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Chapter 21: Problem 42

A gas is at \(0^{\circ} \mathrm{C}\). If we wish to double the rms speed of its molecules, to what temperature must the gas be brought?

Short Answer

Expert verified
The gas must be brought to \(819^{\circ} \mathrm{C}\) to double the rms speed of its molecules.

Step by step solution

01

Convert initial temperature to Kelvin

Given that the initial temperature is \(0^{\circ} \mathrm{C}\), convert it to absolute temperature scale (Kelvin) by adding 273. This means \(T_i=273 \mathrm{K}\)
02

Finding the final temperature

Since the root mean square (rms) speed is directly proportional to the square root of the temperature, and in this case the speed is doubled, hence, the temperature must be quadrupled. Therefore, \(T_f=4*T_i=4*273 \mathrm{K}=1092 \mathrm{K}\)
03

Convert Kelvin to Celsius

Convert the final temperature in Kelvin to Celsius by subtracting 273 from the final temperature in Kelvin. Hence, \(T_f=1092 \mathrm{K}-273=819^{\circ} \mathrm{C}\)

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Key Concepts

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

Temperature Conversion
Understanding temperature conversion is fundamental in many scientific calculations, including the study of gas behavior. The most common scales are Celsius and Kelvin. To convert from Celsius to Kelvin, which is necessary in gas-related calculations due to its direct relationship with kinetic energy, one simply adds 273.15. Precisely, if you have a temperature of 0掳C, its Kelvin equivalent is given by adding 273.15, yielding 273.15K. When you wish to convert back from Kelvin to Celsius, you subtract 273.15. It's crucial because the Kelvin scale starts at absolute zero, the coldest possible temperature, where particles have minimal kinetic energy.
Root Mean Square Velocity
The root mean square (rms) velocity is an important concept in understanding gas molecule behavior. It represents the square root of the average of the squares of the velocities of each molecule in a gas sample, giving us an idea of the speed of particles within a gas. According to the kinetic theory, the rms speed (\( v_{rms} \)) is proportional to the square root of the absolute temperature of the gas when measured in Kelvin. Mathematically, this can be expressed as \( v_{rms} \propto \sqrt{T} \) where \( T \) is the temperature in Kelvin. This means that if you want to double the rms speed, you must quadruple the absolute temperature.
Kinetic Theory of Gases
The kinetic theory of gases explains how particles behave in a gas. The basic premises are that gas consists of a large number of tiny particles in constant, random motion and that these collisions are perfectly elastic. This theory connects macroscopic properties like pressure and temperature to the microscopic motion of molecules. When temperature increases, the average kinetic energy of the gas molecules increases, which, in turn, increases the rms velocity. Temperature, in this theory, is directly proportional to the average kinetic energy per molecule.
Absolute Temperature Scale
The absolute temperature scale, also known as the Kelvin scale, is a pivotal concept in physics and chemistry as it relates directly to the kinetic energy of particles. Unlike the Celsius or Fahrenheit scales, which are based on water's freezing and boiling points, the Kelvin scale starts from absolute zero. At 0 Kelvin (equivalent to -273.15掳C), the particle motion theoretically stops, and no kinetic energy remains. Therefore, the Kelvin scale is essential in our understanding of thermodynamics and the behaviour of gases under different temperatures as it provides a true physical baseline.

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Most popular questions from this chapter

A diatomic ideal gas \((\gamma=1.40)\) confined to a cylinder is put through a closed cycle. Initially the gas is at \(P_{i}, V_{i},\) and \(T_{i} .\) First, its pressure is tripled under constant volume. It then expands adiabatically to its original pressure and finally is compressed isobarically to its original volume. (a) Draw a PV diagram of this cycle. (b) Determine the volume at the end of the adiabatic expansion. Find (c) the temperature of the gas at the start of the adiabatic expansion and \((\mathrm{d})\) the temperature at the end of the cycle. (e) What was the net work done on the gas for this cycle?

A vertical cylinder with a heavy piston contains air at a temperature of \(300 \mathrm{K}\). The initial pressure is \(200 \mathrm{kPa},\) and the initial volume is \(0.350 \mathrm{m}^{3} .\) Take the molar mass of air as \(28.9 \mathrm{g} / \mathrm{mol}\) and assume that \(C_{V}=5 R / 2 .\) (a) Find the specific heat of air at constant volume in units of \(\mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\) A vertical cylinder with a heavy piston contains air at a temperature of \(300 \mathrm{K}\). The initial pressure is \(200 \mathrm{kPa},\) and the initial volume is \(0.350 \mathrm{m}^{3} .\) Take the molar mass of air as \(28.9 \mathrm{g} / \mathrm{mol}\) and assume that \(C_{V}=5 R / 2 .\) (a) Find the specific heat of air at constant volume in units of \(\mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\)

A vessel contains \(1.00 \times 10^{4}\) oxygen molecules at \(500 \mathrm{K}\) (a) Make an accurate graph of the Maxwell-Boltzmann speed distribution function versus speed with points at speed intervals of \(100 \mathrm{m} / \mathrm{s}\). (b) Determine the most probable speed from this graph. (c) Calculate the average and rms speeds for the molecules and label these points on your graph. (d) From the graph, estimate the fraction of molecules with speeds in the range \(300 \mathrm{m} / \mathrm{s}\) to \(600 \mathrm{m} / \mathrm{s}\)

(a) Show that \(1 \mathrm{Pa}=1 \mathrm{J} / \mathrm{m}^{3} .\) (b) Show that the density in space of the translational kinetic energy of an ideal gas is \(3 P / 2.\)

A sealed cubical container \(20.0 \mathrm{cm}\) on a side contains three times Avogadro's number of molecules at a temperature of \(20.0^{\circ} \mathrm{C}\). Find the force exerted by the gas on one of the walls of the container.
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