Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 14: Problem 8

Mark these statements as being True or False for a binary mixture of substances \(A\) and \(B\). (a) The density of a mixture is always equal to the sum of the densities of its constituents. (b) The ratio of the density of component \(A\) to the density of component \(B\) is equal to the mass fraction of component \(A\). (c) If the mass fraction of component \(A\) is greater than \(0.5\), then at least half of the moles of the mixture are component \(A\). (d) If the molar masses of \(A\) and \(B\) are equal to each other, then the mass fraction of \(A\) will be equal to the mole fraction of \(A\). (e) If the mass fractions of \(A\) and \(B\) are both \(0.5\), then the molar mass of the mixture is simply the arithmetic average of the molar masses of \(A\) and \(B\).

Short Answer

Expert verified
In summary: a) The density of a mixture is always equal to the sum of the densities of its constituents. - False b) The ratio of the density of component A to the density of component B is equal to the mass fraction of component A. - False c) If the mass fraction of component A is greater than 0.5, then at least half of the moles of the mixture are component A. - False d) If the molar masses of A and B are equal to each other, then the mass fraction of A will be equal to the mole fraction of A. - True e) If the mass fractions of A and B are both 0.5, then the molar mass of the mixture is simply the arithmetic average of the molar masses of A and B. - True

Step by step solution

01

Statement (a): Maintaining True or False

The density of a mixture is always equal to the sum of the densities of its constituents. The total density of a binary mixture is given by the relation: \(\rho_{mixture} = \frac{m_{A} + m_{B}}{V_{mixture}}\) However, the volume of the mixture may not be equal to the sum of the volumes of its constituents. Therefore, this statement is False.
02

Statement (b): Maintaining True or False

The ratio of the density of component \(A\) to the density of component \(B\) is equal to the mass fraction of component A. Mass fraction of component A (\(\chi_A\)) is defined as the mass of A divided by the total mass of the mixture: \(\chi_A = \frac{m_A}{m_A + m_B}\) But the given statement talks about the ratio of densities, which has no direct relation to the mass fraction. Therefore, the statement is False.
03

Statement (c): Maintaining True or False

If the mass fraction of component \(A\) is greater than \(0.5\), then at least half of the moles of the mixture are component A. Mole fraction of component A (\(x_A\)) is defined as the moles of A divided by the total moles of the mixture: \(x_A = \frac{n_A}{n_A + n_B}\) However, mass fraction and mole fraction have different units and are not directly related. The mass fraction tells us the fraction of mass in terms of substance A, while the mole fraction tells us the number of moles of substance A present. The statement makes an inaccurate comparison, so it is False.
04

Statement (d): Maintaining True or False

If the molar masses of \(A\) and \(B\) are equal to each other, then the mass fraction of \(A\) will be equal to the mole fraction of \(A\). Let's denote the molar masses of A and B as \(M_A\) and \(M_B\). Since \(M_A = M_B\), we can write the relation: \(\chi_A = \frac{m_A}{m_A + m_B} = \frac{n_A M_A}{n_A M_A + n_B M_A}\) Since \(M_A\) is equal to \(M_B\), we can cancel out the molar masses: \(\chi_A = \frac{n_A}{n_A + n_B} = x_A\) Thus, the statement is True.
05

Statement (e): Maintaining True or False

If the mass fractions of \(A\) and \(B\) are both \(0.5\), then the molar mass of the mixture is simply the arithmetic average of the molar masses of \(A\) and \(B\). For a binary mixture, the mass fraction of the two components always sums up to 1: \(\chi_A + \chi_B = 1\) Given mass fractions of A and B are both 0.5, so \(\chi_A = \chi_B = 0.5\) Now, let's find the molar mass of the mixture: \(M_{mixture} = \chi_A M_A + \chi_B M_B = 0.5 M_A + 0.5 M_B\) This is the arithmetic average of the molar masses of A and B, so the statement is True.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

What is the physical significance of the Sherwood number? How is it defined? To what dimensionless number does it correspond in heat transfer? What does a Sherwood number of 1 indicate for a plane fluid layer?

The pressure in a pipeline that transports helium gas at a rate of \(5 \mathrm{lbm} / \mathrm{s}\) is maintained at \(14.5\) psia by venting helium to the atmosphere through a \(0.25\)-in-internal-diameter tube that extends \(30 \mathrm{ft}\) into the air. Assuming both the helium and the atmospheric air to be at \(80^{\circ} \mathrm{F}\), determine \((a)\) the mass flow rate of helium lost to the atmosphere through the tube, (b) the mass flow rate of air that infiltrates into the pipeline, and \((c)\) the flow velocity at the bottom of the tube where it is attached to the pipeline that will be measured by an anemometer in steady operation.

Heat convection is expressed by Newton's law of cooling as \(\dot{Q}=h A_{s}\left(T_{s}-T_{\infty}\right)\). Express mass convection in an analogous manner on a mass basis, and identify all the quantities in the expression and state their units.

The top section of an 8-ft-deep \(100-\mathrm{ft} \times 100-\mathrm{ft}\) heated solar pond is maintained at a constant temperature of \(80^{\circ} \mathrm{F}\) at a location where the atmospheric pressure is \(1 \mathrm{~atm}\). If the ambient air is at \(70^{\circ} \mathrm{F}\) and 100 percent relative humidity and wind is blowing at an average velocity of \(40 \mathrm{mph}\), determine the rate of heat loss from the top surface of the pond by ( \(a\) ) forced convection, \((b)\) radiation, and \((c)\) evaporation. Take the average temperature of the surrounding surfaces to be \(60^{\circ} \mathrm{F}\).

Saturated water vapor at \(25^{\circ} \mathrm{C}\left(P_{\text {sat }}=3.17 \mathrm{kPa}\right)\) flows in a pipe that passes through air at \(25^{\circ} \mathrm{C}\) with a relative humidity of 40 percent. The vapor is vented to the atmosphere through a \(7-\mathrm{mm}\) internal-diameter tube that extends \(10 \mathrm{~m}\) into the air. The diffusion coefficient of vapor through air is \(2.5 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). The amount of water vapor lost to the atmosphere through this individual tube by diffusion is (a) \(1.02 \times 10^{-6} \mathrm{~kg}\) (b) \(1.37 \times 10^{-6} \mathrm{~kg}\) (c) \(2.28 \times 10^{-6} \mathrm{~kg}\) (d) \(4.13 \times 10^{-6} \mathrm{~kg}\) (e) \(6.07 \times 10^{-6} \mathrm{~kg}\)
See all solutions

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Read Explanation

Quantum Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Medical Physics

Read Explanation

Waves Physics

Read Explanation

Electromagnetism

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.

Sign-up for free