Question

According to government standards, the \(8-h\) threshold limit value is \(5000 \mathrm{ppmv}\) for \(\mathrm{CO}_{2}\) and \(0.1\) ppmv for \(\mathrm{Br}_{2}\) (I ppmv is 1 part by volume in \(10^{6}\) parts by volume). Exposure to either gas for \(8 \mathrm{~h}\) above these limits is unsafe. At STP, which of the following would be unsafe for \(8 \mathrm{~h}\) of exposure? (a) Air with a partial pressure of \(0.2\) torr of \(\mathrm{Br}_{2}\) (b) Air with a partial pressure of \(0.2\) torr of \(\mathrm{CO}_{2}\) (c) \(1000 \mathrm{~L}\) of air containing \(0.0004 \mathrm{~g}\) of \(\mathrm{Br}_{2}\) gas (d) \(1000 \mathrm{~L}\) of air containing \(2.8 \times 10^{22}\) molecules of \(\mathrm{CO}_{2}\)

Short answer

Options (a) and (c) are unsafe for 8-hour exposure.

Step-by-step solution

Convert pressure to ppmv for \(\text{Br}_2\) and \(\text{CO}_2\)

First, convert the given partial pressures to ppmv using the formula: \(\text{ppmv} = \frac{P_{\text{gas}}}{P_{\text{total}}} \times 10^6\). At STP, \(P_{\text{total}}\) is 760 torr.

Evaluate partial pressure of \(\text{Br}_2\) in option (a)

For \(\text{Br}_2\) with a partial pressure of 0.2 torr, \(\text{ppmv} = \frac{0.2}{760} \times 10^6 = 263.16 \) ppmv. The limit is 0.1 ppmv, so option (a) is unsafe.

Evaluate partial pressure of \(\text{CO}_2\) in option (b)

For \(\text{CO}_2\) with a partial pressure of 0.2 torr, \(\text{ppmv} = \frac{0.2}{760} \times 10^6 = 263.16 \) ppmv. The limit is 5000 ppmv, so option (b) is safe.

Determine ppmv from mass for \(\text{Br}_2\) in option (c)

Convert mass of \(\text{Br}_2\) gas to moles: \(0.0004 \text{g} / 159.808 \text{g/mol} = 2.502 \times 10^{-6} \text{mol}\). Calculate ppmv using the ideal gas law: \(\text{moles} = \text{volume} \times \text{molar volume at STP}\) thus, \( \text{moles} = 1000 \text{L}/22.4 \text{L/mol} = 44.64 \text{mol} \). Hence, \(\text{ppmv} = \frac{2.502 \times 10^{-6}}{44.64} \times 10^6 = 0.056 \) ppmv, which is still unsafe as the limit is 0.1 ppmv.

Determine ppmv from number of molecules for \(\text{CO}_2\) in option (d)

Convert number of molecules to moles: \(\frac{2.8 \times 10^{22}}{6.022 \times 10^{23}} = 0.0465 \text{mol}\). Find ppmv: \( 0.0465 \text{mol}/44.64 \text{mol} \times 10^6 = 1041 \) ppmv. This is below the 5000 ppmv limit, so option (d) is safe.

Key concepts

Partial Pressure

Partial pressure refers to the pressure exerted by a single component of a mixture of gases. In a mixture of gases, each gas contributes to the total pressure in proportion to its concentration. The total pressure is the sum of the partial pressures of all gases present.
To calculate partial pressure, we can use Dalton's Law of Partial Pressure, which states that the partial pressure of each gas in a mixture is equal to the mole fraction of that gas times the total pressure:
\[P_{i} = X_{i} \times P_{\text{total}}\]
where:

• \(P_{i}\) = partial pressure of gas \(i\)
• \(X_{i}\) = mole fraction of gas \(i\)
• \(P_{\text{total}}\) = total pressure of the gas mixture

For instance, in the given exercise, we calculated the partial pressure of gases (\(\text{Br}_2\) and \(\text{CO}_2\)) in a mixture to determine if the levels were unsafe. We used the formula:
\[\text{ppmv} = \frac{P_{\text{gas}}}{P_{\text{total}}} \times 10^6\] where the total pressure \(P_{\text{total}}\) at standard temperature and pressure (STP) was taken as 760 torr.

Molar Volume

Molar volume is the volume occupied by one mole of a substance (usually a gas) at a given temperature and pressure. At standard temperature and pressure (STP), which is 0°C (273.15 K) and 1 atm (760 torr), the molar volume of an ideal gas is 22.4 liters per mole.
Molar volume can be used to convert between the amount of substance in moles and the volume it occupies. This concept is particularly useful when dealing with gases, as gases expand to fill their containers.
In the exercise, molar volume was used to calculate the concentration of gas in ppmv from its mass. For example, we calculated the molar volume at STP:
\[ \text{moles} = \frac{\text{volume}}{\text{molar volume at STP}} \]
Using this formula, we found the mole conversion for \(\text{Br}_2\) gas and subsequently calculated ppmv values to assess if the concentration exceeded safe thresholds.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the behavior of an ideal gas. It relates the pressure, volume, number of moles, and temperature of a gas using the formula:
\[ PV = nRT \]
where:

• \(P\) = pressure of the gas
• \(V\) = volume of the gas
• \(n\) = number of moles of gas
• \(R\) = ideal gas constant (\(0.0821 \text{ L·atm/(mol·K)}\) or other units as appropriate)
• \(T\) = temperature in Kelvin

The Ideal Gas Law assumes no interactions between gas molecules and that the volume occupied by gas molecules is negligible compared to the container volume.
This law allowed us to convert the volume and temperature conditions to a mole count in the given exercise. For example, for 1000 L of air containing \(\text{Br}_2\), we used ideal gas principles to determine the moles and subsequently the concentration in ppmv. For \(\text{CO}_2\), we converted the given number of molecules to moles and then used the Ideal Gas Law to find ppmv.