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Chapter 14: Problem 36

Two moles of an ideal gas are placed in a container whose volume is \(8.5 \times 10^{-3} \mathrm{m}^{3} .\) The absolute pressure of the gas is \(4.5 \times 10^{5} \mathrm{Pa}\) . What is the average translational kinetic energy of a molecule of the gas?

Short Answer

Expert verified
The average translational kinetic energy is \(4.78 \times 10^{-21} \; \mathrm{J}.\)

Step by step solution

01

Identify Known Values

First, we recognize the given values. We have two moles of gas, a volume of \(V = 8.5 \times 10^{-3} \mathrm{m}^3\), and a pressure of \(P = 4.5 \times 10^5 \mathrm{Pa}\). We need to find the average translational kinetic energy of a gas molecule.
02

Use Ideal Gas Law

The ideal gas law is \(PV = nRT\). Our aim is to find the temperature \(T\) first. We are given \(P\), \(V\), and \(n\) (number of moles = 2), and \(R = 8.314 \; \mathrm{J/(mol \cdot K)} \) is the gas constant.
03

Calculate Temperature (T)

Rearrange the ideal gas law to solve for \(T\): \[T = \frac{PV}{nR}\]. Substitute the known values: \[T = \frac{(4.5 \times 10^5) \times (8.5 \times 10^{-3})}{2 \times 8.314} \approx 230.77 \; \mathrm{K}.\]
04

Average Kinetic Energy Formula

The average translational kinetic energy per molecule of an ideal gas is given by \(\frac{3}{2}kT\), where \(k\) is Boltzmann's constant \((1.38 \times 10^{-23} \mathrm{J/K})\).
05

Calculate Average Kinetic Energy

Using the formula, substitute the values to find: \[\mathrm{KE_{avg}} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 230.77 \approx 4.78 \times 10^{-21} \; \mathrm{J}.\]
06

Verify Units And Calculation

Check if the units correctly cancel to give energy in joules and verify the arithmetic calculation. Ensuring accuracy in each calculation step confirms the correctness of the result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Kinetic Energy
The average kinetic energy of a molecule in a gas gives us insight into its motion. According to the kinetic molecular theory, this energy is tied directly to the temperature of the gas. In simpler terms, as the temperature increases, the average kinetic energy of the molecules also increases.

In the context of an ideal gas, the average translational kinetic energy per molecule can be calculated using the formula \( \frac{3}{2} kT \), where \( k \) is Boltzmann's constant and \( T \) is the temperature in Kelvin. By plugging in values, we can predict how energetic the molecules are on average. This is not only a foundation for understanding gas behavior but also helps in comprehending how temperature affects the motion of molecules.
  • The average kinetic energy depends solely on the temperature.
  • It provides a benchmark for the speed and activity of gas molecules.
  • For all gases at the same temperature, this average energy is identical.
Boltzmann's Constant
Boltzmann's constant, symbolized as \( k \), is a fundamental value in physics, linking temperature to energy. It acts as a bridge between microscopic and macroscopic systems, playing a pivotal role in thermodynamics and statistical mechanics.

The value of Boltzmann's constant is approximately \( 1.38 \times 10^{-23} \mathrm{J/K} \). It appears in the formula for the average kinetic energy and in the ideal gas law when expressed in molecular terms. Whenever you think about the kinetic energy of molecules, Boltzmann's constant is the key component translating thermal temperature into energy.
  • Boltzmann's constant relates energy scales to temperature scales.
  • It helps calculate the average energy per particle in a substance.
  • This constant is foundational for understanding statistical behavior of systems.
Translational Motion
In gases, translational motion is the simplest form of motion and represents straight-line movement from one point to another. This motion is crucial because it directly affects the gas's pressure and temperature.

The translational motion of molecules in a gas is responsible for the spread and mixing of gases, as well as for the exertion of pressure when these molecules collide with the walls of their container. In ideal gases, this type of motion is not hindered by interactions between molecules, leading to predictable behaviors like those described by the ideal gas law.
  • Translational motion contributes significantly to the pressure exerted by a gas.
  • It facilitates the dissemination of gas molecules throughout a space.
  • In ideal gases, translational motion is free and not influenced by intermolecular forces.

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Most popular questions from this chapter

A gas fills the right portion of a horizontal cylinder whose radius is 5.00 \(\mathrm{cm} .\) The initial pressure of the gas is \(1.01 \times 10^{5}\) Pa. A frictionless movable piston separates the gas from the left portion of the cylinder, which is evacuated and contains an ideal spring, as the drawing shows. The piston is initially held in place by a pin. The spring is initially unstrained, and the length of the gas-filled portion is 20.0 \(\mathrm{cm}\) . When the pin is removed and the \(\mathrm{gas}\) is allowed to expand, the length of the gas-filled chamber doubles. The initial and final temperatures are equal. Determine the spring constant of the spring.

When you push down on the handle of a bicycle pump, a piston in the pump cylinder compresses the air inside the cylinder. When the pressure in the cylinder is greater than the pressure inside the inner tube to which the pump is attached, air begins to flow from the pump to the inner tube. As a biker slowly begins to push down the handle of a bicycle pump, the pressure inside the cylinder is \(1.0 \times 10^{5} \mathrm{Pa},\) and the piston in the pump is 0.55 \(\mathrm{m}\) above the bottom of the cylinder. The pressure inside the inner tube is \(2.4 \times 10^{5}\) Pa. How far down must the biker push the handle before air begins to flow from the pump to the inner tube? Ignore the air in the hose connecting the pump to the inner tube, and assume that the temperature of the air in the pump cylinder does not change.

A tube has a length of 0.015 \(\mathrm{m}\) and a cross-sectional area of \(7.0 \times 10^{-4} \mathrm{m}^{2} .\) The tube is filled with a solution of sucrose in water. The diffusion constant of sucrose in water is \(5.0 \times 10^{-10} \mathrm{m}^{2 / \mathrm{s}}\) . A difference in concentration of \(3.0 \times 10^{-3} \mathrm{kg} / \mathrm{m}^{3}\) is maintained between the ends of the tube. How much time is required for \(8.0 \times 10^{-13} \mathrm{kg}\) of sucrose to be transported through the tube?

On the sunlit surface of Venus, the atmospheric pressure is \(9.0 \times 10^{6} \mathrm{Pa},\) and the temperature is 740 \(\mathrm{K}\) . On the earth's surface the atmospheric pressure is \(1.0 \times 10^{5} \mathrm{Pa}\) , while the surface temperature can reach 320 \(\mathrm{K}\) . These data imply that Venus has a "thicker" atmosphere at its surface than does the earth, which means that the number of molecules per unit volume \((N / V)\) is greater on the surface of Venus than on the earth. Find the ratio \((N / V)_{\mathrm{Venus}} /(N / V)_{\mathrm{Earth}}\)

An ideal gas at \(15.5^{\circ} \mathrm{C}\) and a pressure of \(1.72 \times 10^{5}\) Pa occupies a volume of 2.81 \(\mathrm{m}^{3} .\) (a) How many moles of gas are present? (b) If the volume is raised to 4.16 \(\mathrm{m}^{3}\) and the temperature raised to \(28.2^{\circ} \mathrm{C},\) what will be the pressure of the gas?
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