Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 18: Thermal Properties of Matter

Problem 21b

Chapter 18, Problem 21b

Modern vacuum pumps make it easy to attain pressures of the order of $10^{-13}$ atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. How many molecules would be present at the same temperature but at $1.00$ atm instead?

Verified step by step guidance

Start by recalling the ideal gas law, which is given by the equation:

P V = n R T

, where

P

is the pressure,

V

is the volume,

n

is the number of moles,

R

is the ideal gas constant, and

T

is the temperature.

To find the number of molecules, we need to determine the number of moles first. Rearrange the ideal gas law to solve for

n

n = \frac{P V}{R T}

Assume the volume

V

and temperature

T

remain constant. Substitute

P

with 1.00 atm, and use the ideal gas constant

R

in appropriate units (e.g., 0.0821 L atm/mol K).

Calculate the number of moles

n

using the rearranged equation. This will give you the moles of gas at 1.00 atm.

Finally, convert the number of moles to molecules using Avogadro's number, which is approximately

6.022 \times 10^{23}

molecules/mol. Multiply the moles by Avogadro's number to find the total number of molecules.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physics that relates the pressure, volume, and temperature of an ideal gas with the number of molecules. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. This law helps in calculating the number of molecules when conditions such as pressure change.

Mole Concept

The mole concept is essential for quantifying the amount of substance in chemistry and physics. One mole corresponds to Avogadro's number, approximately 6.022 x 10^23 molecules. Understanding this concept allows us to convert between the number of molecules and moles, which is crucial when using the Ideal Gas Law to find the number of molecules at different pressures.

Pressure

Pressure is the force exerted by gas molecules per unit area on the walls of a container. It is measured in units such as atmospheres (atm). In the context of the question, understanding how pressure changes affect the number of molecules in a given volume is key, as the pressure is directly proportional to the number of molecules in the Ideal Gas Law.

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Related Practice

Textbook Question

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare the average kinetic energies of the three types of atoms.

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Textbook Question

Modern vacuum pumps make it easy to attain pressures of the order of $10^{- 13}$ atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. At a pressure of $9.00 \times 10^{- 14}$ atm and an ordinary temperature of $300.0$ K, how many molecules are present in a volume of $1.00$ cm³?

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Textbook Question

What is the total translational kinetic energy of the air in an empty room that has dimensions $8.00$ m $times 12.00$ m $times 4.00$ m if the air is treated as an ideal gas at $1.00$ atm?

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A large organic molecule has a mass of $1.41 \times 10^{- 21}$ kg. What is the molar mass of this compound?

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Textbook Question

How many moles are in a $1.00$ -kg bottle of water? How many molecules? The molar mass of water is $18.0$ g/mol.

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Consider an ideal gas at $27$ °C and $1.00$ atm. To get some idea how close these molecules are to each other, on the average, imagine them to be uniformly spaced, with each molecule at the center of a small cube. What is the length of an edge of each cube if adjacent cubes touch but do not overlap?

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Modern vacuum pumps make it easy to attain pressures of the order of 10−1310^{-13}10−13 atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. How many molecules would be present at the same temperature but at 1.001.00 atm instead?

Verified video answer for a similar problem:

Key Concepts

Ideal Gas Law

Mole Concept

Pressure

Modern vacuum pumps make it easy to attain pressures of the order of $10^{-13}$ atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. How many molecules would be present at the same temperature but at $1.00$ atm instead?