Problem 28 - University Physics with Modern Physics

Question

How Close Together Are Gas Molecules? Consider an ideal gas at \(27^{\circ} \mathrm{C}\) and 1.00 atm pressure. To get some idea how close these molecules are to each other, on the average, imagine them to be uniformly spaced, with each molecule at the center of a small cube. (a) What is the length of an edge of each cube if adjacent cubes touch but do not overlap? (b) How does this distance compare with the diameter of a typical molecule? (c) How does their separation compare with the spacing of atoms in solids, which typically are about 0.3 \(\mathrm{nm}\) apart?

Step-by-Step Solution

Answer

Edge length of the cube is 0.344 nm. It slightly exceeds typical gas molecule diameters and is comparable to atomic spacing in solids.

Step 1: Convert Given Conditions

First, convert the temperature from Celsius to Kelvin. The given temperature is 27°C. The conversion formula is \( T(K) = T(°C) + 273.15 \). So:\[T = 27 + 273.15 = 300.15 \text{ K}\]The pressure is already given in atm, and it is 1.00 atm.

Step 2: Calculate Molar Volume

Use the ideal gas law to find the molar volume of the gas under these conditions. The ideal gas law is \( PV = nRT \). Rearrange to find the volume per mole \( V_m \):\[V_m = \frac{RT}{P}\]Where \( R = 0.0821 \text{ L atm K}^{-1} \text{mol}^{-1} \). Substitute the given values:\[V_m = \frac{0.0821 \times 300.15}{1.00} = 24.64 \text{ L/mol}\]Convert liters to cubic centimeters: 1 L = 1000 cm³, so \( V_m = 24640 \text{ cm}^3/\text{mol} \).

Step 3: Calculate Volume of a Single Molecule Cube

To find the volume of the cube surrounding one molecule, divide the molar volume by Avogadro's number \( N_A \approx 6.022 \times 10^{23} \text{mol}^{-1} \):\[V_{ ext{cube}} = \frac{24640}{6.022 \times 10^{23}} \approx 4.09 \times 10^{-23} \text{ cm}^3\]

Step 4: Find Edge Length of the Cube

The edge length \( a \) of a cube can be found by taking the cube root of the volume we calculated:\[a = (4.09 \times 10^{-23})^{1/3} \approx 3.44 \times 10^{-8} \text{ cm} = 0.344 \text{ nm}\]

Step 5: Compare to Molecular Diameter

Typical small gas molecules, such as oxygen or nitrogen, have diameters approximately 0.3 nm. The edge length is \( 0.344 \text{ nm} \), slightly larger than the diameter of a typical gas molecule. This suggests that the molecules are roughly one diameter apart from each other on average.

Step 6: Compare to Atomic Spacing in Solids

In solids, atoms are typically about 0.3 nm apart. The calculated separation of gas molecules is 0.344 nm, which is comparable to but slightly larger than the spacing in solids. This is consistent with the more widely spaced arrangement in gases compared to solids.

Key Concepts

Molecular Spacing in Gases

In the ideal gas concept, molecules are imagined to be spaced uniformly, with each molecule at the center of its own small cube. The side length of this cube represents the average distance between molecules. In our exercise, this distance is calculated using the ideal gas law, resulting in an average separation of around 0.344 nanometers (nm) for gas molecules at 27°C and 1.00 atm.



To understand what this spacing means, consider that molecules in a gas are constantly moving and colliding, thus they are not perfectly stationary at these calculated distances. However, the calculated edge length of 0.344 nm provides an idea of how close these molecules are in a typical scenario.

• This distance is slightly larger than the diameter of small gas molecules such as nitrogen or oxygen, which are about 0.3 nm in diameter.
• It means that on average, gas molecules are separated by distances approximately equal to their own diameter.

Understanding molecular spacing helps in comprehending how gases expand to fill their containers and how they respond to changes in pressure and temperature.

Molar Volume and Its Importance in Ideal Gases

Molar volume is a key concept when dealing with the ideal gas law as it represents the volume occupied by one mole of a gas at a given temperature and pressure. For the conditions defined in the original exercise, the molar volume is calculated by rearranging the ideal gas law, resulting in approximately 24.64 liters per mole.



The significance of molar volume extends beyond mere calculations. Here's why it matters:

• Molar volume is crucial for calculating densities of gases, which play a vital role in many practical applications like chemical reaction stoichiometry and industrial gas usage.
• Different gases have different molar volumes based on their molecular weights and intermolecular forces, impacting many physical properties.

This value implies that under standard conditions of temperature and pressure, any ideal gas will occupy 24.64 L/mol, demonstrating the predictability of gas behaviors under such conditions. This predictability supports various applications across scientific and engineering fields.

Atomic Spacing in Solids vs. Gases

Atomic spacing in solids is vastly different from that in gases, primarily due to the forces binding atoms or molecules together. Solids have closely packed atoms due to strong intermolecular or covalent bonds. Typically, atoms in a solid are 0.3 nm apart, indicating a tightly bound structure.



This spacing is contrasted by the more significant separation seen in gases, where molecules are about 0.344 nm apart under conditions described in the exercise.

• The relatively large distance between gas molecules allows for compressibility and adaptability to container shapes.
• In solids, restricted movement due to close packing results in fixed shape and volume.

Comparing these types of spacing helps understand why solids have defined structures and gases do not. The understanding of atomic and molecular spacing in different states of matter provides insights into their physical properties and behaviors.