Question

In the average adult male, the residual volume (RV) of the lungs, the volume of air remaining after a forced exhalation, is \(1200 \mathrm{~mL}\). (a) How many moles of air are present in the \(\mathrm{RV}\) at \(1.0 \mathrm{~atm}\) and \(37^{\circ} \mathrm{C} ?\) (b) How many molecules of gas are present under these conditions?

Short answer

Number of moles ≈ 0.0473 moles.Number of molecules ≈ 2.85 \times 10^{22}.

Step-by-step solution

- Understand the Problem

We need to find the number of moles and the number of molecules of air in the residual volume (RV) of the lungs at specific conditions of pressure and temperature.

- List the Given Values

Given: - Residual Volume (RV) = 1200 mL (or 1.2 L)- Pressure (P) = 1.0 atm- Temperature (T) = 37°C (or 310.15 K, since T in Kelvin = T in °C + 273.15)- Ideal Gas Constant (R) = 0.0821 L·atm/(mol·K)

- Use the Ideal Gas Law to Find Moles

We will use the ideal gas law: PV = nRT Where: - P = Pressure - V = Volume - n = Number of moles - R = Ideal gas constant - T = Temperature in Kelvin Rearranging for n (number of moles): n = \(\frac{PV}{RT}\) Substitute the known values: n = \(\frac{(1.0 \, \mathrm{atm})(1.2 \, \mathrm{L})}{(0.0821 \, \mathrm{L} \, \mathrm{atm}/(\mathrm{mol} \, \mathrm{K}))(310.15 \, \mathrm{K})}\) n ≈ 0.0473 moles

- Use Avogadro's Number to Find Molecules

Avogadro's number approximates the number of particles in one mole: - Avogadro's number = \(6.022 \times 10^{23}\) molecules/mol We can calculate the number of molecules by multiplying the number of moles by Avogadro's number: Number of molecules = n \times Avogadro's number Number of molecules = (0.0473 \; \text{moles}) \times (6.022 \times 10^{23} \; \text{molecules/mol}) Number of molecules ≈ 2.85 \times 10^{22} molecules

Key concepts

ideal gas law

The Ideal Gas Law is a fundamental equation in chemistry connecting pressure (P), volume (V), temperature (T), and the number of moles of gas (n). It is expressed as \( PV = nRT \). Here, R is the ideal gas constant, allowing the equation to work in various unit systems. This law assumes gases behave ideally, meaning gas molecules don't interact in any significant way. We use this equation to solve problems involving gas behaviors under various conditions. Remember, temperature must always be in Kelvin for this equation. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature.

moles calculation

Calculating the number of moles of a gas using the Ideal Gas Law involves rearranging the equation to solve for n, resulting in \( n = \frac{PV}{RT} \). By substituting the given values for pressure (P), volume (V), and temperature (T), you can find the number of moles. In our problem, we used P = 1.0 atm, V = 1.2 L, and T = 310.15 K to find:
\( n = \frac{(1.0 \, \text{atm})(1.2 \, \text{L})}{(0.0821 \, \text{L} \, \text{atm}/(\text{mol} \, \text{K}))(310.15 \, \text{K})} \approx 0.0473 \text{ moles} \). This calculation is crucial for further steps, like finding the number of molecules.

Avogadro's number

Avogadro's Number, approximately \( 6.022 \times 10^{23} \), is the number of atoms, molecules, or particles in one mole of a substance. This constant is vital in converting between moles and the actual number of molecules. In our problem, after calculating the moles, we found the number of molecules by multiplying the moles by Avogadro's Number:
Number of molecules = 0.0473 moles \( \times 6.022 \times 10^{23} \text{ molecules/mol} = 2.85 \times 10^{22} \text{ molecules} \). This conversion helps us understand quantities on a microscopic scale.

residual lung volume

Residual lung volume (RV) refers to the amount of air remaining in the lungs after a maximal exhalation, usually around 1200 mL in an average adult male. Understanding RV is essential in respiratory studies and medical fields. It helps gauge lung capacity and function. For our problem, this volume was converted to liters (1.2 L) to use with the Ideal Gas Law. Knowing RV allows us to calculate the amount of gas left in the lungs and its physical behaviors under different conditions, such as pressure and temperature.

gas constant

The Gas Constant (R) is a key part of the Ideal Gas Law, linking the amount of gas to pressure, volume, and temperature. Its value varies based on the units used: for pressure in atm, volume in liters, and temperature in Kelvin, R = 0.0821 L·atm/(mol·K). In our calculations, we used this specific value to find the number of moles from the given conditions. It ensures the equation balances correctly, making accurate theoretical and experimental calculations possible in gas-related problems.