Question

Tetrodotoxin is a toxic chemical found in fugu pufferfish, a popular but rare delicacy in Japan. This compound has an LD\(_{50}\) (the amount of substance that is lethal to \(50 . \%\) of a population sample) of \(10 . \mu g\) per \(\mathrm{kg}\) of body mass. Tetrodotoxin is \(41.38 \%\) carbon by mass, \(13.16\%\) nitrogen by mass, and \(5.37\%\) hydrogen by mass, with the remaining amount consisting of oxygen. What is the empirical formula of tetrodotoxin? If three molecules of tetrodotoxin have a mass of \(1.59 \times 10^{-21} \mathrm{g},\) what is the molecular formula of tetrodotoxin? What number of molecules of tetrodotoxin would be the \(\mathrm{LD}_{50}\) dosage for a person weighing 165 lb?

Short answer

The empirical formula of tetrodotoxin is C4H6NO3. The molecular formula is the same as the empirical formula. The LD50 dosage for a person weighing 165 lb would be approximately 1.41 x 10^18 molecules of tetrodotoxin.

Step-by-step solution

Calculate moles of Carbon, Nitrogen, and Hydrogen in 100g of tetrodotoxin

Assuming a 100g sample of tetrodotoxin, the mass of each element in the sample can be calculated: Carbon: \(41.38\%\) by mass of 100g sample = 41.38g Nitrogen: \(13.16\%\) by mass of 100g sample = 13.36g Hydrogen: \(5.37\%\) by mass of 100g sample = 5.37g Next, we can find moles of each element by dividing the mass by the element's molar mass: Moles of Carbon = \(41.38\div12.01=\) 3.447 Moles of Nitrogen = \(13.36\div14.01=\) 0.953 Moles of Hydrogen = \(5.37\div1.01=\) 5.317

Calculate moles of Oxygen in the sample

The mass of Oxygen can be found by taking the remaining mass of the 100g sample: Oxygen: 100 - (41.38 + 13.16 + 5.37) = 40.09g Now, we find moles of Oxygen: Moles of Oxygen = \( 40.09 \div 16.00 = \) 2.506

Determine the empirical formula

To find the empirical formula, we must determine the ratio of moles between the elements: Divide all mole values by the smallest value (0.953) to find ratios: Carbon: 3.447/0.953 ≈ 3.62 Hydrogen: 5.317/0.953 ≈ 5.58 Nitrogen: 0.953/0.953 = 1 Oxygen: 2.506/0.953 ≈ 2.63 Round the ratios to the nearest whole number: C4H6N1O3 Thus, the empirical formula is C4H6N1O3.

Find the molecular formula

Determine the mass of the empirical unit: Empirical mass = (4 x 12.01) + (6 x 1.01) + (1 x 14.01) + (3 x 16.00) = 227.25g/mol Calculate the mass of one molecule of tetrodotoxin from the given mass of three molecules: Mass of one molecule = \(\frac{1.59 \times 10^{-21} g}{3} = 5.30 \times 10^{-22} g\) Find the molar mass by multiplying the mass of one molecule by Avogadro's number: Molar mass of tetrodotoxin = \(5.30 \times 10^{-22} g \times 6.022 \times 10^{23} particles/mol =\) 319.34g/mol Determine the ratio between the molecular mass and empirical mass: Ratio = \(319.34\div227.25=\) 1.4 (approximately 1) The molecular formula is the same as the empirical formula, C4H6N1O3.

Calculate the LD50 dosage for a person weighing 165 lb

First, convert the person's weight to kg: Weight in kg = \(165 \times \frac{1 kg}{2.205 lb} =\) 74.83kg Determine the LD50 dosage of tetrodotoxin in µg: LD50 dosage = 10 µg of tetrodotoxin/ 1 kg × 74.83 kg = 748.3 µg Convert the LD50 dosage to grams: LD50 dosage = 748.3 µg × \(\frac{1 g}{10^6µg}\) = 7.483 x 10^{-4} g Now we find the number of molecules of LD50 dosage by dividing it by the mass of a single tetrodotoxin molecule: Molecules in LD50 dosage = \(\frac{7.483 \times 10^{-4} g}{5.30 \times 10^{-22} g/molecule} =\) 1.41 x 10^18 molecules So, the LD50 dosage for a person weighing 165 lb would be approximately 1.41 x 10^18 molecules of tetrodotoxin.

Key concepts

Tetrodotoxin

Tetrodotoxin (TTX) is a potent neurotoxin known for its presence in the pufferfish, or 'fugu,' which is considered a unique culinary experience in Japan. This substance earns its infamy from its ability to block voltage-gated sodium channels in nerve cells, effectively preventing the transmission of nerve signals and leading to paralysis. It's so potent that even minute amounts can prove fatal, making the understanding of TTX in fields like toxicology deeply significant.

When talking about tetrodotoxin, it's not just its potency that's interesting, but also its chemical composition. The empirical formula of a molecule like TTX is the simplest ratio of its constituent atoms. This provides insight into the core structure of the molecule, but it does not give the complete picture of its actual molecular complexity. This leads us naturally to the concept of molecular formula, which accounts for the actual number of atoms in a molecule of TTX, giving scientists the necessary information to understand its biological interactions and effects.

Stoichiometry

Stoichiometry is a section of chemistry that encompasses the quantitative relationships between reactants and products in a chemical reaction. It's essentially a counting tool for chemists, enabling them to predict amounts of substances consumed and produced in a given reaction. When we apply stoichiometry to identifying chemical formulas, such as with tetrodotoxin, we use the mass percentages of each element to determine the moles of each.

Illustrating this with tetrodotoxin, we calculate the moles of each element based on the given mass percentages and then find the simplest whole-number ratio of these moles to determine the empirical formula. This technique is fundamental when dealing with chemical equations and understanding the actual amounts of substances involved, whether they're toxic compounds or everyday chemicals.

Molar Mass

In the realm of chemistry, molar mass serves as a critical bridge between the macroscopic world we observe and the microscopic world of molecules and atoms. Defined as the mass of one mole of a substance, it allows chemists to convert between mass and moles efficiently. When it's about compounds like tetrodotoxin, knowing the molar mass enables us to understand how many molecules are involved in a given weight.

In the step-by-step solution to our exercise, we see the use of molar mass in action, calculating the mass of a single molecule from the mass of three molecules of TTX. Determining the molar mass of tetrodotoxin then becomes a straightforward operation of multiplying the mass of one molecule by Avogadro's number, which gives us the substance's weight per mole and is pivotal in many areas of chemistry, including pharmacology and dosing calculations.

LD50 Dosage Calculation

The LD50 value is a crucial measure in toxicology, representing the dose of a substance that is lethal to 50% of a test population. It's a statistical tool to gauge the toxicity of substances, with a lower LD50 indicating higher toxicity. Calculations involving LD50 are vital for safety assessments and determining safe exposure levels.

Regarding tetrodotoxin, the LD50 dosage calculation begins with a pivotal conversion of body weight from pounds to kilograms, ensuring consistent units. Then, using the established LD50 value of 10 micrograms per kilogram, we multiply by the individual's body mass to find the lethal dose in micrograms. To convert to grams, a standard measurement that aligns with most scientific analyses, and then finally establish the number of toxic molecules that constitute the LD50 dose. It's these calculations that underpin the safe handling directives for hazardous materials and facilitate the development of safety guidelines for potentially lethal substances.