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Ch 20: The Micro/Macro Connection

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 20

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Chapter 20, Problem 20

The rms speed of the atoms in a 2.0 g sample of helium gas is 700 m/s. What is the thermal energy of the gas?

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Key Concepts

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

Root Mean Square Speed (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 particularly useful in kinetic theory, as it relates to the temperature and energy of the gas. For an ideal gas, the rms speed can be expressed as v_rms = sqrt(3kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of a gas particle.

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Root-Mean-Square Speed of Ideal Gases

Thermal Energy

Thermal energy is the total kinetic energy of the particles in a substance due to their motion. In the context of an ideal gas, it can be calculated using the formula E = (3/2)nRT, where E is the thermal energy, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. This energy is directly related to the temperature of the gas and reflects the energy available for doing work or transferring heat.

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Volume Thermal Expansion

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that describes the behavior of ideal gases. 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 the temperature. This law allows for the calculation of various properties of gases, including their thermal energy, by relating pressure, volume, and temperature in a single equation.

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Ideal Gases and the Ideal Gas Law

Previous problem

Related Practice

Textbook Question

On earth, STP is based on the average atmospheric pressure at the surface and on a phase change of water that occurs at an easily produced temperature, being only slightly cooler than the average air temperature. The atmosphere of Venus is almost entirely carbon dioxide (CO₂), the pressure at the surface is a staggering 93 atm, and the average temperature is 470℃. Venusian scientists, if they existed, would certainly use the surface pressure as part of their definition of STP. To complete the definition, they would seek a phase change that occurs near the average temperature. Conveniently, the melting point of the element tellurium is 450℃. What are (a) the rms speed and (b) the mean free path of carbon dioxide molecules at Venusian STP based on this phase change in tellurium? The radius of a CO₂ molecule is 1.5 x 10⁻¹⁰ m.

Textbook Question

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

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?

Textbook Question

A 6.0 m ✕ 8.0 m ✕ 3.0 m room contains air at 20℃. What is the room's thermal energy?

Textbook Question

Consider a container like that shown in Figure 20.12, with n₁ moles of a monatomic gas on one side and n₂ moles of a diatomic gas on the other. The monatomic gas has initial temperature T₁ᵢ. The diatomic gas has initial temperature T₂ᵢ. b. Show that the equilibrium temperature is

Textbook Question

A gas of 1.0 x 10²⁰ atoms or molecules has 1.0 J of thermal energy. Its molar specific heat at constant pressure is 20.8 J/ mol K. What is the temperature of the gas?