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: For an ideal gas mixture, the molar concentration (C) remains constant when the temperature (T) and pressure (P) are both kept constant. However, the density (ρ) may change under the same conditions if the molar mass (M) of the gas mixture changes.

Step-by-step solution

Recall the Ideal Gas Law

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.

Define the molar concentration and density

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

Relate the expressions for molar concentration and density to the Ideal Gas Law

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

Relate the expression of density to the Ideal Gas Law

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)

Analyze the derived expressions for molar concentration and density

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.

Key concepts

Molar Concentration

Molar concentration refers to the amount of moles of a substance in a unit volume. For an ideal gas mixture, this can be described using the Ideal Gas Law, which is: \( PV = nRT \). Here, \( P \) represents pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature.
When you solve for molar concentration \( C \), it is defined as \( C = n/V \). So, if you rearrange the Ideal Gas Law, you get \( C = P/(RT) \).

• Pressure \( P \) is directly proportional to molar concentration, meaning if pressure increases, molar concentration does too.
• Temperature \( T \) is inversely proportional, meaning as temperature goes up, molar concentration decreases.
• The universal gas constant \( R \) stays fixed, making it a reliable anchor in this relationship.

This relationship tells us that under constant pressure and temperature, the molar concentration of an ideal gas mixture remains stable.

Density of Gas Mixtures

Density is another important property of gas mixtures. It's the mass of gas per unit of volume. Typically expressed as \( \rho = m/V \), where \( m \) is the mass. For gases, the mass can be tricky as it depends on the number of moles \( n \) and the molar mass \( M \).

To find this, we can express mass as \( m = nM \). Substituting into the density formula, we have \( \rho = (nM)/V \). Connect this to the Ideal Gas Law: \( n = PV/RT \),
and our density formula becomes \( \rho = PM/(RT) \).

• Here, \( P \) represents pressure and remains directly proportional to density, similar to molar concentration.
• Molar mass \( M \) of the gas mixture can vary, impacting density.
• Unlike molar concentration, density can change with different gas types or when their mixture composition shifts, even if \( P \) and \( T \) are constant.

This variation is because density does not only rely on physical parameters like \( P \) and \( T \), but also on the property of the gases in the mix itself, represented by \( M \).

Universal Gas Constant

The universal gas constant \( R \) is a crucial part of the Ideal Gas Law. It's one of the few elements that remain unchanged in calculations. Defined as \( R = 8.314 \, J/(mol\cdot K) \), \( R \) is essential in linking physical properties of gases to their behavior.
It lets us convert moles and Kelvin to grasp how gases react under different conditions. While it varies slightly in units depending on if you are looking at energy or pressure-volume works, its value serves as a constant.

• Temperature influences how gases behave, but \( R \) ensures you can predict these changes. A high \( R \) value shows strong variation in behavior with temperature changes.
• \( R \) is universal, so it applies to all ideal gases, making comparisons and predictions across different gases easier.

Understanding \( R \) is about noticing its role as the bridge between energy, volume, and temperature for gases. Without it, our grasp of the Ideal Gas Law becomes hazy.