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Chapter 22: Problem 12

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

Expert verified
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

01

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.
02

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 \(N = n/V \times Avogadro's number\).
03

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.

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Most popular questions from this chapter

Gold has a molar (atomic) mass of \(197 \mathrm{~g} / \mathrm{mol}\). Consider a 2.56\(g\) sample of pure gold vapor. (a) Calculate the number of moles of gold present. (b) How many atoms of gold are present?

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