/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 List all the degrees of freedom,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 25

List all the degrees of freedom, or as many as you can, for a molecule of water vapor. (Think carefully about the various ways in which the molecule can vibrate.)

Short Answer

Expert verified
Water vapor has 9 degrees of freedom: 3 translational, 3 rotational, and 3 vibrational.

Step by step solution

01

Understand the Basic Structure of Water Vapor

Water vapor, or steam, consists of H2O molecules. Each H2O molecule is composed of two hydrogen atoms and one oxygen atom arranged in a bent shape. This basic geometry is crucial for determining the types of motions the molecule can undergo.
02

Identify Translational Degrees of Freedom

For any molecule in a gas phase, there are 3 translational degrees of freedom. These correspond to the movement of the entire molecule along the x, y, and z axes.
03

Identify Rotational Degrees of Freedom

Since water is a non-linear molecule, it has 3 rotational degrees of freedom. These include rotations around each of its three principal axes.
04

Identify Vibrational Degrees of Freedom

The formula for the number of vibrational degrees of freedom for a non-linear molecule is \(3N-6\), where \(N\) is the number of atoms. For water (H2O), \(N=3\). Thus, it has \(3 \times 3 - 6 = 3\) vibrational degrees of freedom.
05

Describe the Vibrational Modes

For water, the three vibrational modes include symmetric stretching, asymmetric stretching, and bending (scissoring). Each of these modes contributes one degree of freedom.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Translational Motion
Translational motion involves the entire molecule moving as a single unit in space. In a gas such as water vapor, molecules are free to move in all directions.
This is characterized by three translational degrees of freedom:
  • The movement along the x-axis
  • The movement along the y-axis
  • The movement along the z-axis
These translational motions allow the molecule to traverse throughout the container, contributing to diffusion and the random motion characteristic of gases. The translational energy directly relates to the temperature of the gas through kinetic theory; higher temperature means more vigorous translational movement.
Rotational Motion
Molecules can rotate around their axes, which gives rise to rotational motion. This type of motion is especially relevant for molecules in the gas phase. Water, being a non-linear molecule due to its bent shape, possesses three rotational degrees of freedom.
These rotations occur around the principal axes of the molecule:
  • Rotation around an axis that runs through the center of mass and lies in the plane of the molecule
  • Rotation around another in-plane axis
  • Rotation around an axis perpendicular to the plane of the molecule
Rotational motion plays a key role in the distribution of molecular energy, especially in non-linear molecules like water vapor. As these rotations are quantized, they also contribute to the spectrum seen in rotational spectroscopy.
Vibrational Modes
Vibrational modes refer to the different ways in which the atoms within a molecule can vibrate relative to each other. This is especially relevant for complex molecules. A non-linear molecule such as water has vibrational degrees of freedom calculated using the formula: \[3N - 6\]where \(N\) is the number of atoms. For water, \(N = 3\), resulting in 3 vibrational modes. These include:
  • Symmetric stretching: both hydrogen atoms move towards and away from the oxygen atom synchronously.
  • Asymmetric stretching: one hydrogen atom moves in while the other moves out.
  • Bending (scissoring): the angle between the hydrogen atoms changes while the bonds themselves do not stretch or compress significantly.
These modes contribute to the vibrational spectra used in infrared and Raman spectroscopy, helping to identify molecular fingerprints.
Non-linear Molecule
Non-linear molecules, such as water, have a structure that cannot be simplified into a straight line. Water's bent shape with an approximate angle of 104.5 degrees between the hydrogen-oxygen-hydrogen bonds classifies it as non-linear.
This structural configuration significantly affects its physical properties:
  • It allows for three-dimensional rotations, leading to three rotational degrees of freedom
  • The shape influences the molecule鈥檚 polar characteristics, making it a strong solvent in many reactions
  • The geometry also dictates the vibrational modes, which affect its spectral absorption properties
Non-linear molecules exhibit more diversity in their energy states compared to linear molecules, influencing their interactions and reactivity.
Water Vapor Molecule
Water vapor, the gaseous state of water, is composed of H鈧侽 molecules that possess particular motion characteristics due to their structure.
Key aspects of water vapor molecules include:
  • Composed of two hydrogen atoms and one oxygen atom, arranged in a bent, non-linear geometry
  • Three translational, three rotational, and three vibrational degrees of freedom
  • The molecule鈥檚 polarity allows it to interact strongly with infrared light, leading to distinct absorption features in the spectroscopic data
The degrees of freedom define how these molecules move, rotate, and vibrate at the molecular level. Understanding this behavior aids in fields such as meteorology, where water vapor plays a vital role in weather systems and climate.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

When spring finally arrives in the mountains, the snow pack may be two meters deep, composed of \(50 \%\) ice and \(50 \%\) air. Direct sunlight provides about 1000 watts/m \(^{2}\) to earth's surface, but the snow might reflect \(90 \%\) of this energy. Estimate how many weeks the snow pack should last, if direct solar radiation is the only source of energy.

