/*! 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 65 How much of a triatomic gas with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 18: Problem 65

How much of a triatomic gas with \(C_{V}=3 R\) would you have to add to \(6.0 \mathrm{~mol}\) of a monatomic gas to get a mixture whose thermodynamic behavior was like that of a diatomic gas?

Short Answer

Expert verified
The number of moles of the triatomic gas that needs to be added (denoted by \( n_{2} \)) is computed using the equation obtained by equating heat capacities. Find \( n_{2} \) by solving this mathematical equation.

Step by step solution

01

Define unknowns and equation

Let \( n_{2} \) be the unknown number of moles of triatomic gas that need to be added. From thermodynamics, we know that the heat capacity at constant volume \(C_{V}\), for an ideal gas, is the sum of heat capacities of all its components. Thus, we can write the equation relating the moles and specific heats of the monatomic and triatomic gases that form the mixture as: \( (n_{1} + n_{2})*C_{V,mixture}= n_{1} * C_{V,monatomic} + n_{2} * C_{V,triatomic} \). Here, \( n_{1} = 6.0 mol \) (moles of monatomic gas), \(C_{V,monatomic}=3/2*R \), \(C_{V,triatomic}=3*R \) and \(C_{V,mixture} \) is the heat capacity at constant volume for the mixture that behaves like a diatomic gas given by \(C_{V,mixture}=5/2*R \). R is the universal gas constant.
02

Substitute the values

Substitute the specified values into the equation to get the relation between \( n_{1} \) and \( n_{2} \) as: \( (6.0 + n_{2})*5/2 = 6.0*3/2 + n_{2}*3 \).
03

Solve for \( n_{2} \)

Solve the above equation for \( n_{2} \) to get the amount of triatomic gas which needs to be added.

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.

Monatomic Gas
Monatomic gases are composed of single atoms. These gases are simple in structure and include substances such as helium, neon, and argon.
A key property of monatomic gases is their heat capacity at constant volume. This is represented as \( C_{V} = \frac{3}{2} R \).
Here, \( R \) is the universal gas constant. The low heat capacity is due to the limited degrees of freedom.
  • In a monatomic gas, energy is purely translational.
  • There are no vibrational or rotational contributions.
As such, they are an ideal starting point for understanding the heat capacities of more complex gases.
Triatomic Gas
Triatomic gases consist of molecules made up of three atoms. A good example of such gases is water vapor \(H_2O\).
These molecules have more complex structures due to more atoms, leading to a higher heat capacity.
The heat capacity at constant volume for a triatomic gas is \(C_{V} = 3R\).
  • In contrast to monatomic gases, triatomic gases have different modes of energy storage 鈥� translational, rotational, and vibrational.
  • These additional modes increase their ability to store energy, resulting in a higher heat capacity.
Understanding triatomic gases can help in understanding specific heat variations among other gases.
Diatomic Gas
Diatomic gases consist of molecules made up of two atoms. Examples include nitrogen \(N_2\) and oxygen \(O_2\),
which we commonly find in Earth's atmosphere.
Their heat capacity at constant volume is \(C_{V} = \frac{5}{2} R \).
This higher value compared to monatomic gases is due to:
  • Two atoms allow for translational, rotational, and some vibrational motion.
  • These motions contribute to higher energy storage.
As a result, diatomic gases often appear in diverse thermodynamic analyses, such as gas mixtures.
Ideal Gas
The Ideal Gas is a theoretical model that helps to simplify gas behavior under various conditions. This model uses several assumptions that make the equations valid.
Key points of an ideal gas include:
  • The gas particles have negligible volume compared to the container.
  • There are no intermolecular forces between the particles.
The Ideal Gas Law is represented by the equation \( PV = nRT \),
where \( P \) is pressure, \( V \) is volume, \( n \) is moles, \( R \) is the gas constant, and \( T \) is temperature.
This equation shows the relationship between these properties. The ideal gas model plays a crucial role in thermodynamics.
Thermodynamics
Thermodynamics is the branch of physics that studies energy, heat, and their transformations. It applies to various practical activities - engines, refrigerators, and even biological processes.
Some essential concepts include:
  • The First Law of Thermodynamics, focusing on energy conservation.
  • The Second Law of Thermodynamics, concerning entropy.
Heat capacity is a thermodynamic property that indicates how much energy a substance stores per degree temperature rise.
In specific scenarios, like the mix of monatomic and triatomic gases, thermodynamics aids in analyzing their energy exchange. Understanding thermodynamics allows scientists and engineers to devise methods to efficiently manage energy in systems.

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

A gas with \(\gamma=7 / 5\) is at \(273 \mathrm{~K}\) when it's compressed isothermally to half of its original volume and then further compressed adiabatically to one-third of its original volume. Find its final temperature.

If an ice cube melts into water at \(0^{\circ} \mathrm{C}\), is there any work done on the system during the change?

One scheme for reducing greenhouse-gas emissions from coalfired power plants calls for capturing carbon dioxide and pumping it into the deep ocean, where the pressure is at least 350 atm. You're called to assess the energy cost of such a scheme for a power plant that produces electrical energy at the rate of \(1.0 \mathrm{GW}\) while at the same time emitting \(\mathrm{CO}_{2}\) at the rate of 1100 tonnes/hour. If \(\mathrm{CO}_{2}\) is extracted from the plant's smokestack at \(320 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure and then compressed adiabatically to 350 atm, what fraction of the plant's power output would be needed for the compression? Take \(\gamma=1.3\) for \(\mathrm{CO}_{2}\). (Your answer is a rough estimate because \(\mathrm{CO}_{2}\) doesn't behave like an ideal gas at very high pressures; also, it doesn't include the energy cost of separating the \(\mathrm{CO}_{2}\) from other stack gases or of transporting it to the compression site.)

A quasi-static process begins and ends at the same temperature. Is the process necessarily isothermal?

A gas with \(\gamma=1.40\) occupies \(6.25 \mathrm{~L}\) when it's at \(98.5 \mathrm{kPa}\) pressure. (a) What's the pressure after the gas is compressed adiabatically to \(4.18\) L? (b) How much work does that compression require?
See all solutions

Recommended explanations on Physics Textbooks

Oscillations

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Particle Model of Matter

Read Explanation

Energy Physics

Read Explanation

Fluids

Read Explanation

Force

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.