Question

Calculate the pressure exerted by \(0.5000\) mole of \(\mathrm{N}_{2}\) in a 1.0000-L container at \(25.0^{\circ} \mathrm{C}\) a. using the ideal gas law. b. using the van der Waals equation. c. Compare the results.

Short answer

The pressure exerted by 0.5000 moles of N2 in a 1.0000 L container at 25.0°C is approximately 12.24 atm using the ideal gas law and 12.29 atm using the van der Waals equation. The slight difference in the pressures can be attributed to the intermolecular interactions considered in the van der Waals equation that are not accounted for in the ideal gas law. The pressures are quite close, indicating that N2 behaves mostly like an ideal gas under these conditions.

Step-by-step solution

Convert temperature to Kelvin

Since the given temperature is in Celsius, we need to convert it to Kelvin. The formula for converting Celsius to Kelvin is: \(K = °C + 273.15\) For our problem, the temperature is 25.0°C, so we have: \(K = 25.0 + 273.15 = 298.15 K\) Now we have the temperature in Kelvin, which we will use for further calculations.

Part a: Calculate pressure using the ideal gas law

The ideal gas law 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 (\(8.314 J/mol\cdot K\)), and T is the temperature in Kelvin. We want to solve for P, so we can rearrange the formula as: \(P = \frac{nRT}{V}\) Plugging in the given values: \(P = \frac{0.5000\text{ mol} \times 8.314 \frac{\text{J}}{\text{mol} \cdot \text{K}} \times 298.15 \text{K}}{1.0000 \text{L}}\) \(P = \frac{1240.54 \text{J}}{\text{L}}\) We must convert Joules per liter (J/L) to atmospheric pressure (atm) using the conversion factor \(1 \text{atm} = 101.3 \frac{\text{J}}{\text{L}}\): \(P = \frac{1240.54 \text{J}}{1 \text{L}}\times\frac{1 \text{atm}}{101.3 \text{J/L}} = 12.24 \text{ atm}\) Therefore, the pressure exerted by the N2 gas using the ideal gas law is approximately 12.24 atm.

Part b: Calculate pressure using the van der Waals equation

The van der Waals equation is given by: \((P + a\frac{n^2}{V^2})(V-nb) = nRT\) where a and b are the van der Waals constants for the N2 gas. For N2, a = 1.390 L² atm/mol² and b = 0.0391 L/mol. To calculate the pressure, we can rearrange the equation as: \(P = \frac{nRT}{(V - nb)} - a\frac{n^2}{V^2}\) Plugging in the given values: \(P = \frac{0.5000\text{ mol} \times 0.08206 \frac{\text{L}\cdot\text{atm}}{\text{mol} \cdot \text{K}} \times 298.15 \text{K}}{(1.0000 \text{L} - 0.5000\text{ mol}\times 0.0391 \frac{\text{L}}{\text{mol}})} - 1.390 \frac{\text{ L}^2\cdot\text{ atm}}{(\text{mol}^2)}\times\frac{0.5000^2\text{ mol}^2}{1.0000^2\text{ L}^2}\) \(P = \frac{12.253 \text{ atm}}{1 - (0.01955)} - 0.3482 \text{ atm}\) \(P = 12.29 \text{ atm}\) Therefore, the pressure exerted by the N2 gas using the van der Waals equation is approximately 12.29 atm.

Part c: Compare the pressures from the ideal gas law and van der Waals equation

From part a, using the ideal gas law, the pressure was 12.24 atm. From part b, using the van der Waals equation, the pressure was 12.29 atm. The pressure calculated using the van der Waals equation is slightly higher than the pressure calculated using the ideal gas law. This difference can be attributed to the intermolecular interactions between gas molecules considered in the van der Waals equation that are not accounted for in the ideal gas law. Overall, the pressures are quite close, suggesting that N2 behaves mostly like an ideal gas under the given conditions.

Key concepts

Pressure Calculation

Understanding pressure calculation is essential when studying the behavior of gases. Pressure, denoted as 'P', is a measure of the force exerted over a certain area, and in a gaseous context, it represents the force that the gas molecules apply on a container's walls. The Ideal Gas Law, which is often a great approximation for real gases under normal conditions, provides a straightforward approach to calculate pressure.

The formula for the Ideal Gas Law is written as \( PV = nRT \). Here, 'V' represents the volume of the container, 'n' stands for the number of moles of the gas, 'R' is the gas constant, and 'T' is the absolute temperature in Kelvin. To find the pressure, we rearrange the equation to \( P = \frac{nRT}{V} \). By inputting the values for 'n', 'R', 'V', and 'T', we can solve for the pressure 'P'.

It's crucial to note that when using this equation, units must be consistent, so conversions may be necessary, such as converting Celsius to Kelvin for temperature and ensuring the volume is in liters if using the constant \( R = 0.0821 \frac{L \times atm}{mol \times K} \). In practice, real gases might deviate from ideal behavior, and that's when more complex equations like the van der Waals equation come into play.

Van der Waals Equation

The van der Waals equation is a refined model to describe the behavior of real gases, accounting for two main factors: the volume occupied by the molecules and the intermolecular forces. The equation is expressed as:\((P + a\frac{n^2}{V^2})(V-nb) = nRT\), where the constants 'a' and 'b' are specific to each gas and account for the intermolecular attractions and finite size of molecules, respectively.

When calculating pressure using this equation, one needs to consider these additional factors that the Ideal Gas Law overlooks. For example, in the case of nitrogen gas (N2), 'a' and 'b' have specific given values that, when added into the more complex van der Waals equation, result in a slightly different pressure value compared to the Ideal Gas Law.

The ability to navigate the van der Waals equation allows for more accurate predictions of gas behavior, especially at high pressures or low temperatures where deviations from ideal behavior become more pronounced. This highlights the importance of understanding the limitations of the Ideal Gas Law and the conditions under which a more complex model like the van der Waals equation should be used.

Gas Constant (R)

The Gas Constant, symbolized as 'R', is a fundamental component that bridges the microscopic properties of molecules with the macroscopic observable properties of a gas. In the context of the Ideal Gas Law \( PV = nRT \), 'R' is the proportionality constant that relates pressure (P), volume (V), moles (n), and temperature (T).

Its value depends on the units being used for pressure and volume. Commonly, 'R' is used with a value of \( 8.314 \frac{J}{mol \times K} \) when working with energy units in Joules, or \( 0.0821 \frac{L \times atm}{mol \times K} \) when dealing in liters and atmospheres. Knowing the correct value and units for 'R' is crucial for students to solve problems accurately.

Learning how to apply 'R' effectively requires not only memorization of its value but also an understanding of how it functions within the larger framework of gas laws and a clear grasp of unit conversions. This ensures students are prepared for the varied contexts in which the gas constant appears.