# Notes on ideal gases for Cambridge A level and UTME

**Notes on ideal gases for Cambridge A level and UTME**

An ** ideal gas** is one that obeys the gas laws, and equation of state for ideal gas, at all temperature, pressure and volume. This means Ideal gas obeys

*pV *= *nRT *

P = Pressure

V = Volume

T = Temperature

*R* = universal gas constant

*n*= number of moles

**Infer from a Brownian motion experiment the evidence for the movement of molecules**

Brownian motion: random movement of small particles caused be bombardment of invisible molecules

- Smoke (oil droplets) are seen to move randomly
- This motion is evidence that the air particles are also moving randomly and colliding with the smoke droplets
- The air particles cannot be seen but their motion can be understood by the smoke droplets which can be seen

Kinetic theory of gases

- The attraction between molecules is negligible
- The volume of the molecules is negligible compared with the volume occupied by the gas
- The molecules are like perfectly elastic spheres
- The duration of a collision is negligible compared with the time between collisions

*E***xplain how molecular movement causes the pressure exerted by a gas and hence deduce the relationship p = 1/3Nm/V < c 2 >**

**( N = number of molecules)
**

- Consider a cube of space with length L
- Consider a particle moving in one dimension x with velocity c
_{x} - When the particle collides with the wall its velocity is reversed so its change in momentum is equal to…
- Dp
_{x}= 2mc_{x}

- Dp

- The time between collisions with each wall of the cube is equal to…
- Time between collisions = 2L / c
_{x}

- Time between collisions = 2L / c

- The rate at which momentum is transferred to the wall is…
- Rate of change of momentum = 2mc
_{x}/ (2L/c_{x}) = mc_{x}^{2}/ L

- Rate of change of momentum = 2mc

- If there are N particles in the cube the total force is…
- Total force = Nmc
_{x}^{2}/ L

- Total force = Nmc

- Pressure is force over area so pressure is…
- Pressure on one wall is Nmc
_{x}^{2}/ L^{3}

- Pressure on one wall is Nmc

- L
^{3}is the volume so…- Pressure = Nmc
_{x}^{2}/ V

- Pressure = Nmc

- The average of c
_{x}^{2}can be written as < c_{x}^{2}> - As all directions, x, y and z can be considered equal
- < c
_{x}^{2}> = 1/3< c^{2}>

- < c

- Hence
- P = 1/3Nm<c
^{2}> / V

- P = 1/3Nm<c
- p(rho) = density of gas
- <c²> = mean square speed

It should be carefully noted that the pressure p of the gas depends on the “mean square” of the speed. This is because

- The momentum change at a wall is proportional to the speed
- The time interval before this change is repeated is inversely proportional to the speed

**Compare pV = 1/ 3 Nm < c 2 > with pV = NkT and hence deduce that the average translational kinetic energy of a molecule is proportional to T.**

- The average translational E
_{k}of the particles can be expressed as …- <E
_{k}> = 1/2m< c^{2}>

- <E

- Combining with P = 1/3Nm<c
^{2}> / V we get….- pV = 2/3N(1/2m< c
^{2}>) = 2/3N<E_{k}>

- pV = 2/3N(1/2m< c

- Combining this with pV = NkT we get…
- pV = 2/3N<E
_{k}> = NkT - <E
_{k}> =3/2kT

- pV = 2/3N<E

- Therefore, Temperature is proportional to Average translational kinetic energy

Note

R/N = K

K is the Boltzman’s constant

R is the molar gas constant

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