The kinetic theory pressure equation

Overview

Mean square speed of the molecules of a gas and root mean square speed.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   Explain in terms of molecular motion why a gas exerts a pressure on the walls of its container.

   Model answer

   Molecules strike the walls and rebound, and each collision changes the molecule's momentum. The wall must therefore exert a force on the molecules and, by Newton's third law, the molecules exert an equal force on the wall, which spread over the wall's area is the pressure.

2. 2

   State the kinetic theory expression for the pressure of an ideal gas in terms of density, defining every symbol and its unit.

   Model answer

   The expression is p = ⅓ρ⟨c²⟩, where p is the pressure in Pa, ρ is the density of the gas in kg m⁻³ and ⟨c²⟩ is the mean square speed of the molecules in m² s⁻².

3. 3

   State the kinetic theory expression for the pressure of an ideal gas in terms of the number of molecules, defining every symbol.

   Model answer

   The expression is p = ⅓(N/V)m⟨c²⟩, where N is the number of molecules, V is the volume in m³, m is the mass of one molecule in kg and ⟨c²⟩ is the mean square speed in m² s⁻².

4. 4

   Show that p = ⅓ρ⟨c²⟩ and p = ⅓(N/V)m⟨c²⟩ are the same statement.

   Model answer

   The total mass of gas is Nm and it occupies volume V, so the density ρ is Nm/V, and substituting that into p = ⅓ρ⟨c²⟩ gives p = ⅓(N/V)m⟨c²⟩.

5. 5

   Define the mean square speed of the molecules of a gas.

   Model answer

   The mean square speed is the average of the squares of the speeds of all the molecules present.

6. 6

   Define the root mean square speed.

   Model answer

   The root mean square speed is the square root of the mean square speed of the molecules.

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