Please wait for the animation to completely load.
In this animation N = nR (i.e., kB = 1). This, then, gives the ideal gas law as PV = NT. The average values shown, < >, are calculated over intervals of one time unit. Using the ideal gas law, we can make a connection between the macroscopic quantities of temperature (T) and pressure (P) and the individual microscopic properties of a particle of momentum (p = mv) and kinetic energy (1/2 mv2). Restart.
Let's begin with one particle in an enclosed box and bouncing between two walls.
Increase the speed of the particle. The momentum delivered to the wall (and the force it experiences) will increase, thereby increasing the pressure. If the pressure of the gas increases (and if the volume is constant), the temperature of the gas will also increase.
By the same reasoning, increasing the mass of the particle will also increase the pressure, so the temperature should be connected to the mass of the particle as well. Increase the mass of the particle.
One particle in an enclosed box is not very realistic, so let's add a second particle (of the same mass) with a different speed. This time, however, we'll plot the kinetic energy of each particle as a function of time and the change in momentum at any wall (an average of this will give us the pressure). The table now shows the average momentum change at the walls. How does this compare with the pressure you calculate using the ideal gas law?
Now let's add some more particles of the same mass and different speeds. The table gives the momentum delivered to the wall as particles collide with the wall (<dp/dt>), as well as the pressure calculated from the ideal gas law. This time we plot a histogram of the speeds of the particles. Stop the animation at some time and calculate the total kinetic energy of the ensemble of particles. This, divided by the number of particles, should be the same as the temperature of the system. This is the equipartition of energy theorem: The internal energy of a gas (the sum of the energy of all particles) is equal to (f/2)kBNT, where f is the number of degrees of freedom for the atoms or molecules in a gas. In this case, the particles have 2 degrees of freedom; they can move in the x direction and y direction and thus f = 2. Because we are treating the gas particles as "hard spheres" (one of our assumptions in the ideal gas model), the internal energy of the gas is due to the kinetic energy of the particles and is equal to kBNT and for this animation, kB = 1.
Exploration authored by Anne J. Cox.
© 2004 by Prentice-Hall, Inc. A Pearson Company