Question

In a certain particle accelerator the protons travel around a circular path of diameter \(23.5 \mathrm{~m}\) in a chamber of \(1.10 \times 10^{-6}\) \(\mathrm{mm}\) Hg pressure and \(295 \mathrm{~K}\) temperature. ( \(a\) ) Calculate the number of gas molecules per cubic meter at this pressure. (b) What is the mean free path of the gas molecules under these conditions if the molecular diameter is \(2.20 \times 10^{-8} \mathrm{~cm} ?\)

Short answer

To summarize, the number of gas molecules per cubic meter at a pressure of \(1.10 \times 10^{-6}\) mmHg and a temperature of 295 K is approximately equal to \(2.69 \times 10^{24} molecules/m^{3}\). The mean free path of the gas molecules, given a molecular diameter of \(2.20 \times 10^{-8}\) cm, is approximately \(1.11 \times 10^{-5}\) m.

Step-by-step solution

Calculate the number of molecules

First, use the Ideal Gas Law: \(PV = nRT\), where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. The number of moles per volume unit (n/V) can be obtained by dividing the pressure by the product of the gas constant and temperature, \(n/V = P/RT\). However, R herein is the specific gas constant for air: \(R = 8.31 JK^{-1}mol^{-1}\). Also, remember to convert pressure from mmHg to Pascals.

Convert moles to molecules

Use Avogadro’s number (\(6.022 \times 10^{23} molecules/mol\)) to convert moles to molecules. The number of molecules per cubic meter (\(N\)) can be calculated as \(N = n/V \times Avogadro's number\).

Calculate the mean free path

The mean free path (\(ℓ\)) can be calculated with the following equation: \(ℓ = \frac{1}{√2πd^{2}N}\), where d is the molecular diameter given in the problem and N is the number of molecules per cubic meter calculated in step 2. It should be noted that the molecular diameter should be converted from cm to m before calculation.

Key concepts

Molecular Mean Free Path

The molecular mean free path is a crucial concept in understanding how gas molecules behave. It represents the average distance a molecule travels before colliding with another molecule. The mean free path, denoted as \( ℓ \), becomes especially important in scenarios involving low-pressure environments, such as in particle accelerators.

To calculate the mean free path, one typically uses the equation:

• \( ℓ = \frac{1}{√2πd^{2}N} \)

Here, \( d \) is the molecular diameter and \( N \) is the number of molecules per unit volume.

Understanding the mean free path allows us to comprehend how efficiently gases can be pumped out of a vacuum chamber and how often gas molecules undergo collisions in a given environment. In this context, the mean free path provides pivotal insights into the behavior of gas molecules within a pressurized chamber of a particle accelerator.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and physics. It represents the number of atoms or molecules in one mole of a substance, precisely \( 6.022 \times 10^{23} \). This number makes it easier to bridge the gap between the microscopic world of atoms and molecules and the macroscopic world of grams and liters that we can measure.

In exercises involving the Ideal Gas Law, Avogadro's number is essential for converting the number of moles into actual molecule counts. Once we know the number of moles, we can effortlessly calculate the total number of gas molecules by multiplying by Avogadro's number.

• For example, if there are \( n \) moles of gas, the number of molecules is given by \( n \times 6.022 \times 10^{23} \).

This conversion is fundamental when we need the number of molecules to determine properties like the mean free path.

Pressure Conversion

Pressure conversion is often necessary when dealing with gas laws in different units. Pressure in mmHg, also known as Torr, often needs to be converted to other units such as Pascals (Pa) for use in standard physical equations like the Ideal Gas Law.

The relationship between mmHg and Pascals is straightforward:

• 1 mmHg = 133.322 Pa

By applying this conversion factor, you can transform any given pressure from mmHg to Pascals, allowing for consistent use in calculations with the gas constant \( R \), which stands at \( 8.31 \) J/K/mol. This uniformity is crucial, especially in a context that requires precise measurements like the determination of the number of gas molecules or their interactions in accelerator environments.

Having a good grasp of pressure conversion ensures accuracy in the application of the Ideal Gas Law and related calculations.