Hot air feels hot, and cold air feels cold, but have you ever wondered why? What does a hot gas have that a cold gas does not? At the molecular level, what is going on?

## Observing Particle Motion

The temperature of a gas is a measure of the motion of its particles. By watching how particles move and interact, you can see what temperature actually means.

1. In the default settings, the Gizmo™ shows the motion of Hydrogen molecules at a temperature of 300 Kelvin (K). The Kelvin scale is often used in physics because it starts at absolute zero, the lowest possible temperature (−273.15°C). To convert from the Kelvin scale to Celsius, subtract 273.15 from the Kelvin temperature. A temperature of 273.15 K is equivalent to 0°C, and 100°C is the same as 373.15 K.
1. Notice the current temperature, 300 K. What is the equivalent Celsius temperature?
2. Look at the particles carefully. When two particles collide, what happens to both particles?
3. Are all of the particles moving at the same speed?
4. Notice the velocity distribution of the particles shown on the graph at right. What is the most likely particle speed at this temperature?
2. Use the Temperature slider to gradually raise the temperature to 1,000 K.
1. How does the particle motion at 1,000 K compare to the particle motion at 300 K?
2. While the temperature is 1,000 K, look at the particles carefully. At this high temperature, are there any slow-moving particles?
3. Compare the velocity graph for 1,000 K to the velocity graph for 300 K. How are the graphs different? If the two graphs were superimposed, would there be any overlap between the particle velocity distributions? If so, what does that indicate about the range of particle velocities at each temperature?
4. What is the most common particle velocity at 1,000 K?
3. The graph you see is a probability curve known as the Maxwell-Boltzmann distribution. It was derived independently by two of the most accomplished physicists of the 19th century, James Clerk Maxwell and Ludwig Boltzmann. Each of these men had profound influences on the greatest ideas of 20th century physics, relativity and quantum mechanics.
1. Use the Temperature slider to decrease the temperature of the Hydrogen gas. How does this affect the motion of the particles? What happens to the shape of the velocity probability graph?
2. Based on what you have observed, what do you think the particle motion will be at the lowest possible temperature, absolute zero (0 K)?
4. Set the Temperature to 300 K, and observe the Hydrogen atoms. Now, select Helium from the dropdown menu. Notice that the particles are now single atoms of helium. (Helium, a noble gas, does not form molecules.)
1. Compare the motion of the helium atoms to the motion of the hydrogen molecules. At the same temperature, which type of particle tends to move faster?
2. Which has more mass, a molecule of hydrogen or an atom of helium? (Hint: An atom of hydrogen has a single proton and no neutrons. An atom of helium has two protons and two neutrons.)
3. Temperature measures the energy of molecular motion. If two gases are at the same temperature, their particles have the same average energy. However, if you had helium gas and hydrogen gas at equal temperatures, you would find that the hydrogen atoms were moving faster than the helium atoms. Why is this? (Hint: The kinetic energy of an individual particle is given by the formula mv2/2, where m stands for mass, and v stands for velocity.)
4. Of the other gases listed in the dropdown menu, which molecules do you think will move most quickly at a given temperature? Which will move most slowly? Test your answers using the Gizmo.
5. Because particles have a range of velocities at any given temperature, it is useful to find the average velocity. Physicists use three separate quantities to express the average velocity. Select Hydrogen and a temperature of 400 K. Make sure all three checkboxes are unchecked.
1. The most probable velocity is denoted by the symbol vp. To estimate the most probable velocity, find the peak of the probability curve for hydrogen at 400 K. What is your estimate?
2. The mean velocity is denoted by the symbol . Based on the shape of the curve, do you think the mean (average) velocity is less than, the same as, or greater than the most probable velocity? Explain your answer.
3. Check your answers by selecting the Show most probable velocity and Show mean velocity checkboxes. Were your above hypotheses correct?
4. Experiment with other gases and other temperatures. Is the mean velocity always greater than the most probable velocity? Based on the shape of the velocity curve, why is this? (Hint: If the curve were a solid object, the mean velocity would be the point at which it would balance.)
6. You can calculate the exact values of vp and with the formulas below.

In each formula, R stands for the universal gas constant, 8.3144 J / K. M stands for the molar mass, in kilograms. M is equal to the molecular mass divided by 1,000 (so hydrogen has a molar mass of 2 ÷ 1,000, or 0.002 kg). T stands for the Kelvin temperature. Turn off the Show most probable velocity and Show mean velocity checkboxes. Select Hydrogen again, and choose a new temperature.
1. Calculate vp. Check your answer by turning on Show most probable velocity.
2. Calculate . Check your answer by turning on Show mean velocity.
3. According to the formulas given, can vp ever be greater than ? Explain.
7. Another useful quantity is the root mean square velocity, or vrms. It is equal to the square root of the sum of the squared velocities of the particles. It is useful in many calculations, including finding average kinetic energy and frequency of particle collisions.

1. Calculate vrms, and check your answer by turning on Show root mean square velocity.
2. How does this value compare to vp and ?
3. Practice calculating vp, and vrms for other temperatures and other gases. Remember to use the molecular masses rather than the atomic masses for each gas. For example, a nitrogen atom has a mass of 14 amu, but a nitrogen molecule (N2) has a mass of 28 amu.
8. Of course, no discussion of this topic would be complete without presenting the Maxwell-Boltzmann equation. If you like a challenge, use this formula and a spreadsheet program to create a graph like the one seen in the Gizmo. In the formula, f(v) stands for the probability of a particle being at a particular velocity (v), in meters per second.