Interstellar space far from any stars is usually filled with...

Interstellar space far from any stars is usually filled with atomic hydrogen (H) at a density of 1 atom/cm $^{3}$ and a very low temperature of $2.73 \mathrm{~K}$. a) Determine the pressure in interstellar space. b) What is the root-mean-square speed of the atoms? c) What would be the edge length of a cube that would contain atoms with a total of $1.00 \mathrm{~J}$ of energy?

Interstellar space far from any stars is usually filled with atomic hydrogen (H) at a density of 1 atom/cm $^{3}$ and a very low temperature of $2.73 \mathrm{~K}$.
a) Determine the pressure in interstellar space.
b) What is the root-mean-square speed of the atoms?
c) What would be the edge length of a cube that would contain atoms with a total of $1.00 \mathrm{~J}$ of energy?

Key Concepts

Energy Content and Volume Calculations

In a gas, the total thermal energy can be determined by multiplying the number of particles by the average kinetic energy per particle. Understanding the relation between energy content, particle density, and volume allows one to estimate the size of a region (such as a cube) that contains a given total energy. This concept is important for scaling microscopic energy contributions to macroscopic volumes.

Ideal Gas Law

The ideal gas law (PV = NkT) connects macroscopic properties of a gas such as pressure (P), volume (V), and temperature (T) with the microscopic count of particles (N) via the Boltzmann constant (k). This law underpins how low-density gases behave, even under extreme conditions like interstellar space, and provides a way to calculate the pressure when the temperature and particle density are known.

Boltzmann Constant

The Boltzmann constant (k) is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It acts as a bridge between macroscopic thermodynamic quantities and microscopic properties such as energy at the particle level, making it crucial for calculations involving thermal energy, pressure, and speed of gas atoms.

Kinetic Theory of Gases

Kinetic theory provides a microscopic explanation for the behavior of gases by relating the thermal energy of the gas particles to their motion. It explains how temperature is linked to the average kinetic energy of the atoms or molecules, forming the basis for deriving formulas like the root-mean-square (rms) speed, which quantifies the typical speed of particles in a gas.

Root-Mean-Square Speed

The root-mean-square (rms) speed is a statistical measure of the speed of particles in a gas, derived from their kinetic energy. It is calculated from the relation vrms = ?(3kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of an individual gas particle. This concept is crucial in understanding the dynamics of particles in any gaseous system.

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