to achieve a certain concentration of carbon monoxide. This law combines pressure (P), volume (V), the number of moles (n), and temperature (T) with the gas constant (R), written as:\[ PV = nRT \] Using the Ideal Gas Law helps in understanding how much CO is needed to reach a specific concentration. For example, given:

• A garage volume of \(4.1 \times 10^{4}\) liters
• Standard pressure of 1 atm and a temperature of \(27^{\circ} C\)
• Constant production of CO at 0.60 moles per minute

We can calculate how many moles of CO are required to reach 0.4% concentration and subsequently how long it will take using the production rate. Understanding these calculations ensures safety by informing us how quickly a dangerous environment can develop, providing critical insights into needed precautions.