The Fahrenheit temperature scale is defined so that ice melts at\(32^{\circ} \mathrm{F}\) and water boils at \(212^{\circ} \mathrm{F}\) (a) Derive the formulas for converting from Fahrenheit to Celsius and back. (b) What is absolute zero on the Fahrenheit scale?

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion, $$P V=n R T\left(1+\frac{B(T)}{(V / n)}+\frac{C(T)}{(V / n)^{2}}+\cdots\right)$$ where the functions \(B(T), C(T),\) and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it's sufficient to omit the third term and concentrate on the second, whose coefficient \(B(T)\) is called the second virial coefficient (the first coefficient being 1 ). Here are some measured values of the second virial coefficient for nitrogen \(\left(\mathrm{N}_{2}\right):\) $$\begin{array}{cc} T(\mathrm{K}) & B\left(\mathrm{cm}^{3} / \mathrm{mol}\right) \\ \hline 100 & -160 \\ 200 & -35 \\ 300 & -4.2 \\ 400 & 9.0 \\ 500 & 16.9 \\ 600 & 21.3 \end{array}$$ (a) For each temperature in the table, compute the second term in the virial equation, \(B(T) /(V / n),\) for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions. (b) Think about the forces between molecules, and explain why we might expect \(B(T)\) to be negative at low temperatures but positive at high temperatures. (c) Any proposed relation between \(P, V,\) and \(T,\) like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation, $$\left(P+\frac{a n^{2}}{V^{2}}\right)(V-n b)=n R T$$ where \(a\) and \(b\) are constants that depend on the type of gas. Calculate the second and third virial coefficients \((B \text { and } C)\) for a gas obeying the van der Waals equation, in terms of \(a\) and \(b\). (Hint: The binomial expansion says that \((1+x)^{p} \approx 1+p x+\frac{1}{2} p(p-1) x^{2},\) provided that \(|p x| \ll 1 .\) Apply this approximation to the quantity \([1-(n b / V)]^{-1}\).) (d) Plot a graph of the van der Waals prediction for \(B(T),\) choosing \(a\) and \(b\) so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)

Uranium has two common isotopes, with atomic masses of 238 and \(235 .\) One way to separate these isotopes is to combine the uranium with fluorine to make uranium hexafluoride gas, UF \(_{6}\), then exploit the difference in the average thermal speeds of molecules containing the different isotopes. Calculate the rms speed of each type of molecule at room temperature, and compare them.

If you poke a hole in a container full of gas, the gas will start leaking out. In this problem you will make a rough estimate of the rate at which gas escapes through a hole. (This process is called effusion, at least when the hole is sufficiently small.) (a) Consider a small portion (area \(=A\) ) of the inside wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval \(\Delta t\) is \(P A \Delta t /(2 m \overline{v_{x}}),\) where \(P\) is the pressure, \(m\) is the average molecular mass, and \(\overline{v_{x}}\) is the average \(x\) velocity of those molecules that collide with the wall. (b) It's not easy to calculate \(\overline{v_{x}},\) but a good enough approximation is \((\overline{v_{x}^{2}})^{1 / 2}\) where the bar now represents an average over all molecules in the gas. Show that \((\overline{v_{x}^{2}})^{1 / 2}=\sqrt{k T / m}\) (c) If we now take away this small part of the wall of the container, the molecules that would have collided with it will instead escape through the hole. Assuming that nothing enters through the hole, show that the number \(N\) of molecules inside the container as a function of time is governed by the differential equation $$\frac{d N}{d t}=-\frac{A}{2 V} \sqrt{\frac{k T}{m}} N$$ Solve this equation (assuming constant temperature) to obtain a formula of the form \(N(t)=N(0) e^{-t / \tau},\) where \(\tau\) is the "characteristic time" for \(N\) \((\text { and } P)\) to drop by a factor of \(e\) (d) Calculate the characteristic time for a gas to escape from a 1 -liter container punctured by a \(1-\mathrm{mm}^{2}\) hole. (e) Your bicycle tire has a slow leak, so that it goes flat within about an hour after being inflated. Roughly how big is the hole? (Use any reasonable estimate for the volume of the tire.) (f) In Jules Verne's Round the Moon, the space travelers dispose of a dog's corpse by quickly opening a window, tossing it out, and closing the window. Do you think they can do this quickly enough to prevent a significant amount of air from escaping? Justify your answer with some rough estimates and calculations.
See all solutions

Recommended explanations on Physics Textbooks

Wave Optics

Read Explanation

Astrophysics

Read Explanation

Linear Momentum

Read Explanation

Classical Mechanics

Read Explanation

Nuclear Physics

Read Explanation

Measurements

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.