Internal Energy of an Ideal Gas

Derive average kinetic energy of one gas particle.

1. Equate pV = NkT and pV = ⅓Nmc² 2. Simply N and rearrange for mc² = 3kT. 3. Eₖ = ½mv² so multiply by ½. 4. Hence ½mc² = ³/₂kT E = ³/₂kT E - average kinetic energy of a single particle, k - Boltzmann constant, T - temperature.

Derive internal energy of ideal gas.

1. Eₚ = 0 hence U = Eₖ. 2. Multiply by N so U = NE 3. Hence U = ³/₂NkT U - internal energy of ideal gas, N - number of particles of gas, k - Boltzmann constant, T - temperature.

Explain the energy changes between gas particles.

1. Particles collide with each other and as a result energy is transferred between particles. 2. Some particles gain speed whilst others lose speed. 3. Between collisions, particles travel at constant speed. 4. Total energy of the system remains constant. 5. So, average speed remains constant provided temperature remains constant.

Speed distribution changes as gas temperature increases

Average particle speed increases. Maximum particle speed increases. Distribution curve becomes more spread out.

Speed distribution

Proportion of particles with a given speed against particle speed

Maxwell-Boltzmann distribution of kinetic energy graph.

Starts a (0, 0) as molecules have zero energy. First section has slow moving molecules. Middle has most molecules moving at moderate speed so the energies are in this range. Last has relatively few molecules moving quickly.

Maxwell-Boltzmann distribution

The distribution of energies of the molecules in an ideal gas. Or a graph of number of molecules against kinetic energy of an ideal gas.