Question

Interstellar space contains mostly hydrogen atoms at a concentration of about 1 atom/cm \(^{3}\). (a) Calculate the pressure of the \(\mathrm{H}\) atoms. (b) Calculate the volume (in liters) that contains \(1.0 \mathrm{~g}\) of \(\mathrm{H}\) atoms. The temperature is \(3 \mathrm{~K}\)

Short answer

We are given the concentration of hydrogen atoms as 1 atom/cm\(^{3}\), and the temperature as 3 K. To calculate the pressure, we will use the ideal gas law \(PV = nRT\), where \(n\) is the number of moles, \(R\) is the ideal gas constant, \(P\) is the pressure, and \(V\) is the volume. Since \(n = \frac{N}{N_A}\), where \(N\) is the number of hydrogen atoms and \(N_A\) is Avogadro's number, we must convert the concentration from atoms/cm\(^{3}\) to moles/cm\(^{3}\).

Step-by-step solution

Identify Given Values for Pressure Calculation

We are given the concentration of hydrogen atoms as 1 atom/cm\(^{3}\), and the temperature as 3 K. To calculate the pressure, we will use the ideal gas law \(PV = nRT\), where \(n\) is the number of moles, \(R\) is the ideal gas constant, \(P\) is the pressure, and \(V\) is the volume. Since \(n = \frac{N}{N_A}\), where \(N\) is the number of hydrogen atoms and \(N_A\) is Avogadro's number, we must convert the concentration from atoms/cm\(^{3}\) to moles/cm\(^{3}\).

Key concepts

Hydrogen Atoms

Hydrogen atoms are the simplest atoms in the universe, consisting of just one proton and one electron. These atoms are extremely light and are the most abundant element in the cosmos. In interstellar space, such as the region between stars, hydrogen atoms usually exist as neutral hydrogen atoms rather than molecules due to the low temperatures and low densities found there.

Given that outer space contains about 1 hydrogen atom per cubic centimeter, this might seem sparse compared to Earth's atmosphere. However, this is typical of interstellar space where such low densities are normal. This sparse density of hydrogen plays a significant role in understanding various astrophysical phenomena, such as the formation of stars and galaxies.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry and physics, used to solve problems involving gases. The law is expressed as:\[ PV = nRT \]

Here, \(P\) is the pressure of the gas, \(V\) is the volume, \(n\) is the number of moles of the gas, \(R\) is the ideal gas constant (8.314 J/mol·K), and \(T\) is the temperature in Kelvin.

This law assumes that gases are composed of tiny particles in rapid, random motion, which perfectly translate kinetic energy into the pressure exerted by the gas within a container. While it provides a very good approximation for many gases under a wide range of conditions, it performs best under conditions of low pressure and high temperature. Known deviations occur in gases that are either highly compressed or at very low temperatures, but in the vast, low-pressure environment of space, the ideal gas law holds quite well.

Pressure Calculation

To calculate the pressure of hydrogen atoms in interstellar space, we need to switch from thinking about individual atoms to thinking about moles—the standard chemical unit. Here’s a simplified process to follow:

The concentration given is 1 atom/cm\(^3\). First, convert this concentration into moles using Avogadro's number, which is approximately 6.022 x 10\(^{23}\) atoms/mol. This conversion helps because the ideal gas law is naturally defined with moles, not atoms.

Once converted, we can apply the ideal gas law \(PV = nRT\), rearranging it to solve for pressure \(P\):\[ P = \frac{nRT}{V} \]

Plugging in the given temperature (3 K) and the converted concentration, we obtain the pressure of this diffuse interstellar gas. The result will indicate the influence of these sparse hydrogen atoms in the vast emptiness of space.

Moles Conversion

Moles are a central concept in chemistry, serving as the link between particle counts and mass. One mole represents Avogadro's number—approximately 6.022 x 10\(^{23}\) particles, whether they be atoms, molecules, or ions.

To convert between these units, especially when dealing with gases like hydrogen, you need to connect the atomic scale to the macroscopic scale. For instance, if you are given a mass of hydrogen, you can find the number of moles using:\[ n = \frac{m}{M} \] where \(m\) is the mass of the gas, and \(M\) is the molar mass of hydrogen, about 2 g/mol for diatomic hydrogen (H\(_2\)), but 1 g/mol for atomic hydrogen.

Such conversions are vital for using the ideal gas law, as this law depends on moles rather than the mass or a simple count of atoms. Being comfortable with these conversions helps better understand, predict, and manipulate chemical and physical systems.