Question

Consider a rubber membrane separating carbon dioxide gas that is maintained on one side at \(2 \mathrm{~atm}\) and on the opposite at \(1 \mathrm{~atm}\). If the temperature is constant at \(25^{\circ} \mathrm{C}\), determine (a) the molar densities of carbon dioxide in the rubber membrane on both sides and \((b)\) the molar densities of carbon dioxide outside the rubber membrane on both sides.

Short answer

Answer: (a) Inside the rubber membrane: Side 1 (2 atm): 0.0814 mol/L Side 2 (1 atm): 0.0407 mol/L (b) Outside the rubber membrane: Side 1 (2 atm): 0.0814 mol/L Side 2 (1 atm): 0.0407 mol/L

Step-by-step solution

Calculate the constants required

First, let's figure out the values of the constants used in the Ideal Gas Law. We need to convert the temperature from Celsius to Kelvin and find the value of the universal gas constant (R). To convert the temperature from Celsius to Kelvin: $$ T = 25 + 273.15 = 298.15 K $$ Universal gas constant (R) = 0.08206 L·atm/mol·K

Calculate the molar volume of carbon dioxide in the rubber membrane on side 1

For side 1 with a pressure of 2 atm, we will rearrange the Ideal Gas Law to find the molar volume V/n: $$ \frac{V}{n} = \frac{RT}{P} = \frac{0.08206 L·atm/mol·K × 298.15 K}{2 atm} = 12.2794 L/mol $$

Calculate the molar volume of carbon dioxide in the rubber membrane on side 2

For side 2 under 1 atm, we will again rearrange the Ideal Gas Law to find the molar volume V/n: $$ \frac{V}{n} = \frac{RT}{P} = \frac{0.08206 L·atm/mol·K × 298.15 K}{1 atm} = 24.5588 L/mol $$

Determine the molar densities of carbon dioxide in the rubber membrane

The molar densities for CO2 in the rubber membrane are the reciprocals of the molar volumes calculated above. For side 1 (2 atm): $$ Molar \; Density = \frac{1}{12.2794 L/mol} = 0.0814 \;mol/L $$ For side 2 (1 atm): $$ Molar \; Density = \frac{1}{24.5588 L/mol} = 0.0407 \; mol/L $$

Determine the molar densities of carbon dioxide outside the rubber membrane

Since the rubber membrane is at equilibrium, the molar densities outside the rubber membrane should be the same as the molar densities inside the membrane. For side 1 (2 atm): $$ Molar \; Density = 0.0814 \; mol/L $$ For side 2 (1 atm): $$ Molar \; Density = 0.0407 \; mol/L $$ To summarize, the molar densities of carbon dioxide in the rubber membrane and outside the rubber membrane are as follows: (a) Inside the rubber membrane: Side 1 (2 atm): 0.0814 mol/L Side 2 (1 atm): 0.0407 mol/L (b) Outside the rubber membrane: Side 1 (2 atm): 0.0814 mol/L Side 2 (1 atm): 0.0407 mol/L