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Ch 18: A Macroscopic Description of Matter
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 18: A Macroscopic Description of MatterProblem 44b
Chapter 18, Problem 44b

The semiconductor industry manufactures integrated circuits in large vacuum chambers where the pressure is 1.0×10-10 mm of Hg. At T=20°C, how many molecules are in a cylindrical chamber 40 cm in diameter and 30 cm tall?

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Step 1: Convert the pressure from mm of Hg to Pascals (Pa). Use the conversion factor: 1 mm of Hg = 133.322 Pa. Multiply the given pressure (1.0×10^−10 mm of Hg) by this factor to express the pressure in SI units.
Step 2: Calculate the volume of the cylindrical chamber. Use the formula for the volume of a cylinder: V = πr²h, where r is the radius and h is the height. Convert the diameter (40 cm) to radius (20 cm) and height (30 cm) to meters before substituting into the formula.
Step 3: Use the ideal gas law to relate pressure, volume, and the number of molecules. The ideal gas law is given by: PV = Nk_BT, where P is pressure, V is volume, N is the number of molecules, k_B is Boltzmann's constant (1.38×10^−23 J/K), and T is temperature in Kelvin. Convert the temperature from Celsius to Kelvin using T(K) = T(°C) + 273.15.
Step 4: Rearrange the ideal gas law to solve for N, the number of molecules: N = (PV) / (k_BT). Substitute the values for pressure (in Pascals), volume (in cubic meters), Boltzmann's constant, and temperature (in Kelvin) into the equation.
Step 5: Perform the calculation to determine the number of molecules in the chamber. Ensure all units are consistent (e.g., pressure in Pascals, volume in cubic meters, temperature in Kelvin) before substituting into the formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. In this context, it allows us to calculate the number of gas molecules in a given volume at a specific temperature and pressure, assuming the gas behaves ideally.
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Molecular Density

Molecular density refers to the number of molecules per unit volume of a gas. It can be derived from the Ideal Gas Law and is crucial for determining how many molecules are present in a specific volume, such as the cylindrical chamber described in the question.
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Vacuum Pressure

Vacuum pressure is the pressure in a chamber that is lower than atmospheric pressure. In this case, the extremely low pressure of 1.0×10^−10 mm of Hg indicates a near-perfect vacuum, which significantly affects the behavior and density of gas molecules within the chamber.
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