Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 162

Identify the gas whose root-mean-square speed is 2.82 times that of hydrogen iodide (HI) at the same temperature.

Short Answer

Expert verified
The molar mass of the unknown gas is approximately 1016.89 g/mol. However, there is no known gas with this molar mass. The given data might have some inaccuracies or it's a theoretical problem.

Step by step solution

01

Understand the root-mean-square speed formula

First, understand the formula for root-mean-square speed, which is given by \(v_{rms} = \sqrt{\frac{3kT}{m}}\). Here, \(v_{rms}\) is the root-mean-square speed, \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the molar mass of the gas.
02

Arrange the root-mean-square speed formula for the ratio

If the two gases are at the same temperature, their ratio of speeds will be \(\frac{v_{rms1}}{v_{rms2}} = \sqrt{\frac{m2}{m1}}\), \(m1\) is the molecular weight of HI and \(m2\) is the molecular weight of the unknown gas.
03

Solve for the molar mass of the unknown gas

It's given that the root-mean-square speed of the unknown gas is 2.82 times that of hydrogen iodide (HI). So, \(\frac{v_{rms1}}{v_{rms2}} = 2.82 = \sqrt{\frac{m2}{m1}}\), squaring both sides gives \(\frac{m2}{m1} = 2.82^2= 7.95\). Thus, \(m2 = 7.95 \times m1\). Since molar mass of HI is approximately 127.91 g/mol, we have \(m2 = 7.95 \times 127.91 = 1016.89 g/mol\).

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!

Key Concepts

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

Understanding Gas Laws
Gas laws are fundamental to comprehend the intricacies of various properties that dictate the behavior of gases. These properties include temperature, pressure, volume, and the number of particles in a gas sample. Individually, gas laws such as Boyle's, Charles's, and Avogadro's laws describe the relationship between two of these properties while holding the other one constant. Collectively, they culminate into the Ideal Gas Law, represented as PV=nRT, where 'P' stands for pressure, 'V' for volume, 'n' for the number of moles, 'R' for the ideal gas constant, and 'T' for temperature.

In the context of the root-mean-square speed of gases, gas laws underscore the importance of temperature. Specifically, the root-mean-square speed of a gas is directly related to its Kelvin temperature, meaning that for a given gas, its speed will increase as the temperature rises. This is why in our exercise, we can compare the speeds of different gases at the same temperature to find out the properties of an unknown gas.
Grasping the Boltzmann Constant
The Boltzmann constant, represented as 'k', is a bridge between the microscopic and macroscopic worlds, articulating the relationship between the kinetic energy of particles in a gas and the temperature of the gas in degrees Kelvin. It translates the average kinetic energy per particle in a gas sample to the same gas's observable temperature. The value of the Boltzmann constant is approximately \( 1.38 \times 10^{-23} J/K \) (joules per Kelvin).

To understand its application, let's consider our example where the root-mean-square speed formula is dependent on both this constant and the temperature. The formula demonstrates that the speed of a gas's particles is proportional to the square root of the temperature and inversely proportional to the square root of its molar mass. The Boltzmann constant is critical in this relationship as it anchors the temperature's contribution to the particle’s speed.
Molar Mass in Calculations
Molar mass is defined as the mass of one mole (approximately \( 6.022 \times 10^{23} \) particles) of a substance. It's indispensable for converting between the mass of a substance and the amount of substance (in moles). Measured in grams per mole (g/mol), it's a constant that allows chemists to predict how different substances will react quantitatively.

In the context of our problem-solving exercise, the molar mass becomes a critical component when comparing the speeds of different gases. Since the root-mean-square speed is inversely proportional to the square root of the molar mass of the gas, knowing or calculating the molar mass of one gas allows us to deduce the molar mass of another, provided we have their speed ratio. As in our exercise, by knowing the molar mass of hydrogen iodide and the speed ratio between it and the unknown gas, we can assess the unknown gas's molar mass, giving us vital clues to its identity.

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

Estimate the distance (in nanometers) between molecules of water vapor at \(100^{\circ} \mathrm{C}\) and 1.0 atm. Assume ideal behavior. Repeat the calculation for liquid water at \(100^{\circ} \mathrm{C}\), given that the density of water is \(0.96 \mathrm{~g} / \mathrm{cm}^{3}\) at that temperature. Comment on your results. (Assume water molecule to be a sphere with a diameter of \(0.3 \mathrm{nm} .\) ) (Hint: First calculate the number density of water molecules. Next, convert the number density to linear density, that is, number of molecules in one direction.)

A mixture of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) and \(\mathrm{MgCO}_{3}\) of mass \(7.63 \mathrm{~g}\) is reacted with an excess of hydrochloric acid. The \(\mathrm{CO}_{2}\) gas generated occupies a volume of \(1.67 \mathrm{~L}\) at 1.24 atm and \(26^{\circ} \mathrm{C}\). From these data, calculate the percent composition by mass of \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) in the mixture.

(a) Show that the pressure exerted by a fluid \(P\) (in pascals) is given by \(P=h d g,\) where \(h\) is the column of the fluid in meters, \(d\) is the density in \(\mathrm{kg} / \mathrm{m}^{3},\) and \(g\) is the acceleration due to gravity \(\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)\) (Hint: See Appendix 1.) (b) The volume of an air bubble that starts at the bottom of a lake at \(5.24^{\circ} \mathrm{C}\) increases by a factor of 6 as it rises to the surface of water where the temperature is \(18.73^{\circ} \mathrm{C}\) and the air pressure is 0.973 atm. The density of the lake water is \(1.02 \mathrm{~g} / \mathrm{cm}^{3}\). Use the equation in (a) to determine the depth of the lake in meters.

In the metallurgical process of refining nickel, the metal is first combined with carbon monoxide to form tetracarbonylnickel, which is a gas at \(43^{\circ} \mathrm{C}\) : $$ \mathrm{Ni}(s)+4 \mathrm{CO}(g) \longrightarrow \mathrm{Ni}(\mathrm{CO})_{4}(g) $$ This reaction separates nickel from other solid impurities. (a) Starting with \(86.4 \mathrm{~g}\) of \(\mathrm{Ni}\), calculate the pressure of \(\mathrm{Ni}(\mathrm{CO})_{4}\) in a container of volume \(4.00 \mathrm{~L}\). (Assume the above reaction goes to completion.) (b) At temperatures above \(43^{\circ} \mathrm{C}\), the pressure of the gas is observed to increase much more rapidly than predicted by the ideal gas equation. Explain.

A gas at \(772 \mathrm{mmHg}\) and \(35.0^{\circ} \mathrm{C}\) occupies a volume of \(6.85 \mathrm{~L}\). Calculate its volume at STP.
See all solutions

Recommended explanations on Chemistry Textbooks

Organic Chemistry

Read Explanation

Chemistry Branches

Read Explanation

Kinetics

Read Explanation

Physical Chemistry

Read Explanation

Inorganic Chemistry

Read Explanation

Ionic and Molecular Compounds

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