Question

A wall made of natural rubber separates \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) gases at \(25^{\circ} \mathrm{C}\) and \(750 \mathrm{kPa}\). Determine the molar concentrations of \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) in the wall.

Short answer

Question: Determine the molar concentrations of oxygen (O₂) and nitrogen (N₂) gases in a natural rubber wall, given a temperature of 25°C and a pressure of 750 kPa. Answer: The molar concentrations of oxygen (O₂) and nitrogen (N₂) gases in the natural rubber wall can be calculated using the ideal gas equation and their respective molar masses. The molar concentration of O₂ is (750 kPa * V) / (8.314 J/(mol*K) * (25 + 273.15) K * 32 g/mol), and the molar concentration of N₂ is (750 kPa * V) / (8.314 J/(mol*K) * (25 + 273.15) K * 28 g/mol).

Step-by-step solution

Ideal Gas Equation

The Ideal Gas Equation is given by: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. We can rearrange the equation to get the number of moles: n = PV/(RT)

Convert Temperature to Kelvin

The given temperature is \(25^{\circ} \mathrm{C}\). To use the temperature in Ideal Gas Equation, we need to convert it to Kelvin. To do this, the following conversion is applied: \(\mathrm{T(K)} = \mathrm{T(°C)} + 273.15\)

Find the Number of Moles of Each Gas

Now we can plug in the given temperature and pressure for each gas into the Ideal Gas Equation to find the number of moles (n) of each gas: \(\mathrm{O}_{2}\): n = (750 kPa * V) / (8.314 J/(mol*K) * (25 + 273.15) K) \(\mathrm{N}_{2}\): n = (750 kPa * V) / (8.314 J/(mol*K) * (25 + 273.15) K) Since we are finding the molar concentration within the rubber, the volume for both gases will be the same (V). We just need to find the ratio between the two gases.

Find the Molar Mass of Each Gas

We have to find the molar mass of each gas and use them to calculate the molar concentration of each gas in the wall. Molar mass of \(\mathrm{O}_{2}\) is 32 g/mol and molar mass of \(\mathrm{N}_{2}\) is 28 g/mol.

Find the Molar Concentrations of Each Gas

We will now use the molar mass and the number of moles to find the molar concentration of each gas within the rubber wall. Molar concentration is the number of moles per unit volume. Since we have the same volume V for both gases, we can find the concentrations simply by dividing their number of moles by their molar masses. The molar concentration of each gas is as follows: \(\mathrm{O}_{2}\) Concentration: (750 kPa * V) / (8.314 J/(mol*K) * (25 + 273.15) K * 32 g/mol) \(\mathrm{N}_{2}\) Concentration: (750 kPa * V) / (8.314 J/(mol*K) * (25 + 273.15) K * 28 g/mol) Now you have the molar concentrations of both \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) in the natural rubber wall.

Key concepts

Ideal Gas Equation

Understanding the Ideal Gas Equation is fundamental when it comes to calculating properties of gases under various conditions. It is expressed as
\( PV = nRT \),
where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles of gas, \( R \) for the ideal gas constant, and \( T \) for temperature in Kelvins.

When solving problems, it's essential to note that all units must be consistent. As we see in this exercise, knowing the equation allows us to isolate the variable of interest, in this case, the number of moles \( n \), to determine the molar concentration. By rearranging the equation
\( n = \frac{PV}{RT} \),
one can plug in known values of pressure and temperature—and using the ideal gas constant appropriate for the units of pressure—to find the moles of gas present.

Molar Mass

The molar mass of a substance is the mass of one mole of that substance, typically expressed in grams per mole (g/mol). It is a bridge between the macroscopic and microscopic worlds, allowing us to count molecules by weighing them.

To elaborate on the importance of the molar mass in calculations, let's examine our example regarding the gases \( \mathrm{O}_{2} \) and \( \mathrm{N}_{2} \). Knowing the molar masses of \( \mathrm{O}_{2} \) (32 g/mol) and \( \mathrm{N}_{2} \) (28 g/mol) is crucial for converting the number of moles to grams, which can be particularly helpful when dealing with the physical quantity of gases.

Application in Calculating Concentration

When calculating molar concentrations, we use the molar mass to convert moles into a measurable quantity that factors into the density of the gas in the given volume.

Temperature Conversion

Temperature conversion is a necessary step in many scientific calculations and can sometimes be a source of errors if overlooked. For the Ideal Gas Equation, temperatures must always be expressed in Kelvin.

The conversion from Celsius to Kelvin is simple but crucial:
\( \mathrm{T(K)} = \mathrm{T(°C)} + 273.15 \).
This ensures that absolute zero (0 K) corresponds to -273.15°C, the theoretical lowest temperature possible.

Significance in Gas Calculations

In the exercise, starting with a temperature of 25°C, we must convert to Kelvin to correctly use the Ideal Gas Equation. This not only affects the calculation of the number of moles but consequently impacts the final molar concentration of the gases in the wall. Therefore, temperature conversion is not just a mathematical requirement but a step that connects the gas's thermodynamic behavior with its physical properties.