Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 13: Problem 52

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?

Short Answer

Expert verified
Answer: The approximate absolute pressure in intergalactic space is 4.14 x 10⁻¹⁸ Pa.

Step by step solution

01

Data given in the question

We are given the following data: - Average density of hydrogen atoms: 1 atom/\(\mathrm{cm}^{3}\) - Temperature: \(3 \mathrm{K}\) We will use the ideal gas law to calculate the absolute pressure.
02

Ideal gas law and constants

The ideal gas law can be expressed as: \(P V = n R T\) where \(P\) - Pressure (in Pascals) \(V\) - Volume (in cubic meters) \(n\) - number of moles of gas \(R\) - Ideal gas constant (\(8.314 \mathrm{J\, mol^{-1} K^{-1}}\)) \(T\) - Temperature (in Kelvin) Also, we need to recall: - Avogadro's number: \(N_A = 6.022 \times 10^{23}\, \mathrm{atoms/mol}\) - Boltzmann's constant: \(k_B = 1.38 \times 10^{-23}\, \mathrm{J/K}\) The pressure we want to find is: \(P = \frac{n R T}{V} = \frac{N k_B T}{V' * V}\) where \(N\) - number of hydrogen atoms per unit volume \(V'\) - Volume of unit hydrogen atom \(V\) - total volume of the space
03

Number of hydrogen atoms per cm³

We are given that there is one hydrogen atom per cm³. Hence, the number of hydrogen atoms per unit volume is: \(N = \frac{1\, \mathrm{atom}}{\mathrm{cm}^{3}}\).
04

Convert volume to meters³

It's important to note that all units in the formula should be in SI (International System) units. So, we need to convert volume from cubic centimeters to cubic meters: \(V' = \frac{1\, \mathrm{cm}^3}{10^6} = 10^{-6}\, \mathrm{m}^3\)
05

Apply the formula for pressure

Now we can use the ideal gas law to calculate the absolute pressure in intergalactic space: \(P = \frac{N k_B T}{V' * V} = \frac{1\, \mathrm{atom/cm^3} \cdot 1.38 \times 10^{-23}\, \mathrm{J/K} \cdot 3 \mathrm{K}}{10^{-6}\, \mathrm{m^3}}\) \(P = 4.14 \times 10^{-18} \, \mathrm{Pa}\) In intergalactic space, the absolute pressure is approximately \(4.14 \times 10^{-18}\, \mathrm{Pa}\).

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

Incandescent light bulbs are filled with an inert gas to lengthen the filament life. With the current off (at \(T=\) \(\left.20.0^{\circ} \mathrm{C}\right),\) the gas inside a light bulb has a pressure of \(115 \mathrm{kPa} .\) When the bulb is burning, the temperature rises to \(70.0^{\circ} \mathrm{C} .\) What is the pressure at the higher temperature?
As a Boeing 747 gains altitude, the passenger cabin is pressurized. However, the cabin is not pressurized fully to atmospheric $\left(1.01 \times 10^{5} \mathrm{Pa}\right),$ as it would be at sea level, but rather pressurized to \(7.62 \times 10^{4} \mathrm{Pa}\). Suppose a 747 takes off from sea level when the temperature in the airplane is \(25.0^{\circ} \mathrm{C}\) and the pressure is \(1.01 \times 10^{5} \mathrm{Pa} .\) (a) If the cabin temperature remains at \(25.0^{\circ} \mathrm{C},\) what is the percentage change in the number of moles of air in the cabin? (b) If instead, the number of moles of air in the cabin does not change, what would the temperature be?
At \(0.0^{\circ} \mathrm{C}\) and \(1.00 \mathrm{atm}, 1.00 \mathrm{mol}\) of a gas occupies a volume of \(0.0224 \mathrm{m}^{3} .\) (a) What is the number density? (b) Estimate the average distance between the molecules. (c) If the gas is nitrogen \(\left(\mathrm{N}_{2}\right),\) the principal component of air, what is the total mass and mass density?
A temperature change \(\Delta T\) causes a volume change \(\Delta V\) but has no effect on the mass of an object. (a) Show that the change in density $\Delta \rho\( is given by \)\Delta \rho=-\beta \rho \Delta T$. (b) Find the fractional change in density \((\Delta \rho / \rho)\) of a brass sphere when the temperature changes from \(32^{\circ} \mathrm{C}\) to \(-10.0^{\circ} \mathrm{C}\)
What is the kinetic energy per unit volume in an ideal gas at (a) \(P=1.00\) atm and (b) \(P=300.0\) atm?
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Waves Physics

Read Explanation

Electrostatics

Read Explanation

Scientific Method Physics

Read Explanation

Classical Mechanics

Read Explanation

Dynamics

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