Question

In the year 2000 , the Guinness Book of World Records called ethyl mercaptan, \(\mathrm{C}_{2} \mathrm{H}_{6} \mathrm{S}\), the smelliest substance known. The average person can detect its presence in air at levels as low as \(9 \times 10^{-4} \mu \mathrm{mol} / \mathrm{m}^{3} .\) Express the limit of detectability of ethyl mercaptan in parts per billion (ppb). (Note: 1 ppb \(\mathrm{C}_{2} \mathrm{H}_{6} \mathrm{S}\) means there is \(1 \mathrm{g}\) \(\mathrm{C}_{2} \mathrm{H}_{6} \mathrm{S}\) per billion grams of air.) The density of air is approximately \(1.2 \mathrm{g} / \mathrm{L}\) at room temperature.

Short answer

The limit of detectability of ethyl mercaptan in air is approximately 67.4 ppb

Step-by-step solution

Determine Molar Mass of Ethyl Mercaptan

Ethyl mercaptan is composed of Carbon (C), Hydrogen (H) and Sulfur (S). The molar masses of these elements are about 12 g/mol, 1g/mol and 32g/mol respectively. Therefore, the molar mass of ethyl mercaptan, \(C_{2} H_{6} S\), is calculated as \(2(12g/mol) + 6(1g/mol) + 32g/mol = 62 g/mol \)

Convert Micro-moles to Grams

To convert from moles to grams, we multiply the quantity in moles by molar mass. Given that ethyl mercaptan can be detected at \(9 × 10^{-4} µmol/m^{3}\), subtract 6 from the exponent of 10 to convert µmol to mol, and multiply by molar mass to convert to grams. Hence the detectability limit of ethyl mercaptan in air is \( (9 × 10^{-4} mol/m^{3}) × (62 g/mol) = 0.0558 g/m^{3} \)

Convert Cubic Meters to Liters

To convert \(m^{3}\) to liters (L), we use the equivalence \(1 m^{3} = 1000 L\). Therefore, \(0.0558 g/m^{3} = 0.0558 g/1000 L = 5.58 × 10^{-5} g/L \)

Convert Grams per Liter to Parts per Billion

Understanding that 1 ppb means 1g of substance per billion grams of air, to convert \(g/L\) (grams of ethyl mercaptan per Liter of air) into ppb, we multiply \(5.58 × 10^{-5} g/L\) by the density of air \(1.2 g/L\). Therefore our answer is \(5.58 × 10^{-5} g/L * 1 billion L/billion g = 67.4 ppb\)

Key concepts

Molar Mass Calculation

To determine the molar mass of a compound, we need to know the molar masses of the individual elements that make up the compound. Each element has its own unique molar mass, measured in grams per mole (g/mol), which reflects the mass of one mole of atoms of that element.

For example, ethyl mercaptan, represented as \( C_{2}H_{6}S \), consists of carbon (C), hydrogen (H), and sulfur (S). The molar masses are approximately:

• Carbon (C): \(12\, \text{g/mol}\)
• Hydrogen (H): \(1\, \text{g/mol}\)
• Sulfur (S): \(32\, \text{g/mol}\)

To calculate the molar mass of ethyl mercaptan, multiply the molar mass of each element by the number of atoms of that element in the compound, and then sum these values. Thus, the molar mass of ethyl mercaptan is computed as:\[ (2 \times 12\, \text{g/mol}) + (6 \times 1\, \text{g/mol}) + (1 \times 32\, \text{g/mol}) = 62\, \text{g/mol}\]This calculation gives the total mass of one mole of ethyl mercaptan, an important step in converting quantities between moles and grams.

Parts per Billion Conversion

Parts per billion (ppb) is a unit of measurement used to express the concentration of a substance in a solution. It represents one part of the substance per billion parts of the entire mixture, often using grams for convenience in such calculations.

Understanding units like ppb is crucial when dealing with extremely small concentrations, such as detecting trace amounts of ethyl mercaptan in air. In this exercise, we converted from grams per liter (\( g/L \)) to parts per billion, building on how concentration relates to both mass and volume. To achieve this, we first used the known detectability limit of ethyl mercaptan in \( \mu mol/m^{3} \).

To convert this detectability into grams, the molar mass was used. Then, converting cubic meters to liters allowed us to find the measurement in \( g/L \).

Finally, to convert \( g/L \) of ethyl mercaptan to ppb, we account for the density of air by multiplying \( 5.58 \times 10^{-5} \ g/L \) by the standard conversion factor of a billion (as there's no unit change in mass), resulting in \( 67.4 \ ppb \). This unit helps us understand trace concentrations in environmental or laboratory settings.

Density of Air

The concept of density is essential in many scientific and engineering calculations. Density is defined as mass per unit volume, commonly expressed as grams per liter (\( g/L \)) when referring to gases like air. In the context of gases, density can vary with changes in temperature or pressure.

For everyday conditions, we typically use a standardized value which approximates the density of air at room temperature and standard atmospheric pressure. This density is approximately \( 1.2\, \text{g/L} \), a value that simplifies calculations when working with air.

When converting concentrations from grams per cubic meter (\( g/m^3 \)) to parts per billion (ppb), understanding the density of air is a keystone concept. This density factor allows us to bridge the gap between volume-based measures (like liters) and mass-based measures (like grams).

Thus, when converting \(5.58 \times 10^{-5} \ g/L \) to ppb, knowing the density of air as \( 1.2\, \text{g/L} \), helps us precisely express ethyl mercaptan's detectability in air, ensuring our calculations accurately represent real-world conditions.