Problem 79 - University Physics with Modern Physics

Question

(a) For what mass of molecule or particle is \(v_{\mathrm{rms}}\) equal to 1.00 \(\mathrm{mm} / \mathrm{s}\) at 300 \(\mathrm{K} ?\) (b) If the particle is an ice crystal, how many molecules does it contain? The molar mass of water is 18.0 \(\mathrm{g} / \mathrm{mol}\) . (c) Calculate the diameter of the particle if it is a spherical piece of ice. Would it be visible to the naked eye?

Step-by-Step Solution

Answer

The mass is \(1.24 \times 10^{-13}\text{ kg}\). It contains \(4.15 \times 10^{10}\) molecules. The diameter is \(9 \times 10^{-7}\text{ m}\); not visible.

Step 1: Understanding the RMS Speed Formula

The root mean square speed \( v_{\text{rms}} \) for gas molecules is given by the formula: \( v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \), where \( k \) is Boltzmann's constant \( 1.38 \times 10^{-23} \text{ J/K} \), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a particle in kilograms.

Step 2: Solve for Mass of Particle

Re-arrange the equation \( v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \) to solve for the mass \( m \): \[ m = \frac{3kT}{v_{\text{rms}}^2}.\]Plug in the values: \( k = 1.38 \times 10^{-23} \text{ J/K} \), \( T = 300 \text{ K} \), and \( v_{\text{rms}} = 1.00 \times 10^{-3} \text{ m/s} \). Calculate \( m \). The result is \( m = 1.24 \times 10^{-13} \text{ kg} \).

Step 3: Calculate Number of Molecules in Ice Crystal

Start from the mass of this conceivable ice particle : \( 1.24 \times 10^{-13} \text{ kg} \). Convert it to grams: \( 1.24 \times 10^{-10} \text{ g} \). Knowing that the molar mass of water \( \text{M}_{\text{H}_2\text{O}} = 18 \text{ g/mol} \), applying the formula number of moles \( n = \frac{\text{mass}}{\text{molar mass}} \).Then, find the number of molecules: \( n \cdot N_A \), where \( N_A \) is Avogadro's number \( 6.022 \times 10^{23} \text{mol}^{-1} \).The number of molecules is approximately \( 4.15 \times 10^{10} \).

Step 4: Estimate the Sphere's Diameter and Visibility

Assume the ice crystal is a sphere. Use the formula for volume \( V = \frac{4}{3} \pi r^3 \) with the density of ice \( \rho = 917 \text{ kg/m}^3 \) to find radius: \( V = \frac{\text{mass}}{\rho} \).Then, \( r = \left( \frac{3V}{4\pi} \right)^{1/3} \).Calculate the diameter \( d = 2r \).For \( m = 1.24 \times 10^{-13} \text{ kg} \), the diameter is about \( 9 \times 10^{-7} \text{ m} \). Since it is smaller than 0.1 mm, it isn't visible to the naked eye.

Key Concepts

Root Mean Square Speed

The root mean square speed (\(v_{\text{rms}}\)) is a crucial concept in the kinetic theory of gases that measures how fast particles move on average within a substance. It is expressed by the formula:

• \( v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \)

Here, \( k \) is Boltzmann's constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a molecule or particle in kilograms.
This formula helps us understand the relationship between the temperature of a gas and the speed of its particles. In simpler terms, as the temperature increases, particles move faster. This increase in speed is due to the extra energy particles achieve at higher temperatures.
Understanding \( v_{\text{rms}} \) is essential for predicting how particles behave when conditions such as temperature change. This concept is not only vital for gases but also extends to estimating the motion in liquid and solid particles.

Ice Crystal

An ice crystal is a solid particle formed from water molecules that align themselves in a unique hexagonal lattice structure. This alignment gives ice its rigid shape and distinct properties, such as being less dense than liquid water, allowing it to float.
In our exercise, the ice crystal is imagined as a small particle that moves with a specific speed at a given temperature. This scenario provides insight into how even solids can have kinetic energy, though significantly less than in gaseous or liquid water.
Calculating the number of water molecules in such an ice particle enables us to appreciate the molecular composition involved in everyday substances like ice. When we count all the molecules in an ice crystal, we see the intricate construction that takes place even in commonplace ice.

Molar Mass of Water

The molar mass of water (\(\text{M}_{\text{H}_2\text{O}}\)) is a fundamental quantity that represents the mass of one mole of water molecules.For water, this mass is 18 grams per mole.
This value is crucial in different calculations, such as converting between mass and the number of molecules using Avogadro's number. The molar mass helps in understanding the scale of chemical reactions and how substances are transformed at the molecular level.
Understanding molar mass allows us to calculate the number of water molecules in a given mass of water or ice, linking the macroscopic scale to the molecular scale. In this context, it aids in determining the number of water molecules in the ice crystal when combined with other constants and formulas, such as Avogadro's number.

Spherical Particle Volume

The concept of spherical particle volume is integral to understanding the physical space a particle occupies. For spheres, the volume (\(V\)) is determined by:

• \( V = \frac{4}{3}\pi r^3 \)

Here \(r\) is the radius of the sphere.
In the exercise, this formula is essential to estimate the size of an ice crystal. To find the radius, we rearrange the volume formula, considering the particle's mass and density:

• \( r = \left( \frac{3V}{4\pi} \right)^{1/3} \)

This allows us to determine its diameter, which is twice the radius (\(d = 2r\)).
Calculating the volume and dimensions of microscopic particles helps in estimating their physical properties and visibility. It shows whether they can be observed by the naked eye, based on size thresholds.

Avogadro's Number

Avogadro's number (\(N_A\)) is a fundamental constant used to quantify the number of entities in one mole of a substance. It is approximately \(6.022 \times 10^{23}\) per mole.
This number allows us to make quantitative connections between the macroscopic amounts we see and measure and the microscopic world of atoms and molecules. In the context of the exercise, Avogadro's number helps translate the number of moles of water into the actual number of water molecules within the ice crystal.
Understanding Avogadro’s number is central for converting masses to molecules and atoms in chemical and physical calculations. It underscores the immense scale of microscopic particles present in even the smallest amounts of material.