Problem 122 - General Chemistry: Principles and Modern Applications

Question

According to the CRC Handbook of Chemistry and Physics (83rd ed.), the molar volume of \(\mathrm{O}_{2}(\mathrm{g})\) is \(0.2168 \mathrm{Lmol}^{-1}\) at \(280 \mathrm{K}\) and \(10 \mathrm{MPa}\). (Note: \(1 \mathrm{MPa}=\) \(\left.1 \times 10^{6} \mathrm{Pa} .\right)\)(a) Use the van der Waals equation to calculate the pressure of one mole of \(\mathrm{O}_{2}(\mathrm{g})\) at \(280 \mathrm{K}\) if the volume is 0.2168 L. What is the \% error in the calculated pressure? The van der Waals constants are \(a=1.382 \mathrm{L}^{2}\) bar \(\mathrm{mol}^{-2}\) and \(b=0.0319 \mathrm{L} \mathrm{mol}^{-1}\) (b) Use the ideal gas equation to calculate the volume of one mole of \(\mathrm{O}_{2}(\mathrm{g})\) at \(280 \mathrm{K}\) and \(10 \mathrm{MPa}\). What is the \% error in the calculated volume?

Step-by-Step Solution

Answer

The calculated pressure using the Van der Waals equation is 10.3 MPa with a 3% error. The calculated volume using the ideal gas equation is 0.232 L with a 7.04% error.

Step 1: Solve using the Van der Waals Equation

The Van der Waals equation can be used to calculate the pressure of one mole of \(O_{2}(g)\) at 280 K if the volume is 0.2168 L. Inserting the given values, we find: \[P = \frac{(1)(8.3145)(280)}{(0.2168 - (1)(0.0319))} - \frac{(1^{2})(1.382)}{(0.2168^{2})}\]

Step 2: Calculate the Pressure and % error

After calculation, the pressure is found to be 10.3 MPa. The % error in the calculated pressure can then be found by \(\frac{|10 - 10.3|}{10} \times 100% = 3%\).

Step 3: Solve using the Ideal Gas Equation

The ideal gas equation can be used to calculate the volume of one mole of \(O_{2}(g)\) at 280 K and 10 MPa. Inserting the given values, we find: \[V = \frac{(1)(8.3145)(280)}{10 \times 10^{6}}\]

Step 4: Calculate the Volume and % error

After calculation, the volume is found to be 0.232 L. The % error in the calculated volume can then be found by \(\frac{|0.2168 - 0.232|}{0.2168} \times 100% = 7.04%\).

Key Concepts

Ideal Gas Law

The ideal gas law is a simple equation that connects four important properties of gases: pressure (P), volume (V), temperature (T), and the number of moles (n). It is expressed as:\[ PV = nRT \]Where R is the ideal gas constant, which is typically 8.314 J/mol·K. This equation assumes that gas molecules do not exert any forces upon each other and occupy no volume themselves. In practice, this is not true for real gases, particularly under high pressure or low temperature conditions. However, the ideal gas law serves as a good approximation for many gases under moderate conditions. In the context of this exercise, you'll use the ideal gas law to solve for the volume when other properties such as pressure and temperature are specified, illustrating its utility in practical calculations.

Molar Volume

Molar volume is the volume occupied by one mole of a gas at a given temperature and pressure. In this exercise, the molar volume is given as 0.2168 L/mol for oxygen (O\(_2\)) at 280 K and 10 MPa. Molar volume changes with different temperatures and pressures, and this change can be predicted using gas laws. When you compare the ideal gas law calculations with given molar volumes for real gases, differences arise due to the assumptions made in the ideal gas law. Calculating molar volumes using ideal conditions often results in errors as real gas interactions are neglected. Understanding molar volume helps in analyzing these discrepancies and allows for comparisons between the behavior of ideal and real gases.

Calculation of Pressure

In order to calculate the pressure of a gas using the Van der Waals equation, you account for the interactions between gas molecules and the volume occupied by them:\[ \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]Here, 'a' and 'b' are Van der Waals constants unique to each gas, which correct for these interactions and the volume. For oxygen, these constants are given as \(a = 1.382\, \text{L}^2 \text{bar} \text{mol}^{-2}\) and \(b = 0.0319 \text{L}\, \text{mol}^{-1}\). By substituting the relevant values of temperature, volume, and the Van der Waals constants into the equation, you can find the corrected pressure for a mole of oxygen at a specific volume and temperature. This more accurately reflects real pressure conditions as opposed to the ideal gas law. This calculation gives insights into the effects of molecular interactions and finite volume on pressure, especially under non-ideal conditions.

Calculation of Volume

For calculating the volume of a gas under specific conditions, you can use the ideal gas law when real gas interactions are negligible, or the Van der Waals equation if more precise results are needed. For oxygen, calculating the volume using the ideal gas formula \[ V = \frac{nRT}{P} \]involves using absolute conditions like temperature and pressure provided. While this offers an estimate, the % error calculation compared to the known molar volume demonstrates that under higher pressure and specific volumes, discrepancies can occur due to real gas behavior. In this exercise, calculating volume and then noting the % error helps in recognizing the limitations of the ideal gas formula when applied to real gases. It is a crucial learning point for understanding the conditions where ideal assumptions fall short, leading into a more comprehensive understanding of gas behavior in varied environments.