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 23a

Chapter 18, Problem 23a

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?

Verified step by step guidance

Convert the temperature from Celsius to Kelvin. The formula for conversion is:

T = 27 + 273.15

. This gives the temperature in Kelvin, which is necessary for calculations involving gases.

Use the ideal gas law to find the volume occupied by one mole of gas. The ideal gas law is given by:

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 in Kelvin.

Calculate the volume occupied by one mole of gas at 1 atm and the converted temperature using the ideal gas law. Rearrange the formula to solve for

V

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

. Use

R

= 0.0821 L·atm/mol·K.

Determine the volume occupied by a single molecule. Since one mole of gas contains Avogadro's number of molecules, divide the volume of one mole by Avogadro's number (

6.022 \times 10^23

molecules/mol) to find the volume occupied by one molecule.

Calculate the length of an edge of the cube. Since the volume of the cube is equal to the volume occupied by one molecule, use the formula for the volume of a cube:

V = a^{3}

, where

a

is the length of an edge. Solve for

a

by taking the cube root of the volume occupied by one molecule.

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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, PV = nRT, relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the number of moles (n) and the gas constant (R). It is crucial for calculating the volume occupied by the gas molecules, which helps in determining the average spacing between them.

Molecular Volume

Molecular volume refers to the space occupied by a single molecule in a given volume of gas. By assuming each molecule is at the center of a cube, the volume of the cube can be calculated using the total volume of the gas divided by the number of molecules, providing insight into the average distance between molecules.

Avogadro's Number

Avogadro's Number (approximately 6.022 x 10^23) is the number of molecules in one mole of a substance. It is essential for converting moles of gas into the number of molecules, which is necessary to calculate the volume of each cube surrounding a molecule in the gas.

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Related Practice

Textbook Question

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare the root-mean-square speeds. (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

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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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Textbook Question

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

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?

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