Ch.4 - Reactions in Aqueous Solution

Problem 97b

Chapter 4, Problem 97b

Neurotransmitters are molecules that are released by nerve cells to other cells in our bodies, and are needed for muscle motion, thinking, feeling, and memory. Dopamine is a common neurotransmitter in the human brain and is a weak base. Its molecular weight is 153.2 g/mol. b. Experiments with rats show that if rats are dosed with 3.0 mg/kg of cocaine (that is, 3.0 mg cocaine per kg of animal mass), the concentration of dopamine in their brains increases by 0.75 𝜇𝑀 after 60 seconds. Calculate how many molecules of dopamine would be produced in a rat (average brain volume 5.00mm3) after 60 seconds of a 3.0 mg/kg dose of cocaine.

Verified step by step guidance

Step 1: Convert the concentration of dopamine from micromolar (\(\mu M\)) to moles per liter (M). Since 1 \(\mu M\) is equal to \(1 \times 10^{-6}\) M, multiply 0.75 \(\mu M\) by \(1 \times 10^{-6}\) to convert it to M.

Step 2: Calculate the number of moles of dopamine in the rat's brain. Use the formula: \(\text{moles} = \text{concentration (M)} \times \text{volume (L)}\). First, convert the brain volume from \(\text{mm}^3\) to liters. Since 1 \(\text{mm}^3\) is equal to \(1 \times 10^{-3}\) mL and 1 mL is \(1 \times 10^{-3}\) L, convert 5.00 \(\text{mm}^3\) to liters.

Step 3: Use Avogadro's number to convert moles of dopamine to molecules. Avogadro's number is \(6.022 \times 10^{23}\) molecules/mol. Multiply the number of moles of dopamine by Avogadro's number to find the number of molecules.

Step 4: Review the calculations to ensure all unit conversions are correct and consistent, especially the conversion from \(\mu M\) to M and \(\text{mm}^3\) to L.

Step 5: Summarize the process and ensure the final answer is expressed in terms of the number of molecules of dopamine produced.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Molarity and Concentration

Molarity is a measure of concentration defined as the number of moles of solute per liter of solution. In this context, the concentration of dopamine is given in micromolar (𝜇M), which indicates the number of micromoles of dopamine per liter of brain volume. Understanding how to convert between different units of concentration is essential for calculating the total amount of dopamine produced.

Molecular Weight and Moles

Molecular weight is the mass of one mole of a substance, expressed in grams per mole (g/mol). For dopamine, the molecular weight is 153.2 g/mol, which allows us to convert between grams and moles. This conversion is crucial for determining the number of molecules present in a given mass of dopamine, especially when calculating the total amount produced in the rat's brain.

Volume and Density Relationships

The volume of a substance is the amount of space it occupies, and in this case, the average brain volume of a rat is given as 5.00 mm³. To relate this volume to the concentration of dopamine, it is important to convert the volume into liters (1 mm³ = 1 x 10^-6 L). This conversion is necessary to accurately calculate the total number of dopamine molecules produced in the specified brain volume after cocaine administration.

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Relationship of Volume and Moles Example

Related Practice

Textbook Question

The commercial production of nitric acid involves the following chemical reactions:

4 NH₃(g) + 5 O₂(g) → 4 NO(g) + 6 H₂O(g)

2 NO(g) + O₂(g) → 2 NO₂(g)

3 NO₂(g) + H₂O(l) → 2 HNO₃(aq) + NO(g)

(a) Which of these reactions are redox reactions?

Textbook Question

The commercial production of nitric acid involves the following chemical reactions:

4 NH₃(g) + 5 O₂(g) → 4 NO(g) + 6 H₂O(g)

2 NO(g) + O₂(g) → 2 NO₂(g)

3 NO₂(g) + H₂O(l) → 2 HNO₃(aq) + NO(g)

(b) Identify the element undergoing oxidation and the element undergoing reduction.

Textbook Question

The commercial production of nitric acid involves the following chemical reactions:

4 NH₃(g) + 5 O₂(g) → 4 NO(g) + 6 H₂O(g)

2 NO(g) + O₂(g) → 2 NO₂(g)

3 NO₂(g) + H₂O(l) → 2 HNO₃(aq) + NO(g)

(c) How many grams of ammonia must you start with to make 1000.0 L of a 0.150 M aqueous solution of nitric acid? Assume all the reactions give 100% yield.

Textbook Question

Hard water contains Ca2+, Mg2+, and Fe2+, which interfere with the action of soap and leave an insoluble coating on the insides of containers and pipes when heated. Water softeners replace these ions with Na+. Keep in mind that charge balance must be maintained. (a) If 1500 L of hard water contains 0.020 M Ca2+ and 0.0040 M Mg2+, how many moles of Na+ are needed to replace these ions?

Textbook Question

Hard water contains Ca2+, Mg2+, and Fe2+, which interfere with the action of soap and leave an insoluble coating on the insides of containers and pipes when heated. Water softeners replace these ions with Na+. Keep in mind that charge balance must be maintained. (b) If the sodium is added to the water softener in the form of NaCl, how many grams of sodium chloride are needed?

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Tartaric acid, H2C4H4O6, has two acidic hydrogens. The acid is often present in wines, and a salt derived from the acid precipitates from solution as the wine ages. A solution containing an unknown concentration of the acid is titrated with NaOH. It requires 24.65 mL of 0.2500 M NaOH solution to titrate both acidic protons in 50.00 mL of the tartaric acid solution. Write a balanced net ionic equation for the neutralization reaction