Question

At \(0^{\circ} \mathrm{C}\) the density of nitrogen at 1 atm is \(1.25 \mathrm{~kg} / \mathrm{m}^{3}\). The nitrogen which occupied \(1500 \mathrm{ml}\) at \(0^{\circ} \mathrm{C}\) and 1 atm was compressed at \(0^{\circ} \mathrm{C}\) and 575 atm and the gas volume was observed to be \(3.92 \mathrm{ml}\), in violation of Boyle's law. What was the final density of this non-ideal gas? (a) \(278 \mathrm{~kg} / \mathrm{m}^{3}\) (b) \(378 \mathrm{~kg} / \mathrm{m}^{3}\) (c) \(478 \mathrm{~kg} / \mathrm{m}^{3}\) (d) \(578 \mathrm{~kg} / \mathrm{m}^{3}\)

Short answer

The final density of the non-ideal compressed nitrogen is 478 kg/m^3.

Step-by-step solution

Determine the Initial Mass of Nitrogen

To find the initial mass of nitrogen, use the density and volume at initial conditions. The density is given as 1.25 kg/m^3. Convert the volume from milliliters (ml) to cubic meters (m^3) by dividing by 1,000,000 because there are 1,000,000 milliliters in a cubic meter: Volume = 1500 ml / 1,000,000 = 0.0015 m^3. Now, use the formula: mass = density x volume to find the initial mass of nitrogen: mass = 1.25 kg/m^3 x 0.0015 m^3 = 0.001875 kg.

Convert the Final Volume to Cubic Meters

The final volume of the gas after compression is given as 3.92 ml, which we also need to convert to cubic meters: Final volume = 3.92 ml / 1,000,000 = 3.92 x 10^-6 m^3.

Calculate the Final Density of Nitrogen

The mass of nitrogen remains constant after compression because no nitrogen is added or removed. Therefore, use the initial mass for the final mass. To find the final density, use the formula: final density = mass/volume. Plugging in the values: final density = 0.001875 kg / 3.92 x 10^-6 m^3 = 478.06 kg/m^3, which rounds to 478 kg/m^3.

Key concepts

Boyle's Law

Boyle's Law describes the relationship between the pressure and volume of a gas at a constant temperature. In its simplest form, it can be expressed as: \( P_1V_1 = P_2V_2 \)where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume, respectively. This inverse relationship states that as pressure increases, volume decreases proportionally, and vice versa, assuming we are dealing with an ideal gas and a constant temperature. However, in the case of non-ideal gas behavior, discrepancies from this law can occur due to intermolecular forces and the finite volume of gas particles, both of which are not accounted for in Boyle's Law. This is why the exercise mentions the violation of Boyle's Law, indicating the presence of non-ideal behavior.

Gas Density Calculations

Calculating the density of a gas involves understanding the relationship between its mass and volume. The density (\( \rho \)) can be calculated using the formula: \( \rho = \frac{mass}{volume} \)In the context of the exercise, obtaining the initial mass involves multiplying the given density by the initial volume. To find the gas's final density after compression, we use the same mass (since mass is conserved) and the new, smaller volume. Density calculations become more complicated with non-ideal gases. Real gases don't always comply with straightforward equations due to their interaction and particle volume, which must be accounted for in advanced calculations.

Non-Ideal Gas Behavior

Non-ideal gas behavior is typically observed under conditions of high pressure, low temperature, or when gas particles have significant volume or interactions. For non-ideal gases, the simple prediction of Boyle's Law may not hold true because gas particles deviate from the assumptions of the Ideal Gas Law. Real gases exhibit attractions or repulsions that can affect their pressure, volume, and temperature relationships. In the problem provided, nitrogen is subjected to a high pressure of 575 atm, under which it would not behave ideally. This explains why the exercise specifies that there was a violation of Boyle's Law. In such instances, modifications to the Ideal Gas Law, such as the Van der Waals equation, are required to accurately describe the behavior of the gas.

To explore the nuances of non-ideal gas behavior and improve understanding, students may consider studying the Van der Waals equation or engaging in experiments that illustrate deviations from ideal behavior, such as observing the compression of a gas at various pressures and temperatures.