Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 13: Problem 66

If 2.0 mol of nitrogen gas \(\left(\mathrm{N}_{2}\right)\) are placed in a cubic box, \(25 \mathrm{cm}\) on each side, at 1.6 atm of pressure, what is the rms speed of the nitrogen molecules?

Short Answer

Expert verified
Answer: The root-mean-square speed of the nitrogen molecules is approximately 515.6 m/s.

Step by step solution

01

Write down the given information.

The given information is: - Number of moles of nitrogen gas (\(n\)) = 2.0 moles - Volume of the container (\(V\)) = 25 cm³ on each side = \(25 \times 10^{-3}\) m³ - Pressure of the nitrogen gas (\(P\)) = 1.6 atm = \(1.6 \times 10^5\) Pa (1 atm ≈ \(10^5\) Pa)
02

Use the ideal gas law to find temperature.

Using the ideal gas law \(PV = nRT\), we can find the temperature \(T\): \(T = \frac{PV}{nR}\) where \(R\) is the universal gas constant (\(8.314 \frac{J}{mol\cdot K}\)). Plug in the given values: \(T = \frac{(1.6 \times 10^5 \, Pa)(25 \times 10^{-3} \, m^3)}{(2.0 \, mol)(8.314 \frac{J}{mol \cdot K})}\) Now, calculate the temperature: \(T \approx 241.2 \, K\)
03

Calculate molar mass of nitrogen

Nitrogen has an atomic mass of 14 g/mol. Since each nitrogen molecule is diatomic (\(N_2\)), the molar mass of nitrogen gas is: \(M = 2 \times (14 \frac{g}{mol}) = 28 \frac{g}{mol}\) Convert this molar mass to kilograms per mole: \(M = 28 \times 10^{-3} \, \frac{kg}{mol}\)
04

Use rms speed formula to find the speed of nitrogen molecules.

Now, use the rms speed formula: \(v_{rms} = \sqrt{\frac{3RT}{M}}\) Plug in the values for \(R\), \(T\), and \(M\): \(v_{rms} = \sqrt{\frac{(3) (8.314 \frac{J}{mol \cdot K}) (241.2 \, K)}{28 \times 10^{-3} \, \frac{kg}{mol}}}\) And finally, find the rms speed: \(v_{rms} \approx 515.6 \, \frac{m}{s}\) Hence, the rms speed of the nitrogen molecules in the container is approximately \(515.6 \, \frac{m}{s}\).

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

Estimate the number of \(\mathrm{H}_{2} \mathrm{O}\) molecules in a human body of mass \(80.2 \mathrm{kg} .\) Assume that, on average, water makes up about $62 \%$ of the mass of a human body.
A flight attendant wants to change the temperature of the air in the cabin from \(18^{\circ} \mathrm{C}\) to \(24^{\circ} \mathrm{C}\) without changing the number of moles of air per \(\mathrm{m}^{3} .\) What fractional change in pressure would be required?
In intergalactic space, there is an average of about one hydrogen atom per \(\mathrm{cm}^{3}\) and the temperature is \(3 \mathrm{K}\) What is the absolute pressure?
A mass of \(0.532 \mathrm{kg}\) of molecular oxygen is contained in a cylinder at a pressure of \(1.0 \times 10^{5} \mathrm{Pa}\) and a temperature of \(0.0^{\circ} \mathrm{C} .\) What volume does the gas occupy?
Show that, for an ideal gas, $$ P=\frac{1}{3} \rho v_{\mathrm{rms}}^{2} $$ where \(P\) is the pressure, \(\rho\) is the mass density, and \(v_{\mathrm{rms}}\) is the rms speed of the gas molecules.
See all solutions

Recommended explanations on Physics Textbooks

Quantum Physics

Read Explanation

Electromagnetism

Read Explanation

Circular Motion and Gravitation

Read Explanation

Classical Mechanics

Read Explanation

Torque and Rotational Motion

Read Explanation

Fluids

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