Question

Use the ideal gas and van der Waals equations to calculate the pressure when \(2.25 \mathrm{mol} \mathrm{H}_{2}\) are confined to a volume of \(1.65 \mathrm{L}\) at \(298 \mathrm{K}\). Is the gas in the repulsive or attractive region of the molecule-molecule potential?

Short answer

Using the ideal gas equation, we have calculated the pressure as \(P_{1} = (2.25 \ \mathrm{mol})(0.08206 \ \mathrm{L} \cdot \mathrm{atm}/\mathrm{K} \cdot \mathrm{mol})(298 \ \mathrm{K}) / (1.65 \ \mathrm{L}) \approx 27.22 \ \mathrm{atm}\). After substituting the known values in the van der Waals equation and solving for P, we find that the van der Waals pressure \(P_{2}\) is approximately 26.69 atm. Since \(P_{1} > P_{2}\), the hydrogen gas is in the repulsive region of the molecule-molecule potential.

Step-by-step solution

Calculate pressure with 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 (0.08206 L.atm/K.mol) and T is the temperature. We can rearrange the equation to solve for P: P = nRT/V Plugging in the values, we get: P = (2.25 mol)(0.08206 L.atm/K.mol)(298 K) / (1.65 L) Now let's calculate the pressure P:

Calculate pressure with van der Waals Equation

The van der Waals equation is given by: ( P + (a * (n² / V²)) )( V - n * b ) = n * R * T We need to solve for P in the van der Waals equation. First, let's substitute the known values: ( P + (0.244 * (2.25² / 1.65²)) )(1.65 - 2.25 * 0.02661) = 2.25 * 0.08206 * 298 Now, we will solve for P.

Compare ideal gas pressure with van der Waals pressure

Now that we have the pressures calculated using both the ideal gas equation and the van der Waals equation, we can compare them. - Ideal gas pressure: P₁ - Van der Waals pressure: P₂ If P₁ > P₂, the gas is in the repulsive region If P₁ < P₂, the gas is in the attractive region By comparing these, we can determine whether the gas is in the repulsive or attractive region of the molecule-molecule potential.

Determine the repulsive or attractive region of the molecule-molecule potential

Using the calculated pressures from Steps 1 and 2, we can now determine if the gas is in the repulsive or attractive region.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in physical chemistry and is written as \( PV = nRT \). It describes the relationship between pressure (P), volume (V), the amount of gas in moles (n), the ideal gas constant (R), and the temperature in Kelvin (T). This equation assumes that the gas particles are point masses with no volume and no intermolecular forces. This is a good approximation only under conditions of low pressure and high temperature where the gas molecules are far apart and not influenced by the presence of the others.

When calculating the pressure using the ideal gas law, one simply rearranges the equation to \( P = \frac{nRT}{V} \), and by inserting the respective values for n, R, T, and V, the calculation gives us an idea of what the pressure of an ideal gas would be under those specific conditions.

Molecule-Molecule Potential

Molecule-molecule potential refers to the interactions that occur between molecules within a gas. These forces play a critical role in understanding real gases, as they deviate from ideal behavior due to attractions or repulsions between particles. When molecules are far apart, the forces are negligible, but as they approach each other, these potentials become significant.

Attractive forces, often witnessed as molecules clump, can reduce the pressure we observe compared to an ideal gas prediction. In contrast, repulsive forces occur when molecules are very close to each other, typically when a gas is compressed, leading to a larger pressure than what would be predicted under the ideal gas law. Assessing whether a gas follows the attractive or repulsive part of the potential curve is essential when describing its behavior.

Gas Pressure Calculation

Gas pressure calculation is crucial in physical chemistry for understanding how gases behave under different conditions. The ideal gas law allows us to calculate the theoretical pressure of a gas, assuming no interactions between molecules. However, to calculate the pressure of real gases, we use the van der Waals equation which includes corrections for molecular volume and intermolecular forces.

By comparing the pressures calculated using the ideal gas law and the van der Waals equation, we can understand the impact of these corrections on gas behavior. If the calculated pressure using the ideal gas law (\( P_1 \) is greater than the pressure using the van der Waals equation (\( P_2 \)), the gas molecules are experiencing repulsive interactions. Conversely, if \( P_1 \) is less than \( P_2 \), the molecules are experiencing attractive forces. This comparison helps students grasp why and how real gases diverge from ideal behavior.

Physical Chemistry

Physical chemistry is the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. Within this discipline, understanding the behavior of gases is crucial. By learning about different gas laws, such as the ideal gas law and the van der Waals equation, students gain insight into the factors affecting gas properties like pressure, volume, and temperature.

In the context of our problem, physical chemistry helps explain why hydrogen gas at certain conditions may not follow ideal gas behavior due to interactions between the molecules. Furthermore, it illustrates the importance of considering real-world applications when studying theoretical concepts. For example, real gases like hydrogen behave more like an ideal gas when under conditions of high temperature and low pressure, whereas under high pressures or low temperatures, deviations become considerable.