Question

Calculate the number of molecules in a deep breath of air whose volume is \(2.25 \mathrm{~L}\) at body temperature, \(37^{\circ} \mathrm{C}\), and a pressure of 735 torr.

Short answer

First, convert the temperature to Kelvin: \(T(K) = 37^{\circ} \mathrm{C} + 273.15 = 310.15 \mathrm{K}\). Convert pressure to atm: \(P = \frac{735 \mathrm{~torr}}{760 \mathrm{~torr/atm}} = 0.967 \mathrm{~atm}\). Use the Ideal Gas Law to calculate the number of moles: \(n = \frac{(0.967 \mathrm{~atm})(2.25 \mathrm{~L})}{(0.0821 \mathrm{~L~atm/K~mol})(310.15 \mathrm{~K})} = 0.089 \mathrm{~mol}\). Convert moles to molecules using Avogadro's number: \(number~of~molecules = (0.089 \mathrm{~mol})(6.022 \times 10^{23} \mathrm{molecules/mol}) = 5.36 \times 10^{22} \mathrm{molecules}\).

Step-by-step solution

Convert the given values to appropriate units

Before we can use the Ideal Gas Law equation, it's important to convert all given values into the appropriate units. The Ideal Gas Law equation uses the following unit system: Pressure (P) in atmospheres (atm), Volume (V) in liters (L), and Temperature (T) in Kelvin (K). The given values are as follows: - Volume (V) = 2.25 L - Temperature (T) = 37°C - Pressure (P) = 735 torr First, convert the temperature from Celsius to Kelvin: T(K) = T(°C) + 273.15 Second, convert pressure from torr to atm: 1 atm = 760 torr

Use the Ideal Gas Law equation to calculate the number of moles

The Ideal Gas Law equation is: PV = nRT Where P is the pressure (in atm), V is the volume (in L), n is the number of moles, R is the gas constant (0.0821 L atm / K mol), and T is the temperature (in K). Rearrange the equation to solve for n: n = PV / RT Plug in the values we calculated in Step 1: n = (P * V) / (R * T)

Convert the number of moles to the number of molecules using Avogadro's number

Now that we have the number of moles (n), we can convert that into the number of molecules using Avogadro's number, which is 6.022 x 10^23 molecules per mole. number of molecules = n * Avogadro's number Plug in the value for n calculated in Step 2: number of molecules = n * 6.022 x 10^23

Calculate the final answer

Calculate the number of molecules using the values obtained in the previous steps. Remember to round off your answer according to significant figures.

Key concepts

Avogadro's number

Understanding Avogadro's number is crucial for chemists and students of chemistry as it provides the bridge between macroscopic measurements and the microscopic world of atoms and molecules. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Matteo, it is a fundamental constant of physical science.

A mole is a unit of measurement in chemistry that indicates an amount of substance that contains as many elementary entities as there are atoms in exactly 12 g of carbon-12. Avogadro's number is incredibly large, being equal to approximately 6.022 x 10^23. When dealing with gases, we often use Avogadro's hypothesis, which states that at the same temperature and pressure, equal volumes of all gases contain the same number of molecules. This makes Avogadro's number a pivotal part of calculating volumes in reactions when using the Ideal Gas Law.

Gas constant

The gas constant, often denoted as R, is another cornerstone of the Ideal Gas Law. This constant provides the relationship between energy and temperature for a mole of gas particles. The value of the gas constant is 0.0821 liter-atmospheres per mole-Kelvin ( (L atm) / (mol K) ).

It's a universal value, meaning it does not change regardless of the type of gas. This constant arises from the combination of several constants: Avogadro's number and Boltzmann's constant, which refer to the energy per particle per degree Kelvin. The gas constant is pivotal in calculating the behavior of an ideal gas because it allows us to measure the internal energy involved in reactions, encapsulating both the amount of substance and the temperature in the calculations.

Molecular count

The molecular count talks about the actual number of molecules present in a substance. Leveraging from Avogadro's number, when we determine the number of moles from the Ideal Gas Law, we can readily convert that figure into the actual count of molecules by multiplying the number of moles by Avogadro's number.

This process shifts the concept from a rather abstract mole concept, which is convenient for balancing equations and understanding reactions, to a more tangible number of particles that can be used to envision the number of gas molecules involved in our everyday activities, such as taking a deep breath.

Temperature conversion

When working with gas laws, it is imperative to convert the temperature from Celsius to Kelvin. Kelvin is the base unit of temperature in the International System of Units (SI) and is used in nearly all scientific calculations. The conversion formula is simple: T(K) = T(°C) + 273.15. This equation doesn't just shift the starting point of measurement from Celsius - it reflects absolute temperatures which are essential in ensuring accurate and meaningful calculations according to the Ideal Gas Law.

Understanding how to properly convert temperature units ensures that the volume, pressure, and amount of a gas can be related in a way that accurately describes its physical behavior. This step is not to be overlooked as using the incorrect temperature scale can dramatically affect the outcome of calculations involving gases.