Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 18

Chapter 20, Problem 18

The rms speed of molecules in a gas is 600 m/s. What will be the rms speed if the gas pressure and volume are both halved?

Verified step by step guidance

The root mean square (rms) speed of gas molecules is given by the formula:

\sqrt{\frac{3 k T}{m}}

, where

k

is the Boltzmann constant,

T

is the temperature, and

m

is the mass of a single molecule.

The rms speed depends on the temperature

T

, so we need to determine how the temperature changes when the pressure and volume are halved.

Using the ideal gas law,

P V = n R T

, where

P

is pressure,

V

is volume,

n

is the number of moles, and

R

is the gas constant, we can express the temperature as

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

. If both

P

and

V

are halved, their product remains constant, so

T

does not change.

Since the temperature

T

remains constant, the rms speed of the gas molecules also remains unchanged. This is because the rms speed is directly proportional to the square root of the temperature, as shown in the formula in Step 1.

Thus, the rms speed of the gas molecules will still be 600 m/s, even after the pressure and volume are halved.

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Key Concepts

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

Root Mean Square (RMS) Speed

The root mean square speed is a measure of the average speed of particles in a gas, calculated as the square root of the average of the squares of the speeds of the individual particles. It is a crucial concept in kinetic theory, reflecting the energy and temperature of the gas. The RMS speed is directly related to the temperature and molecular mass of the gas.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law helps in understanding how changes in pressure and volume affect the behavior of gases.

Kinetic Theory of Gases

The Kinetic Theory of Gases explains the behavior of gases in terms of the motion of their molecules. It posits that gas pressure results from collisions of molecules with the walls of a container and that temperature is a measure of the average kinetic energy of the molecules. This theory provides insights into how changes in pressure and volume influence the speed and energy of gas molecules.

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Introduction to Kinetic-Molecular Theory

Related Practice

Textbook Question

At 100℃ the rms speed of nitrogen molecules is 576 m/s. Nitrogen at 100℃ and a pressure of 2.0 atm is held in a container with a 10 cm x 10 cm square wall. Estimate the rate of molecular collisions (collisions/s) on this wall.

Textbook Question

By what factor does the rms speed of a molecule change if the temperature is increased from 10℃ to 1000℃?

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Textbook Question

The molecules in a six-particle gas have velocities:

$\begin{aligned}\[\vec{v}\)_1 &= (20$\hat{i}$ - 30$\hat{j}$) $\text{ m/s}$ $\vec{v}$_2 &= (40$\hat{i}$ + 70$\hat{j}$) $\text{ m/s}$ $\vec{v}$_3 &= (-80$\hat{i}$ + 20$\hat{j}$) $\text{ m/s}$ $\vec{v}$_4 &= 30$\hat{i}$ $\text{ m/s}$ $\vec{v}$_5 &= (40$\hat{i}$ - 40$\hat{j}$) $\text{ m/s}$ $\vec{v}$_6 &= (-50$\hat{i}$ - 20$\hat{j}$) \(\text{ m/s}\]\end{aligned}$

Calculate (a) ${\vec{v}}_{avg}$ , (b) $v_{avg}$ , and (c) $v_{rms}$ .

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Textbook Question

A cylinder contains gas at a pressure of 2.0 atm and a number density of 4.2 x 10²⁵ m^-3. The rms speed of the atoms is 660 m/s. Identify the gas.

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Textbook Question

Liquid helium boils at 4.2 K. In a flask, the helium gas above the boiling liquid is at the same temperature. What are (a) the mean free path in the gas, (b) the rms speed of the atoms, and (c) the average energy per atom?

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Textbook Question

1.0 mol of argon has 3100 J of thermal energy. What is the gas temperature in °C?

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