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. (c) 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 mM after 60 seconds. Calculate how many molecules of dopamine would be produced in a rat (average brain volume 5.00 mm3) after 60 seconds of a 3.0 mg/kg dose of cocaine.

Verified step by step guidance

1. Convert the brain volume from mm³ to liters. Since 1 mm³ = 1 x 10⁻³ mL and 1 mL = 1 x 10⁻³ L, multiply the brain volume by 1 x 10⁻³ twice to convert to liters.
2. Use the concentration change of dopamine (0.75 mM) to find the number of moles of dopamine produced. Remember that molarity (M) is moles per liter, so multiply the concentration by the volume in liters.
3. Calculate the number of molecules of dopamine using Avogadro's number (6.022 x 10²³ molecules/mol). Multiply the moles of dopamine by Avogadro's number to find the number of molecules.
4. Ensure all units are consistent throughout the calculations, especially when converting between volume units and using Avogadro's number.
5. Review the steps to ensure that each conversion and calculation aligns with the given data and the desired outcome of finding the number of molecules.

Key Concepts

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 millimoles per liter (mM), which indicates how much dopamine is present in a specific volume of the rat's brain. Understanding how to convert between concentration and the number of molecules is essential for solving the problem.

Volume and Density Relationships

The volume of the rat's brain is provided in cubic millimeters (mm³), which must be converted to liters for consistency with molarity calculations. Knowing that 1 mm³ equals 1 x 10^-6 liters allows for the correct conversion of brain volume, enabling the calculation of the total amount of dopamine present in the brain after cocaine administration.

Avogadro's Number

Avogadro's number (approximately 6.022 x 10²³) is the number of molecules in one mole of a substance. This concept is crucial for converting moles of dopamine, derived from the concentration and volume calculations, into the actual number of molecules. By applying Avogadro's number, one can determine how many dopamine molecules are produced in the rat's brain after the cocaine dose.