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

Verified step by step guidance

1. Convert the temperature from Celsius to Kelvin by using the formula T(K) = T(°C) + 273.15. This is necessary because the ideal gas law requires temperature in Kelvin.
2. Calculate the volume of the cylindrical chamber using the formula V = \pi r^2 h, where r is the radius of the cylinder and h is the height. Convert all dimensions to meters before substituting into the formula.
3. Convert the pressure from mm of Hg to Pascals (Pa) using the conversion factor 1 mm Hg = 133.322 Pa, as the ideal gas law requires pressure in SI units.
4. Use the ideal gas law, PV = nRT, to find the number of moles of gas (n) in the chamber. Here, P is the pressure, V is the volume, R is the universal gas constant (8.314 J/(mol·K)), and T is the temperature in Kelvin.
5. Convert the number of moles of gas to the number of molecules by using Avogadro's number (6.022×10^23 molecules/mol). Multiply the number of moles by Avogadro's number to find the total number of molecules in the chamber.

Key Concepts

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 helps determine the number of gas molecules in the chamber by rearranging the formula to find n (number of moles) and then converting it to molecules using Avogadro's number.

Pressure Conversion

Understanding pressure units is crucial for this problem. The given pressure of 1.0×10^−10 mm of Hg must be converted to a more standard unit, such as Pascals (Pa), to apply the Ideal Gas Law effectively. This conversion ensures that all variables in the equation are compatible.

Volume of a Cylinder

To find the number of molecules in the chamber, we first need to calculate its volume. The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. This volume will be used in conjunction with the pressure and temperature to find the number of gas molecules present.