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Chapter 20: Problem 36

Twenty-two particles have speeds as follows \(\left(N_{i}\right.\) represents the number of particles that have speed \(\left.v_{i}\right)\) : $$ \begin{array}{lccccc} \hline N_{i} & 2 & 4 & 6 & 8 & 2 \\ v_{i}(\mathrm{~cm} / \mathrm{s}) & 1.0 & 2.0 & 3.0 & 4.0 & 5.0 \\ \hline \end{array} $$ (a) Compute their average speed \(\langle v\rangle .\) (b) Compute their root- meansquare speed \(v^{\mathrm{ms}} .\) (c) Of the five speeds shown, which is the most probable speed \(v^{\text {prob } ?}\)

Short Answer

Expert verified
Average speed: 3.18 cm/s. RMS speed: 3.37 cm/s. Most probable speed: 4.0 cm/s.

Step by step solution

01

- Setting Up the Data

Given data is the number of particles with corresponding speeds: \[ \begin{array}{lccccc} \hline N_{i} & 2 & 4 & 6 & 8 & 2 \ v_{i}(\mathrm{~cm} / \mathrm{s}) & 1.0 & 2.0 & 3.0 & 4.0 & 5.0 \ \hline \end{array} \]
02

- Compute the Total Number of Particles

Add up all the particles: \[ N_{\text{total}} = 2 + 4 + 6 + 8 + 2 = 22 \]
03

- Compute the Average Speed \( \langle v \rangle \)

The formula for the average speed is: \[ \langle v \rangle = \frac{ \sum (N_i * v_i) } { N_{\text{total}} } \] Calculate the sum of the products of \(N_i\) and \(v_i\): \[ \sum (N_i * v_i) = 2*1.0 + 4*2.0 + 6*3.0 + 8*4.0 + 2*5.0 = 2 + 8 + 18 + 32 + 10 = 70 \] Now, divide by the total number of particles: \[ \langle v \rangle = \frac{70}{22} \approx 3.18 \text{ cm/s} \]
04

- Compute the Root-Mean-Square Speed \( v^{ms} \)

The formula for the root-mean-square speed is: \[ v^{ms} = \sqrt{ \frac{ \sum (N_i * v_i^2) } { N_{\text{total}} } } \] Calculate the sum of the products of \(N_i\) and \(v_i^2\): \[ \sum (N_i * v_i^2) = 2*1.0^2 + 4*2.0^2 + 6*3.0^2 + 8*4.0^2 + 2*5.0^2 = 2*1 + 4*4 + 6*9 + 8*16 + 2*25 = 2 + 16 + 54 + 128 + 50 = 250 \] Now, divide by the total number of particles and take the square root: \[ v^{ms} = \sqrt{ \frac{250}{22} } \approx 3.37 \text{ cm/s} \]
05

- Determine the Most Probable Speed \( v^{\text{prob}} \)

The most probable speed is given by the speed that occurs most frequently. From the data, the speed 4.0 cm/s is associated with the highest number of particles (8 particles). Thus, the most probable speed \( v^{\text{prob}} \) is 4.0 cm/s.

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Key Concepts

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

particle speeds
Particle speeds are essential concepts in physics, particularly in the study of kinetic theory and thermodynamics. When dealing with a group of particles, their speeds can vary significantly. By examining the distribution of these speeds, we can understand more about the system's behavior. For example, in our exercise, we had 22 particles with different speeds.
  • Each particle had a speed (\(v_i\)) that was recorded, and the quantity of particles at each speed (\(N_i\)) was noted too.
  • We used this information to compute the average speed, the root-mean-square speed, and the most probable speed.
Breaking this down step by step allows us to comprehensively analyze the system's dynamics and interpret data meaningfully.
root-mean-square speed
The root-mean-square (RMS) speed is a statistical measure that gives us insight into the energy possessed by particles in a system. Essentially, it is the square root of the average of the squares of the particle speeds.
  • To compute the RMS speed, we first squared each particle's speed.
  • Next, we multiplied each squared speed by the number of particles at that speed (\(N_i\)).
  • Summing these products gives us the total squared speeds.
Finally, we divided by the total number of particles and took the square root to get the RMS speed. In our exercise, the RMS speed came out to be approximately 3.37 cm/s, providing a useful measure of the overall kinetic energy in the system.
most probable speed
The most probable speed is the speed at which the largest number of particles are traveling in a system. This measure can help us identify the mode of the speed distribution.
  • From our exercise data, we noted the frequency of each speed.
  • The speed with the highest frequency is considered the most probable speed.
For our set of particles, the most probable speed was determined to be 4.0 cm/s, as it was associated with the highest number of particles, specifically 8 particles. This indicates that more particles are traveling at this speed than at any other speed in the system.

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Most popular questions from this chapter

A certain gas occupies a volume of \(4.3 \mathrm{~L}\) at a pressure of \(1.2 \mathrm{~atm}\) and a temperature of \(310 \mathrm{~K}\). It is compressed adiabatically to a volume of \(0.76 \mathrm{~L}\). Determine (a) the final pressure and (b) the final temperature, assuming the gas to be an ideal gas for which \(\gamma=1.4\).

An intensive variable is one that can be defined locally within a system. Its magnitude does not depend on whether we select the whole system or a part of the system. An extensive variable is one that is defined for the system as a whole; its magnitude does depend on how much of the system we choose to select. Which of the following variables are intensive and which are extensive? Explain your reasoning in each case. (a) density, (b) pressure, (c) volume, (d) temperature, (e) mass, (f) internal energy, (g) number of moles, and (h) molecular weight.

We know that for an adiabatic process \(P V^{\gamma}=\) a constant. Evaluate the constant for an adiabatic process involving exactly \(2.0 \mathrm{~mol}\) of an ideal gas passing through the state having exactly \(P=1.0 \mathrm{~atm}\) and \(T=300 \mathrm{~K}\). Assume a diatomic gas whose molecules have rotation but not oscillation.

Determine the average value of the translational kinetic energy of the molecules of an ideal gas at (a) \(0.00^{\circ} \mathrm{C}\) and (b) \(100^{\circ} \mathrm{C}\). What is the translational kinetic energy per mole of an ideal gas at (c) \(0.00^{\circ} \mathrm{C}\) and \((\mathrm{d}) 100^{\circ} \mathrm{C} ?\)

When a molecule of a liquid approaches the surface, it experiences a force barrier that tries to keep it in the liquid. Thus it has to do work to escape and loses some of its kinetic energy when it leaves. (a) Assume that a water molecule can evaporate from the liquid if it hits the surface from the inside with a kinetic energy greater than the thermal energy corresponding to the temperature of boiling water, \(100^{\circ} \mathrm{C}\). Use this to estimate the numerical value of the work \(W\) required to remove a water molecule from the liquid. (b) Even though the average speed of a molecule in water below the boiling point corresponds to a kinetic energy less than \(W\), some molecules leave anyway and the water evaporates. Explain why this happens.
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