Integrated circuits are manufactured in vacuum chambers in which the air pressure is 1.0 x 10^-10 of Hg. What are (a) the number density and (b) the mean free path of a molecule? Assume T = 20℃.

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Convert the given pressure from mmHg to Pascals (Pa) using the conversion factor: 1 mmHg = 133.322 Pa. The pressure in Pascals is given by \( P = (1.0 \times 10^{-10}) \times 133.322 \).

Use the ideal gas law \( PV = nRT \) to find the number density \( n/V \), where \( n/V = P / (RT) \). Here, \( R \) is the universal gas constant (8.314 J/(mol·K)), and \( T \) is the temperature in Kelvin. Convert the temperature from Celsius to Kelvin using \( T(K) = T(°C) + 273.15 \).

Substitute the values of \( P \), \( R \), and \( T \) into the formula \( n/V = P / (RT) \) to calculate the number density of molecules per unit volume.

To calculate the mean free path \( \lambda \), use the formula \( \lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P} \), where \( k_B \) is the Boltzmann constant (1.38 \(\times\) 10^{-23} J/K), \( d \) is the diameter of a molecule (assume a typical value for air molecules, approximately 3.7 \(\times\) 10^{-10} m), and \( P \) is the pressure in Pascals.

Substitute the values of \( k_B \), \( T \), \( d \), and \( P \) into the formula for \( \lambda \) to determine the mean free path of a molecule in the vacuum chamber.

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Number Density

Number density refers to the number of particles (molecules, atoms, etc.) per unit volume in a given space. It is typically expressed in units such as particles per cubic meter. In the context of the question, calculating the number density of air molecules at a specific pressure involves using the ideal gas law, which relates pressure, volume, and temperature to the number of particles.

Mean Free Path

The mean free path is the average distance a molecule travels between collisions with other molecules. It is influenced by the number density of the molecules and their effective cross-sectional area for collisions. In low-pressure environments, such as the vacuum chamber mentioned, the mean free path increases significantly, allowing molecules to travel longer distances without colliding.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that describes the behavior of ideal gases. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law is essential for calculating properties like number density and mean free path, especially when dealing with gases at various pressures and temperatures.