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Chapter 16: Problem 29

Five gas molecules chosen at random are found to have speeds of \(500,600,700,800\) and \(900 \mathrm{~m} / \mathrm{s}\). Then : (a) the rms speed and the average speed are the same (b) the rms speed is \(14 \mathrm{~m} / \mathrm{s}\) higher than the average speed (c) the rms speed is \(14 \mathrm{~m} / \mathrm{s}\) lower than that the average speed (d) the rms speed is \(\sqrt{14} \mathrm{~m} / \mathrm{s}\) higher than that the average speed.

Short Answer

Expert verified
(b) The RMS speed is 14 m/s higher than the average speed.

Step by step solution

01

Calculate the average speed

To calculate the average speed, add all the given speeds together and divide by the number of molecules. The speeds are \(500, 600, 700, 800,\) and \(900\, \text{m/s}\). \[\text{Average Speed} = \frac{500 + 600 + 700 + 800 + 900}{5}\]Calculate it to get:\[\text{Average Speed} = \frac{3500}{5} = 700 \, \mathrm{m/s}\]
02

Calculate the RMS speed

The root mean square (RMS) speed is calculated by taking the square root of the average of the squared speeds. First, square each speed and then find the average:\[\text{RMS Speed} = \sqrt{\frac{500^2 + 600^2 + 700^2 + 800^2 + 900^2}{5}}\]Calculate each squared term and find their sum: \[250000 + 360000 + 490000 + 640000 + 810000 = 2550000\]Divide by 5:\[\frac{2550000}{5} = 510000\]Take the square root:\[\text{RMS Speed} = \sqrt{510000} \approx 714.14 \, \mathrm{m/s}\]
03

Compare RMS and average speeds

The average speed calculated was \(700 \, \mathrm{m/s}\), and the RMS speed calculated was approximately \(714.14 \, \mathrm{m/s}\). Now, compute the difference:\[714.14 - 700 = 14.14 \, \mathrm{m/s}\]\(\sqrt{14} \, \mathrm{m/s} \approx 3.74 \, \mathrm{m/s}\), so the RMS speed is not equal to \(\sqrt{14} \, \mathrm{m/s}\) above the average speed, but actually it is approximately \(14 \, \mathrm{m/s}\) higher.

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Key Concepts

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

Average Speed
When you think about how fast something is moving over a period, you're thinking about average speed. In physics, average speed is a simple computation that's quite intuitive. You simply sum up the speeds you observe and divide by the number of observations.
For instance, in our exercise:
  • Add the speeds: 500, 600, 700, 800, and 900 m/s together. That gets you 3500 m/s total.
  • Next, divide by 5 (because there are 5 gas molecules): \ \( \frac{3500}{5} = 700 \, \mathrm{m/s} \).
Consider this as taking the mean of values, just like you would do in everyday situations, with things like test scores or prices. Average speed gives you a single, neat figure that describes a general tendency of movement. In this exercise, that tidy figure came out to be 700 m/s.
Gas Molecules
Gas molecules are constantly in motion, zipping around at various speeds. They don't all move at the same rate because of their individual kinetic energy. For example, in our scenario, the molecules had speeds ranging from 500 m/s to 900 m/s.
Why isn't there uniformity in their speed?
This variation is often because these molecules are always bumping into each other, exchanging energy, and making the situation really dynamic. Let's consider:
  • Gas molecules sometimes accelerate or decelerate when they collide.
  • In a sample of gas, there's a whole range of speeds, not just one fixed value.
Understanding that gas molecules move chaotically is fundamental for understanding kinetic theory, which in turn helps explain many properties of gases, like pressure and temperature.
Speed Calculation
Speed calculations for gas molecules often involve determining values like average speed and RMS speed. While average speed tells us about the central tendency, the Root Mean Square Speed provides insight into kinetic energy distribution among molecules.
Calculations for these values are straightforward if broken down:
  • For average speed: Add numbers up then divide by the count of deviations (simple arithmetic mean).
  • For RMS speed: Square each number, find the mean of these squares, and take the square root of that mean.
These calculations not only provide insights into molecule behavior but also reveal details about energy distribution in gases. Each method highlights different aspects but is connected through kinetic molecular theory.
Root Mean Square
Root Mean Square Speed (RMS speed) is a bit more complex but vital for describing gases. In physics, it's about understanding the relationship of energy with molecular speed.
Here's a simplified break down of what's going on when you calculate it:
  • Square each observed speed鈥攖his gets rid of negative values since energy can't be negative.
  • Average those squared values. You still take the sum of these squares and divide, like with average speed, but it's for squared numbers.
  • Finally, take the square root of that average. This step transforms it back into speed units, making the result more meaningful.
RMS speed helps in determining a gas's kinetic energy, which is more relevant than just knowing how fast they usually go. In our problem, we found the RMS speed to be \( 714.14 \, \mathrm{m/s} \), illustrating how active, on average, these molecules are due to the energy they possess.

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Most popular questions from this chapter

In the case of saturated vapour: (a) pressure 'depends upon volume at constant temperature (b) pressure varies non-linearly with temperature at constant volume (c) pressure becomes less than one atmosphere at boiling point (d) pressure varies linearly with temperature at constant volume

A gas behaves more closely as an ideal gas at: (a) low pressure and low temperature (b) low pressure and high temperature (c) high pressure and low temperature (d) high pressure and high temperature

Four molecules of a gas have speeds \(1,2,3\) and \(4 \mathrm{~km} / \mathrm{s}\). The value of the root-mean square speed of the gas molecules is : (a) \(\frac{1}{2} \sqrt{15} \mathrm{~km} / \mathrm{s}\) (b) \(\frac{1}{2} \sqrt{10} \mathrm{~km} / \mathrm{s}\) (c) \(2.5 \mathrm{~km} / \mathrm{s}\) (d) \(\sqrt{\frac{15}{2}} \mathrm{~km} / \mathrm{s}\)

The temperature of the mixture, if two perfectly monoatomic gases at absolute temperatures \(\overline{1}_{1}\) and \(T_{2}\) and number of moles in the gases \(n_{1}\) and \(n_{2}\), respectively are mixed, is: (Assume no loss of energy) (a) \(T=\frac{n_{1} T_{2}+n_{2} T_{1}}{n_{1}+n_{2}}\) (b) \(T=\frac{n_{1} T_{2}-n_{2} T_{1}}{n_{1}+n_{2}}\) (c) \(T=\frac{n_{1} T_{1}+n_{2} T_{2}}{n_{1}+n_{2}}\) (d) \(T=\frac{n_{1} T_{1}-n_{2} T_{2}}{n_{1}-n_{2}}\)

Increase of pressure: (a) always increases the boiling point of a liquid (b) always increases the melting point of a solid (c) increases the melting point of solid which expand on melting (d) always decreases the melting point of solid
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