Final answer:
The equation of state for a homogeneous mixture of inert monatomic ideal gases at an absolute temperature T and volume V is derived using the ideal gas law, confirming that the total pressure is the sum of the partial pressures and is proportional to the total moles of gas and temperature.
Step-by-step explanation:
To derive the equation of state for a homogeneous mixture of inert monatomic ideal gases at an absolute temperature T in a container of volume V, we will apply the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
Considering a mixture of gases with v1 moles of gas 1, v2 moles of gas 2, ..., and vk moles of gas k, the total number of moles n is the sum of moles of each individual gas, hence n = v1 + v2 + ... + vk.
Since the gases are ideal and inert, there are no interactions between the particles and the individual gases in the mixture each exert a partial pressure. Let P1, P2, ..., Pk be the partial pressures of each gas. The total pressure p exerted by the mixture is simply the sum of individual partial pressures: p = P1 + P2 + ... + Pk. By the ideal gas law for each component, we know that PiV = viRT, and thus, pV = (v1 + v2 + ... + vk)RT.
Finally, the equation of state for the mixture is pV = nRT, which shows that the total mean pressure p is directly proportional to the total number of moles of gas n and the absolute temperature T, and inversely proportional to the volume V.