Show that for an ideal gas mixture maintained at a constant temperature and pressure, the molar concentration \(C\) of the mixture remains constant but this is not necessarily the case for the density \(\rho\) of the mixture.

The Ideal Gas Law is given by the equation: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature.

The molar concentration (C) of a gas mixture can be defined as the moles of particles (n) per unit volume (V): C = n/V The density (ρ) of a gas mixture can be defined as the mass (m) per unit volume (V): ρ = m/V

We can rewrite the Ideal Gas Law in terms of molar concentration: PV = (CV)RT Thus: C = P/(RT) To relate the density to the Ideal Gas Law, we need to express the mass of the gas mixture (m) in terms of moles (n). This can be done by multiplying the number of moles by the molar mass of the gas mixture (M): m = nM Now, substitute the mass expression into the density expression: ρ = (nM)/V

Substitute the expression for moles (n) from the Ideal Gas Law into the density expression, and solve for density: ρ = [(PV)/(RT)M]/V Thus: ρ = PM/(RT)

We have found expressions for molar concentration and density: C = P/(RT) ρ = PM/(RT) We can see that the molar concentration (C) is directly proportional to the pressure (P) and inversely proportional to the temperature (T) and the universal gas constant (R). Given that R is always constant, if the pressure (P) and temperature (T) are both kept constant, the molar concentration (C) will also remain constant. However, for the density expression, we see that it is also dependent on the molar mass (M) of the gas mixture. If the molar mass of the gas changes, the density (ρ) can change even if the pressure (P) and temperature (T) are kept constant. This demonstrates that while the molar concentration (C) remains constant under constant temperature and pressure for an ideal gas mixture, the density (ρ) does not necessarily. In conclusion, for an ideal gas mixture maintained at a constant temperature and pressure, the molar concentration (C) remains constant, while the density (ρ) may change if the molar mass (M) of the gas mixture changes.