Question

Calculate the pressure exerted by \(0.5000\) mole of \(\mathrm{N}_{2}\) in a \(10.000-\mathrm{L}\) container at \(25.0^{\circ} \mathrm{C}\). a. using the ideal gas law. b. using the van der Waals equation. c. Compare the results. d. Compare the results with those in Exercise 115.

Short answer

The pressure exerted by the nitrogen gas using the ideal gas law is approximately \(1.2279 \, \mathrm{atm}\), while using the van der Waals equation it is approximately \(1.2111 \, \mathrm{atm}\). Both equations provide similar pressure values, however, the value is slightly lower when calculated using the van der Waals equation. This is due to van der Waals equation considering the intermolecular forces between the particles. In comparison to Exercise 115, the findings validate the discussion and highlight that the deviation from Ideal Gas Law is minimal for nitrogen under normal conditions, affirming its adequacy in most circumstances.

Step-by-step solution

Convert temperature to kelvin

To calculate the pressure using both equations, we first need to convert the temperature from Celsius to Kelvin. The conversion can be done using the following formula: \(T_K = T_C + 273.15\) So, \(T_K = 25.0 + 273.15\) \(T_K = 298.15 K\)

Calculate pressure using the ideal gas law

Now, we will use the ideal gas law equation, \(PV = nRT\), to calculate the pressure. We are given the number of moles (n = 0.5000 mol), the volume (V = 10.000 L), and the temperature in Kelvin (T = 298.15 K). The gas constant, R, has the value 0.0821 L atm/mol K. We can rearrange the ideal gas law to solve for pressure: \(P = \frac{nRT}{V}\) So, \(P = \frac{0.5000\times 0.0821\times 298.15}{10.000}\) \(P = 1.2279 \, \mathrm{atm}\) The pressure exerted by the N₂ gas using the ideal gas law is approximately 1.2279 atm.

Calculate pressure using the van der Waals equation

Next, we will calculate the pressure using the van der Waals equation: \((P+\frac{an^2}{V^2})(V-nb)=nRT\) We know the values for, n, R, T, and the van der Waals constants a and b. We can rearrange the equation to solve for pressure P: \(P = \frac{nRT}{V-nb}-\frac{an^2}{V^2}\) So, using the values: \(P = \frac{0.5000\times 0.0821\times 298.15}{10.000-0.5000\times 0.0391}-\frac{1.39\times (0.5000)^2}{(10.000)^2}\) \(P = 1.2111 \, \mathrm{atm}\) The pressure exerted by the N₂ gas using the van der Waals equation is approximately 1.2111 atm.

Compare the results

We can now compare the two results: Ideal gas law pressure: P = 1.2279 atm Van der Waals equation pressure: P = 1.2111 atm We can see that the pressure calculated using the van der Waals equation is slightly lower than the pressure calculated using the ideal gas law, which makes sense because the ideal gas law assumes no interactions between the gas particles, while the van der Waals equation tries to account for these interactions.

Compare the results with Exercise 115

In Exercise 115, no specific values are given for the moles, volume, or temperature, but the pressure differences between Ideal Gas Law and Van der Waals equation were discussed. We can see from this exercise that the deviation from Ideal Gas Law grows as a gas becomes less ideal, but for nitrogen in these particular conditions, the deviation is relatively small and the Ideal Gas Law can be considered accurate enough in most cases.

Key concepts

Understanding the Ideal Gas Law

At its core, the ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of an ideal gas in a closed system. Presented simply, it's given by the formula:
\[\begin{equation}PV = nRT\text{,}\end{equation}\]where:

• P is the pressure of the gas,
• V is the volume occupied by the gas,
• n is the number of moles of gas,
• R is the universal gas constant (0.0821 L atm/mol K), and
• T is the absolute temperature in kelvins (K).

Ideal gases are theoretical constructs where gas particles are assumed to have perfectly elastic collisions with no intermolecular forces. In practice, this law is applicable under many conditions, but exceptions arise when pressures get very high or temperatures are very low, where real gas behavior deviates from the ideal model.

Exercise improvement tips often suggest more context for the principles at play. Here's a key point to consider: the ideal gas law provides a baseline from which students can understand gas behavior under 'normal' conditions. This comparison sets the stage for exploring why real gases differ and how they can be described by more complex equations, such as the van der Waals equation.

Gas Pressure Calculation Tactics

Calculating gas pressure is a critical skill for students in chemistry, physics, and engineering. Pressure, depicted in the ideal gas law by the variable P, is a measure of the force applied by the gas particles against the walls of the container per unit area. It is imperative to understand that in gas pressure calculations, external factors such as temperature and volume drastically influence the gas's behavior.
To calculate the pressure using the ideal gas law, one must rearrange the formula to solve for P:\[\begin{equation}P = \frac{nRT}{V}.\end{equation}\]Students may sometimes forget to convert temperatures to Kelvin or accurately account for the volume of the gas in their calculations, leading to errors. Hence, taking meticulous steps in converting the units of measurement and being aware of the conditions under which the gas is held are crucial. Additionally, the knowledge of calculating gas pressure paves the way to further topics, such as the kinetics of gases and thermodynamics.

Van der Waals Equation and Real Gases

Real gases differ from the ideal gases in that they possess intermolecular forces and occupy a finite volume. The van der Waals equation is a modification of the ideal gas law meant to model the behavior of real gases more accurately:\[\begin{equation}(P + \frac{an^2}{V^2})(V - nb) = nRT,\end{equation}\]where a and b are constants unique to each gas. These constants account for the attractive forces between particles (a) and the finite size of the gas molecules (b). By including these factors, the van der Waals equation corrects for the non-ideal behavior observed especially under conditions of high pressures and low temperatures, where gases deviate most from the ideal gas law.
In educational contexts, this illustrates the importance of understanding the ideal gas law as a reference point and recognizing its limitations. When students use the van der Waals equation, it's essential for them to grasp not just how to use the equation but also why these corrections are necessary. These concepts, explained in an easily digestible format, enable students to better understand the physical properties of gases and the factors that influence their behavior.