Kinetic theory of gases
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State the basic assumptions of the kinetic theory of gases.
1. Gas consists of identical molecules in random, continuous motion. 2. Volume of molecules is negligible compared to gas volume. 3. No intermolecular forces except during collisions. 4. Collisions are perfectly elastic. 5. Duration of collision is negligible compared to time between collisions.
Explain how molecular movement causes gas pressure.
Gas pressure arises from the multitude of collisions of gas molecules with the walls of the container. Each collision exerts a force, and the sum of these forces over the area of the wall results in pressure. Higher molecular speeds or more frequent collisions lead to higher pressure.
State the relationship between pressure (p), volume (V), number of molecules (N), mass of molecule (m), and mean-square speed (<c²>).
The relationship is given by pV = (1/3)Nm<c²>, where <c²> represents the average of the squares of the speeds of the gas molecules.
What is the root-mean-square speed (c<sub>rms</sub>)?
The root-mean-square speed (c<sub>rms</sub>) is the square root of the mean (average) of the squares of the speeds of the molecules in a gas. It is calculated as c<sub>rms</sub> = √<c²>.
How is the average translational kinetic energy of a molecule related to absolute temperature (T)?
The average translational kinetic energy of a molecule is directly proportional to the absolute temperature. It is given by (1/2)m<c²> = (3/2)kT, where k is the Boltzmann constant.
How can you derive the relationship between root-mean-square speed and temperature?
Starting with pV = (1/3)Nm<c²> and pV = NkT, equate to obtain (1/3)Nm<c²> = NkT. Simplify to <c²> = 3kT/m. Therefore, c<sub>rms</sub> = √(3kT/m).
How does increasing the temperature of a gas affect the root-mean-square speed of its molecules?
Increasing the temperature of a gas increases the root-mean-square speed of its molecules. Since c<sub>rms</sub> = √(3kT/m), a higher temperature (T) results in a higher c<sub>rms</sub>, indicating faster-moving molecules.
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