Question

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. (a) Predict what kind of reaction dopamine is most likely to undergo in water: redox, acid-base, precipitation, or metathesis? Explain your reasoning. (b) Patients with Parkinson's disease suffer from a shortage of dopamine and may need to take it to reduce symptoms. An IV (intravenous fluid) bag is filled with a solution that contains 400.0 mg dopamine per 250.0 mL. of solution. What is the concentration of dopamine in the IV bag in units of molarity? (c) Experiments with rats show that if rats are dosed with 3.0 $\mathrm{mg}/\mathrm{kg}$ of cocaine (that is, 3.0 mg cocaine per kg of animal mass), the concentration of dopamine in their brains increases by 0.75$\mu M$ after 60 seconds. Calculate how many molecules of dopamine would be produced in a rat (average brain volume 5.00 ${\mathrm{mm}}^3)$ after 60 seconds of a 3.0 $\mathrm{mg}/\mathrm{kg}$ dose of cocaine.

Short answer

(a) Dopamine is most likely to undergo an acid-base reaction in water due to the presence of hydroxyl (-OH) and amine (-NH2) functional groups in its structure. (b) The concentration of dopamine in the IV bag is $\frac{400.0\text{mg}\cdot (1\text{g}/1000\text{mg})}{153.18\text{g/mol}\cdot (250.0\text{mL}\cdot (1\text{L}/1000\text{mL}))}\approx 0.0104\text{M}$. (c) The number of dopamine molecules produced in a rat's brain after 60 seconds of a 3.0 mg/kg dose of cocaine is approximately $0.75\mu M\cdot 5.00\times {10}^{-6}\text{L}\cdot 6.022\times {10}^{23}\text{molecules/mol}\approx 2.26\times {10}^{15}$ molecules.

Step-by-step solution

(a) Analyzing Dopamine's Reactions

Dopamine has the molecular formula C8H11NO2. It contains hydroxyl (-OH) and amine (-NH2) functional groups. The presence of these functional groups suggests that dopamine can act as either a weak acid, donating its OH- group, or as a weak base, accepting an H+ ion for its NH2 group. Further, the structure of dopamine does not suggest any redox, precipitation, or metathesis reactions. Thus, the most likely reaction dopamine will undergo in water is an acid-base reaction.

(b) Calculating Dopamine Concentration

To find the concentration of dopamine in moles per liter (M), we start by dividing the mass of dopamine by its molar mass to find the moles of dopamine. The molar mass of dopamine is 153.18 g/mol (C8H11NO2). Moles of dopamine = $\frac{400.0\text{mg}}{153.18\text{g/mol}}\cdot \frac{1\text{g}}{1000\text{mg}}$ Next, we convert the volume of the solution from mL to L: Volume in liters = $250.0\text{mL}\cdot \frac{1\text{L}}{1000\text{mL}}$ Now we can find the concentration (in M) by dividing the moles of dopamine by the volume of the solution in liters: Concentration = $\frac{\text{moles of dopamine}}{\text{volume in liters}}$

(c) Calculating Dopamine Molecules Produced

The increase in dopamine concentration is given as 0.75μM. To find the number of dopamine molecules produced, we need to use the increased concentration, rat's brain volume, and Avogadro's number. First, convert the volume of the rat's brain from mm³ to L: Volume in liters = $5.00{\text{mm}}^3\cdot \frac{1\text{L}}{1000\text{mL}}\cdot \frac{1000{\text{cm}}^3}{1\text{L}}\cdot \frac{1\text{mL}}{1{\text{cm}}^3}\cdot \frac{1{\text{cm}}^3}{1000{\text{mm}}^3}$ Next, we calculate the moles of dopamine produced by multiplying the increased concentration and the brain volume in liters: Moles of dopamine = increase in concentration × brain volume in liters Finally, we calculate the number of dopamine molecules produced by multiplying the moles of dopamine and Avogadro's number: Number of dopamine molecules = moles of dopamine × Avogadro's number (6.022 × 10²³ molecules/mol)

Key concepts

Acid-Base Reaction

Dopamine is a neurotransmitter that plays a vital role in sending signals between nerve cells. When we look at its molecular structure, it contains both hydroxyl (-OH) and amine (-NH2) groups. These functional groups define the possible types of reactions dopamine might undergo. In the presence of water, dopamine is likely to participate in an acid-base reaction. This occurs because:

• The hydroxyl group can donate a hydrogen ion (H+) and behave as an acid.
• The amine group can accept a hydrogen ion, allowing it to act as a base.

Hence, with these functionalities, dopamine can exhibit characteristics of both weak acids and weak bases, engaging primarily in acid-base reactions rather than in redox, precipitation, or metathesis reactions. This understanding is crucial when considering how dopamine operates in biological systems.

Molarity Calculation

Molarity is a way to express the concentration of a solution. It indicates how many moles of a substance are present in one liter of solution. To calculate molarity, you should:

• Determine the number of moles of the solute (in this case, dopamine).
• Measure the volume of the solution in liters.

In the given problem, dopamine's mass is converted to moles using its molar mass, which is 153.18 g/mol. The solution's volume is measured in milliliters but converted to liters. Finally, the molarity is found using the formula:$$\text{Molarity}=\frac{\text{moles of dopamine}}{\text{volume of solution in liters}}.$$This gives a complete yet concise method for finding how concentrated the dopamine solution is.

Neurotransmitter Functions

Neurotransmitters are chemical messengers that facilitate the communication between neurons in the nervous system. Dopamine, one of these neurotransmitters, is involved in several critical functions such as:

• Regulating mood and feelings of pleasure.
• Controlling muscle movements and coordination.
• Influencing learning and memory retention.

A careful balance in dopamine levels is vital, as an excess or deficit can lead to conditions like Parkinson's disease or schizophrenia. Understanding the complex role dopamine plays can help recognize its importance in maintaining neurological health and function.

Molecular Formula

A molecular formula denotes the number and type of atoms in a molecule. For dopamine, the formula is $C_8{H}_{11}N{O}_2$. This tells us it contains:

• 8 carbon (C) atoms.
• 11 hydrogen (H) atoms.
• 1 nitrogen (N) atom.
• 2 oxygen (O) atoms.

This formula is critical in understanding not just the composition of dopamine, but also in calculating properties like molar mass and determining how it may react chemically. Knowing how to interpret molecular formulas is a foundation of organic chemistry and pivotal in deducing the chemical behavior of substances.

Avogadro's Number

Avogadro's number, $6.022\times {10}^{23}$, is the number of particles—atoms or molecules—in one mole of a substance. To calculate how many molecules are present when given a certain molar concentration, Avogadro's number is indispensable.

• First, calculate the moles of a substance using its concentration and volume.
• Then, multiply the moles by Avogadro's number to find the number of molecules.

In the exercise, Avogadro’s number helps determine the number of dopamine molecules increased in a rat's brain following cocaine dosage. This link between microscopic particles and macroscopic properties is fundamental for bridging experimental data with molecular-scale understanding.