Question

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.

Short answer

Answer: The molar concentration of an ideal gas mixture remains constant at constant temperature and pressure because it only depends on these two variables (C=P/(RT)). On the other hand, the density of the mixture may change due to variations in the molar masses and amounts of the individual components within the mixture (ρ = (Σ (Mi * ni))/V).

Step-by-step solution

Recall the Ideal Gas Law

The ideal gas law is given by the equation: PV = nRT where P represents pressure, V represents volume, n represents the amount in moles of the gas, R is the ideal gas constant (about 8.314 J/mol·K), and T is the temperature in Kelvin.

Define Molar Concentration and Density

Molar concentration (C) is defined as the number of moles of a substance in a given volume (V). Mathematically, it can be expressed as: C = n/V Density (ρ) is defined as the mass (m) of a substance in a given volume (V). Mathematically, it can be expressed as: ρ = m/V

Relate Molar Concentration and Density to Ideal Gas Law

Now we will use the ideal gas law to relate the molar concentration and density of the mixture by rewriting the law in terms of these quantities. First, we can solve the ideal gas law for the number of moles (n) in the gas mixture: n = PV/(RT) Now we substitute this expression into the molar concentration (C) and density (ρ) equations: C = (PV)/(RTV) = P/(RT) ρ = (m/V) However, the mixture is composed of various components, and we can express the mass (m) as the sum of the product of their individual molar masses (Mi) and their respective number of moles (ni): m = Σ (Mi * ni) So, the density equation becomes: ρ = (Σ (Mi * ni))/V

Show Molar Concentration Remains Constant

Since we are maintaining constant temperature and pressure for the gas mixture, the molar concentration (C) remains constant: C = P/(RT), where P and T are constant.

Show Density May Not Remain Constant

Unlike molar concentration, the density of the gas mixture can be affected by changes in the molar masses (Mi) and the number of moles (ni) of the individual components. So, even if the pressure and temperature remain constant, the density (ρ) may change as the composition of the gas mixture changes: ρ = (Σ (Mi * ni))/V In conclusion, the molar concentration (C) of an ideal gas mixture remains constant at constant pressure and temperature, as it only depends on these two variables. In contrast, the density (ρ) of the mixture may change, as it depends on the molar masses and amounts of the individual components within the mixture.