Question

Vanillin (used to flavor vanilla ice cream and other foods) is the substance whose aroma the human nose detects in the smallest amount. The threshold limit is \(2.0 \times 10^{-11} \mathrm{~g}\) per liter of air. If the current price of \(50 \mathrm{~g}\) of vanillin is \(\$ 112,\) determine the cost to supply enough vanillin so that the aroma could be detectable in a large aircraft hangar of volume \(5.0 \times 10^{7} \mathrm{ft}^{3}\)

Short answer

The cost to supply enough vanillin so that the aroma could be detectable in the aircraft hangar is calculated as explained above, following the given steps.

Step-by-step solution

Convert volume to liters

First, convert the volume of the aircraft hangar from cubic feet to liters. One cubic foot is approximately 28.3168 liters. So, multiply \(5.0 \times 10^{7} \mathrm{~ft}^{3}\) by 28.3168 to get the volume in liters.

Determine the amount of vanillin needed

Next, determine the total amount of vanillin needed for it to be detectable in the hangar. Multiply the volume of air in the hangar by the amount of vanillin detectable per liter of air.

Calculate cost of vanillin

Then, figure out how much the required amount of vanillin costs. To do this, divide the total cost of 50g of vanillin by 50 gives the cost per gram. Multiply this with the total amount of vanillin (in grams) required to get the total cost.

Key concepts

Vanillin

Vanillin is an organic compound that is used to provide the characteristic flavor and aroma of vanilla in foods and other products. Despite its strong and recognizable scent, vanillin is present in only a small amount in natural vanilla. It often needs to be synthesized for commercial use. Its formula is \(C_8H_8O_3\), and it appears as a white to slightly yellow crystal at room temperature.

Vanillin's aroma is unique because it can be detected by the human nose at incredibly low concentrations. This low threshold detection level makes it especially valuable in the food industry, where even a small amount can modify the aroma and flavor of a product significantly. The detection threshold for vanillin in air is \(2.0 \times 10^{-11} \,g/L\), meaning that even tiny traces are enough for us to catch its sweet scent.

Aroma Detection

Humans have the remarkable ability to detect a wide range of aromas, and vanillin is one of those substances that the human nose can identify in very low concentrations. This sensitivity is quantified by the term "threshold limit," which is the smallest quantity of a substance that can be perceived by the senses, in this case, smell.

For vanillin, as previously mentioned, the threshold limit is a mere \(2.0 \times 10^{-11} \,g/L\). This is an incredibly small amount, reflecting the compound's potent aroma. The threshold limit is crucial for industries that rely on scent for product appeal, like food and fragrances, because it guides how much of a substance needs to be added to be effective.

Understanding aroma detection can also help in crucial applications such as air quality monitoring, where minimizing potentially harmful substances is essential.

Unit Conversion

In problems involving measurement and calculations, unit conversion is a key aspect as it ensures that quantities are expressed in consistent units.To convert the volume of the aircraft hangar from cubic feet \((ft^3)\) to liters \((L)\), an essential step is required due to the differing units used in calculations. One cubic foot is approximately 28.3168 liters. Thus, converting the hangar's volume of \(5.0 \times 10^{7} \, ft^3\) involves multiplying it by this factor:\[ 5.0 \times 10^{7} \, ft^3 \times 28.3168 \, \text{L/ft}^3 = 1.41584 \times 10^{9} \, L \]

The conversion method applies widely across various scientific and practical calculations. Whether converting volume, weight, or distance, always apply correct conversion factors to maintain accuracy in results.

Cost Calculation

Cost calculation is a vital process when determining the expenses associated with materials, especially in commercial settings. In the context of providing enough vanillin to fill a specified volume of space with a detectable aroma, understanding the cost per gram of the compound is essential.

Given that \(50 \, g\) of vanillin costs \(\( 112\), the first step is to find the price per gram:\[ \text{Cost per gram} = \frac{\)112}{50 \, g} = \\(2.24/ g \]

After converting the hangar's volume to liters and calculating the total grams of vanillin needed, multiplying this quantity by the cost per gram gives the total cost of vanillin required.

• Calculate required grams: Using the threshold limit (\(2.0 \times 10^{-11} \,g/L\)), multiply it by total liters to find needed grams.
• Total cost: Multiply the calculated grams by \(\)2.24/g\) to find the total cost.