Modern vacuum pumps make it easy to attain pressures of the...

Modern vacuum pumps make it easy to attain pressures of the order of $10^{-13}$ atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (a) At a pressure of $9.00 \times 10^{-14}$ atm and an ordinary temperature of $300.0 \mathrm{K},$ how many molecules are present in a volume of 1.00 $\mathrm{cm}^{3} ?$ (b) How many molecules would be present at the same temperature but at 1.00 atm instead?

Key Concepts

Low-Pressure Conditions and Vacuum Technology

Low-pressure conditions, such as those achieved with modern vacuum pumps, allow scientists and engineers to study gas behavior in environments where only a very small number of molecules are present. This concept is important for applications in high-vacuum technology and experiments requiring extreme reduction of molecular interactions, and it underscores the need to carefully apply gas laws under such conditions.

Ideal Gas Law

The ideal gas law is a fundamental equation in physical chemistry and thermodynamics that relates the pressure, volume, temperature, and amount (in moles) of an ideal gas through the equation PV = nRT. This law allows one to predict how a gas will behave when one or more of these parameters are changed, and it is particularly useful for calculating quantities such as the number of molecules present in a given volume at a specified pressure and temperature.

Mole Concept and Avogadro's Number

The mole is a central concept in chemistry that provides a bridge between the microscopic world of atoms and molecules and the macroscopic quantities that we can measure. Avogadro's number, approximately 6.02×10^23, defines the number of particles (atoms or molecules) in one mole of a substance. This concept is critical when converting between the amount of substance (in moles) and the actual number of particles present in a sample.

Pressure and Unit Conversions

Pressure is a measure of the force exerted per unit area and is commonly measured in atmospheres (atm), pascals (Pa), or other units. In many practical problems, it is necessary to convert units and adjust the volume dimensions (such as converting cubic centimeters to liters) to ensure consistency when applying equations like the ideal gas law. Understanding these conversions is essential for accurate computation of gas properties.

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