Edexcel A Level (IAL) Physics-5.8 Average Molecular Kinetic Energy- Study Notes- New Syllabus
Edexcel A Level (IAL) Physics -5.8 Average Molecular Kinetic Energy- Study Notes- New syllabus
Edexcel A Level (IAL) Physics -5.8 Average Molecular Kinetic Energy- Study Notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
Deriving and Using \( \tfrac{1}{2} m \langle c^2 \rangle = \tfrac{3}{2} kT \)
This equation links the microscopic motion of gas molecules to the absolute temperature of a gas. It shows how temperature is a direct measure of the average kinetic energy of molecules.
Meaning of the Symbols
- \( m \) = mass of one molecule (kg)
- \( \langle c^2 \rangle \) = mean square speed of molecules (m² s⁻²)
- \( k \) = Boltzmann constant \( (1.38\times10^{-23}\ \mathrm{J\,K^{-1}}) \)
- \( T \) = absolute temperature (K)
Important: \( \langle c^2 \rangle \) is the average of the square of the speed, not the square of the average speed.
Starting Point: Ideal Gas Equation
For an ideal gas:
\( pV = NkT \)
- \( N \) = number of molecules
Kinetic Theory Expression for Pressure
From kinetic theory, the pressure of an ideal gas is given by:
\( pV = \tfrac{1}{3} Nm \langle c^2 \rangle \)
This comes from considering molecular collisions with the container walls.
Derivation
Equate the two expressions for \( pV \):
\( NkT = \tfrac{1}{3} Nm \langle c^2 \rangle \)
Cancel \( N \) from both sides:
\( kT = \tfrac{1}{3} m \langle c^2 \rangle \)
Multiply both sides by \( \tfrac{3}{2} \):
\( \tfrac{3}{2} kT = \tfrac{1}{2} m \langle c^2 \rangle \)
This is the required result.
Physical Interpretation
- The left-hand side represents thermal energy per molecule.
- The right-hand side represents average kinetic energy per molecule.
- Temperature is directly proportional to average molecular kinetic energy.
Key conclusion: The average kinetic energy of gas molecules depends only on absolute temperature, not on pressure, volume, or gas type.
Using the Equation
- To find average kinetic energy:
\( \text{Average KE} = \tfrac{3}{2} kT \)
- To find mean square speed:
\( \langle c^2 \rangle = \dfrac{3kT}{m} \)
As temperature increases, molecular speeds increase.
Example (Easy)
Find the average kinetic energy of a molecule at \( 300\,\mathrm{K} \).
▶️ Answer / Explanation
\( \text{Average KE} = \tfrac{3}{2} kT \)
\( = \tfrac{3}{2} \times 1.38\times10^{-23} \times 300 \)
\( = 6.21\times10^{-21}\ \mathrm{J} \)
Example (Medium)
A molecule has mass \( 4.65\times10^{-26}\,\mathrm{kg} \) at \( 400\,\mathrm{K} \). Calculate \( \langle c^2 \rangle \).
▶️ Answer / Explanation
\( \langle c^2 \rangle = \dfrac{3kT}{m} \)
\( = \dfrac{3 \times 1.38\times10^{-23} \times 400}{4.65\times10^{-26}} \)
\( = 3.56\times10^{5}\ \mathrm{m^2\,s^{-2}} \)
Example (Hard)
Explain why gases at the same temperature have the same average kinetic energy per molecule, even if their molecules have different masses.
▶️ Answer / Explanation
- Average kinetic energy is given by \( \tfrac{3}{2} kT \).
- It depends only on temperature.
- Heavier molecules move more slowly, lighter molecules move faster.
- This ensures the average kinetic energy is the same.