(a) In the deep space between galaxies, the density of atoms...

(a) In the deep space between galaxies, the density of atoms is as low as $10^{6}$ atoms $/ \mathrm{m}^{3},$ and the temperature is a frigid $2.7 \mathrm{K}$. What is the pressure? (b) What volume (in $\mathrm{m}^{3}$ ) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?

Key Concepts

Cube Geometry and Unit Conversion

When a gas occupies a cubic volume, the length of the sides of the cube can be determined by taking the cube root of the total volume. Unit conversion is an essential skill in this process, as it ensures that all quantities are expressed in compatible units, often requiring conversion between units such as meters and kilometers. This geometric reasoning connects algebraic manipulations with physical measurements in problems involving spatial dimensions.

Molar Volume

Molar volume is the volume occupied by one mole of a gas at a given temperature and pressure. Under standard conditions (commonly used in calculations) the molar volume is a well?defined value. Understanding molar volume is important because it links the number of particles in a gas to its macroscopic volume and allows conversion from microscopic density to observable macroscopic quantities.

Pressure

Pressure is the force exerted per unit area by gas molecules colliding with the walls of a container. In the context of the ideal gas law, pressure is directly proportional to both the number of molecules (or moles) and the absolute temperature. Calculating pressure from given density and temperature, using both the kinetic theory of gases and the ideal gas law, reveals how even a sparse, low?temperature gas can exert a measurable pressure.

Ideal Gas Law

The ideal gas law, usually written as PV = nRT, relates the pressure, volume, temperature, and the amount (in moles) of a gas. It is a central concept in thermodynamics and statistical mechanics that approximates the behavior of gases under many conditions. By expressing quantities in consistent units, one can calculate an unknown variable if the other three are known.

Mole Concept

The mole is a fundamental unit in chemistry that represents a specific number of particles, typically atoms or molecules, defined by Avogadro’s number (approximately 6.022×10^23 particles per mole). This concept allows scientists to convert between the microscopic scale (individual particles) and the macroscopic scale (measurable amounts of substance), making it essential for calculating quantities such as the molar volume of a gas.

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