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Recommended Videos Problem 2. Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. Problem 8. Problem 9. Problem Video Transcript Hello Today we'll be talking about Chapter 12 Question 59 which tells us that oxygen molecules have an average speed. Numerade Educator. We can now give an equation for the internal energy of a monatomic ideal gas.
Therefore, denoting the internal energy by we simply have or. We can solve for a typical speed of a molecule in an ideal gas in terms of temperature to determine what is known as the root-mean-square rms speed of a molecule. The root-mean-square rms speed of a molecule, or the square root of the average of the square of the speed , is. We digress for a moment to answer a question that may have occurred to you: When we apply the model to atoms instead of theoretical point particles, does rotational kinetic energy change our results?
To answer this question, we have to appeal to quantum mechanics. The rotational inertia of an atom is tiny because almost all of its mass is in the nucleus, which typically has a radius less than. Thus the minimum rotational energy of an atom is much more than for any attainable temperature, and the energy available is not enough to make an atom rotate. We will return to this point when discussing diatomic and polyatomic gases in the next section.
Calculating Kinetic Energy and Speed of a Gas Molecule a What is the average kinetic energy of a gas molecule at room temperature?
Strategy a The known in the equation for the average kinetic energy is the temperature:. Before substituting values into this equation, we must convert the given temperature into kelvin: We can find the rms speed of a nitrogen molecule by using the equation.
Obtaining the molar mass of nitrogen from the periodic table, we find. Significance Note that the average kinetic energy of the molecule is independent of the type of molecule. The average translational kinetic energy depends only on absolute temperature. The kinetic energy is very small compared to macroscopic energies, so that we do not feel when an air molecule is hitting our skin.
On the other hand, it is much greater than the typical difference in gravitational potential energy when a molecule moves from, say, the top to the bottom of a room, so our neglect of gravitation is justified in typical real-world situations. The rms speed of the nitrogen molecule is surprisingly large. These large molecular velocities do not yield macroscopic movement of air, since the molecules move in all directions with equal likelihood.
The mean free path the distance a molecule moves on average between collisions, discussed a bit later in this section of molecules in air is very small, so the molecules move rapidly but do not get very far in a second. The higher the rms speed of air molecules, the faster sound vibrations can be transferred through the air. The speed of sound increases with temperature and is greater in gases with small molecular masses, such as helium see Figure.
This speed is called the escape velocity. At what temperature would helium atoms have an rms speed equal to the escape velocity? Strategy Identify the knowns and unknowns and determine which equations to use to solve the problem. Significance This temperature is much higher than atmospheric temperature, which is approximately K at high elevation.
Very few helium atoms are left in the atmosphere, but many were present when the atmosphere was formed, and more are always being created by radioactive decay see the chapter on nuclear physics. Heavier molecules, such as oxygen, nitrogen, and water, have smaller rms speeds, and so it is much less likely that any of them will have speeds greater than the escape velocity.
In fact, the likelihood is so small that billions of years are required to lose significant amounts of heavier molecules from the atmosphere. Figure shows the effect of a lack of an atmosphere on the Moon. Because the gravitational pull of the Moon is much weaker, it has lost almost its entire atmosphere. Check Your Understanding If you consider a very small object, such as a grain of pollen, in a gas, then the number of molecules striking its surface would also be relatively small.
Would you expect the grain of pollen to experience any fluctuations in pressure due to statistical fluctuations in the number of gas molecules striking it in a given amount of time? Such fluctuations actually occur for a body of any size in a gas, but since the numbers of molecules are immense for macroscopic bodies, the fluctuations are a tiny percentage of the number of collisions, and the averages spoken of in this section vary imperceptibly. Roughly speaking, the fluctuations are inversely proportional to the square root of the number of collisions, so for small bodies, they can become significant.
This was actually observed in the nineteenth century for pollen grains in water and is known as Brownian motion. If two or more gases are mixed, they will come to thermal equilibrium as a result of collisions between molecules; the process is analogous to heat conduction as described in the chapter on temperature and heat.
As we have seen from kinetic theory, when the gases have the same temperature, their molecules have the same average kinetic energy. Thus, each gas obeys the ideal gas law separately and exerts the same pressure on the walls of a container that it would if it were alone. Therefore, in a mixture of gases, the total pressure is the sum of partial pressures of the component gases , assuming ideal gas behavior and no chemical reactions between the components.
In a mixture of ideal gases in thermal equilibrium, the number of molecules of each gas is proportional to its partial pressure. This result follows from applying the ideal gas law to each in the form Because the right-hand side is the same for any gas at a given temperature in a container of a given volume, the left-hand side is the same as well.
An important application of partial pressure is that, in chemistry, it functions as the concentration of a gas in determining the rate of a reaction. Breathing air that has a partial pressure of oxygen below 0. Lower partial pressures of have more serious effects; partial pressures below 0. Safety engineers give considerable attention to this danger.
Another important application of partial pressure is vapor pressure , which is the partial pressure of a vapor at which it is in equilibrium with the liquid or solid, in the case of sublimation phase of the same substance. At any temperature, the partial pressure of the water in the air cannot exceed the vapor pressure of the water at that temperature, because whenever the partial pressure reaches the vapor pressure, water condenses out of the air.
Dew is an example of this condensation. The temperature at which condensation occurs for a sample of air is called the dew point. It is easily measured by slowly cooling a metal ball; the dew point is the temperature at which condensation first appears on the ball.
The vapor pressures of water at some temperatures of interest for meteorology are given in Figure. The relative humidity R. A relative humidity of means that the partial pressure of water is equal to the vapor pressure; in other words, the air is saturated with water. Calculating Relative Humidity What is the relative humidity when the air temperature is and the dew point is? Strategy We simply look up the vapor pressure at the given temperature and that at the dew point and find the ratio.
Significance R. The value of is within the range of recommended for comfort indoors. As noted in the chapter on temperature and heat, the temperature seldom falls below the dew point, because when it reaches the dew point or frost point, water condenses and releases a relatively large amount of latent heat of vaporization.
We now consider collisions explicitly. If we assume all the molecules are spheres with a radius r , then a molecule will collide with another if their centers are within a distance 2 r of each other. As the particle moves, it traces a cylinder with that cross-sectional area.
The mean free path is the length such that the expected number of other molecules in a cylinder of length and cross-section is 1. Taking the motion of all the molecules into account makes the calculation much harder, but the only change is a factor of The result is.
In an ideal gas, we can substitute to obtain. The mean free time is simply the mean free path divided by a typical speed, and the usual choice is the rms speed.
Calculating Mean Free Time Find the mean free time for argon atoms at a temperature of and a pressure of 1. Take the radius of an argon atom to be. Significance We can hardly compare this result with our intuition about gas molecules, but it gives us a picture of molecules colliding with extremely high frequency.
Average kinetic energy of gas molecules Maxwell-Boltzmann distribution curve. The kinetic theory states that "The average kinetic energy of gas molecules is proportional to the absolute temperature of the gas". Not all molecules will have the same kinetic energy and hence the same speed. An increase in temperature increases the average speed of the molecules and the range of speeds.
Look at the image on the left. It represents the Maxwell-Boltzmann distribution curve of kinetic energy of oxygen molecules at three different temperatures.
The average kinetic energy of oxygen molecules and hydrogen molecules will be the same at any particular temperature. If we keep in mind that the formula for kinetic energy is 0. Since the average kinetic energy of oxygen and hydrogen are equal we can write the expression below.
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