Problem 24 - University Physics with Modern Physics

Question

Modern vacuum pumps make it easy to attain pressures of the order of \(10^{-13}\) atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (a) At a pressure of \(9.00 \times 10^{-14}\) atm and an ordinary temperature of \(300.0 \mathrm{K},\) how many molecules are present in a volume of 1.00 \(\mathrm{cm}^{3} ?\) (b) How many molecules would be present at the same temperature but at 1.00 atm instead?

Step-by-Step Solution

Answer

(a) At low pressure, there are approximately \(2.20 \times 10^6\) molecules. (b) At 1.00 atm, there are approximately \(2.45 \times 10^{19}\) molecules.

Step 1: Define given variables and constants

We are given: - Pressure, \( P = 9.00 \times 10^{-14} \) atm.- Temperature, \( T = 300.0 \, \text{K} \).- Volume, \( V = 1.00 \, \text{cm}^3 \).Convert \( V \) to liters because the ideal gas law uses liters:\[ V = 1.00 \, \text{cm}^3 = 1.00 \times 10^{-3} \, \text{L} \]We'll need the gas constant in terms of \( \, \text{L} \, \text{atm} \, \text{K}^{-1} \, \text{mol}^{-1} \), \( R = 0.0821 \). We will also use Avogadro's number, \( N_A = 6.022 \times 10^{23} \, \text{molecules per mol} \).

Step 2: Apply Ideal Gas Law to find moles at low pressure

The ideal gas law is given by:\[ PV = nRT \]Rearrange to solve for \( n \), the number of moles:\[ n = \frac{PV}{RT} \]Substitute the values:\[ n = \frac{(9.00 \times 10^{-14} \, \text{atm})(1.00 \times 10^{-3} \, \text{L})}{(0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1})(300.0 \, \text{K})} \approx 3.66 \times 10^{-18} \, \text{mol} \]

Step 3: Calculate the number of molecules at low pressure

Use Avogadro's number to find the number of molecules:\[ \text{Number of molecules} = n \times N_A \]\[ \text{Number of molecules} = 3.66 \times 10^{-18} \, \text{mol} \times 6.022 \times 10^{23} \, \frac{\text{molecules}}{\text{mol}} \approx 2.20 \times 10^6 \, \text{molecules} \]

Step 4: Calculate number of moles at normal pressure

Now at 1.00 atm pressure and the same temperature and volume:\[ n = \frac{(1.00 \, \text{atm})(1.00 \times 10^{-3} \, \text{L})}{(0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1})(300.0 \, \text{K})} \approx 4.06 \times 10^{-5} \, \text{mol} \]

Step 5: Calculate the number of molecules at normal pressure

Again use Avogadro's number:\[ \text{Number of molecules} = n \times N_A \]\[ \text{Number of molecules} = 4.06 \times 10^{-5} \, \text{mol} \times 6.022 \times 10^{23} \, \frac{\text{molecules}}{\text{mol}} \approx 2.45 \times 10^{19} \, \text{molecules} \]

Key Concepts

Avogadro's Number

Avogadro's Number is a fundamental concept in chemistry and physics. It represents the number of atoms, ions, or molecules found in one mole of a substance. The value of Avogadro's Number is approximately \( 6.022 \times 10^{23} \), which means one mole of any substance contains \( 6.022 \times 10^{23} \) entities. This large number helps scientists understand and perform calculations involving amounts of substances at the microscopic scale.

This number is crucial for converting between moles and the actual number of atoms or molecules. For instance, if we have a certain number of moles of a gas, we can use Avogadro's Number to determine how many molecules of the gas are present.

This was exactly what was done in the exercise - by knowing the moles of gas present under given conditions and applying Avogadro's Number, the number of molecules was calculated. Understanding this process allows us to connect the macroscopic quantities we can measure, like moles, to microscopic entities like molecules.

Mole Concept

The Mole Concept is an essential part of chemistry that allows us to quantify atoms and molecules in practical amounts. A mole is a unit that represents \( 6.022 \times 10^{23} \) particles of a substance. This measure bridges the gap between microscopic particles and macroscopic amounts we can observe and manipulate.

Variants of this concept are applicable in different scenarios. In the exercise, to calculate the number of molecules in a sample of air at different pressures but the same temperature, the number of moles was first determined using the Ideal Gas Law. The Ideal Gas Law, which you may remember as \( PV = nRT \), allows you to find the number of moles \( n \) when you have pressure \( P \), volume \( V \), and temperature \( T \) known, alongside the gas constant \( R \).

By finding the number of moles, we were able to use Avogadro's Number to convert to actual numbers of molecules. This use of the mole is fundamental in chemical calculations, providing both a count of particles and a way to relate different types of quantities, like mass and volume.

Pressure in Gases

Pressure in gases is one of the critical concepts covered by the kinetic molecular theory and the Ideal Gas Law. It describes the force exerted by gas molecules colliding with the walls of a container. Pressure is usually measured in atmospheres (atm), and it varies directly with changes in the number of gas molecules and temperature.

In our exercise, the pressure of a gas was manipulated to assess the number of molecules present at different states. At a very low pressure, such as \( 9.00 \times 10^{-14} \) atm, the number of molecules in a given volume is considerably lower compared to when the pressure is at 1.00 atm. This difference can be attributed to Boyle's Law, a part of the Ideal Gas Law, which tells us that if the temperature is constant, pressure is inversely related to volume. Therefore, fewer molecules mean less pressure.

Understanding pressure is vital because it influences the behavior of gases and their interactions. It also plays a critical role in practical applications like airbags in vehicles or the principles behind refrigeration and air conditioning systems. Studying pressure not only helps in scientific calculations but also in everyday technological applications.