Question

Researchers at Anna Gudmundsdottir's laboratory at the University of Cincinnati have been studying extremely reactive chemicals known as radicals. One of the interesting phenomena they have discovered is that these radicals can be chemically attached to fragrance molecules, effectively tethering them to a solution. When light strikes these tethered molecules, the fragrance is released. This property would allow us to produce perfumes, cleansers, and other consumer products that release fragrance only when exposed to light. If limonene, \(\mathrm{C}_{10} \mathrm{H}_{16}\), the molecule that gives fruits their citrus scent, were able to be tethered to one of these radicals and every photon of light would release one molecule of limonene, calculate the time in seconds required to release \(1.00\) picogram of limonene if ambient light releases \(2.64 \times 10^{18} \mathrm{photons} / \mathrm{sec}\).

Short answer

1.67 nanoseconds.

Step-by-step solution

Determine the molar mass of limonene

Limonene has the molecular formula \(\text{C}_{10}\text{H}_{16}\). Calculate the molar mass by adding the atomic masses of carbon and hydrogen: \[ M(\text{C}_{10}\text{H}_{16}) = 10 \times M(\text{C}) + 16 \times M(\text{H}) = 10 \times 12.01 \text{ g/mol} + 16 \times 1.01 \text{ g/mol} = 136.26 \text{ g/mol}\text. \]

Convert mass of limonene to grams

The problem gives the mass of limonene as 1.00 picogram. Convert this to grams: \[ 1.00 \text{ pg} = 1.00 \times 10^{-12} \text{ g} \]

Calculate the number of moles of limonene

Use the molar mass to convert the mass of limonene to moles: \[ \text{Number of moles} = \frac{ \text{mass} }{ \text{molar mass} } = \frac{ 1.00 \times 10^{-12} \text{ g} }{ 136.26 \text{ g/mol} } = 7.34 \times 10^{-15} \text{ moles} \]

Determine the number of limonene molecules

Use Avogadro's number \(6.022 \times 10^{23}\text{ molecules/mol}\) to find the number of molecules: \[ \text{Number of molecules} = 7.34 \times 10^{-15} \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} = 4.42 \times 10^{9} \text{ molecules} \]

Calculate the time required to release the molecules

According to the problem, ambient light releases \(2.64 \times 10^{18} \text{ photons/sec}\). Since one photon releases one molecule, use the number of molecules to find the time: \[ \text{Time} = \frac{ \text{Number of molecules} }{ \text{photons/sec} } = \frac{ 4.42 \times 10^{9} \text{ molecules} }{ 2.64 \times 10^{18} \text{ photons/sec} } = 1.67 \times 10^{-9} \text{ seconds} \]

Key concepts

Molar Mass Calculation

Molar mass is a key concept in chemistry. It represents the mass of one mole of a given substance, usually expressed in grams per mole (g/mol). To calculate the molar mass, we add up the atomic masses of all the atoms in a molecule. In this exercise, we are dealing with limonene, which has a molecular formula of \( \text{C}_{10}\text{H}_{16} \).

To find limonene's molar mass, follow these steps:

• Identify the atomic masses of carbon (\( M(\text{C}) = 12.01 \text{ g/mol} \)) and hydrogen (\( M(\text{H}) = 1.01 \text{ g/mol} \)).
• Multiply the atomic masses by the respective number of each atom in the molecule.
• Sum up the results:

\[ M(\text{C}_{10}\text{H}_{16}) = 10 \times M(\text{C}) + 16 \times M(\text{H}) = 10 \times 12.01 \text{ g/mol} + 16 \times 1.01 \text{ g/mol} = 136.26 \text{ g/mol} \]
This tells us that one mole of limonene weighs 136.26 grams. Understanding molar mass is essential for converting between mass and moles in any chemical calculation.

Photon Interaction

Photon interaction is where photons (particles of light) interact with matter, causing various effects. In this exercise, we focus on how photons interact with tethered fragrance molecules, like limonene, to release them.

The exercise states that each photon releases one molecule of limonene. Here, ambient light releases \( 2.64 \times 10^{18} \text{ photons/sec} \). We use this to determine the time needed to release a specific mass of limonene:

• Calculate the number of limonene molecules we need to release from the given mass (1.00 picogram in this case).
• Use the ratio of photons to molecules to find the time.

Since the relationship is one-to-one, calculating this is straightforward. For example, to release \( 4.42 \times 10^9 \text{ molecules} \) with each photon releasing one molecule, it would take:
\[ \text{Time} = \frac{ 4.42 \times 10^{9} \text{ molecules} }{ 2.64 \times 10^{18} \text{ photons/sec} } = 1.67 \times 10^{-9} \text{ seconds} \]
This rapid process shows how light can efficiently trigger chemical reactions at the molecular level.

Molecular Conversion

Molecular conversion involves transforming one unit of matter to another form, usually involving calculations of moles, molecules, and mass. In this exercise, molecular conversion is used several times.

• First, we convert the mass of limonene from picograms to grams:
  \[ 1.00 \text{ pg} = 1.00 \times 10^{-12} \text{ g} \]
• Next, we convert the mass in grams to moles using the molar mass:
  \[ \text{Number of moles} = \frac{ \text{mass} }{ \text{molar mass} } = \frac{ 1.00 \times 10^{-12} \text{ g} }{ 136.26 \text{ g/mol} } = 7.34 \times 10^{-15} \text{ moles} \]
• Finally, we use Avogadro's number (\(6.022 \times 10^{23}\)) to convert moles to molecules:
  \[ \text{Number of molecules} = 7.34 \times 10^{-15} \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} = 4.42 \times 10^{9} \text{ molecules} \]

These conversions are crucial for accurate calculations in chemistry, ensuring that you work with consistent and meaningful units across different steps of a calculation.