Question

The ideal gas law is least accurate under conditions of high pressure and low temperature. In those situations, using the van der Waals equation is advisable. (a) Calculate the pressure exerted by \(12.0 \mathrm{g}\) of \(\mathrm{CO}_{2}\) in a \(500-\mathrm{mL}\) vessel at \(298 \mathrm{K},\) using the ideal gas equation. Then, recalculate the pressure using the van der Waals equation. Assuming the pressure calculated from van der Waal's equation is correct, what is the percent error in the answer when using the ideal gas equation? (b) Next, cool this sample to \(-70^{\circ} \mathrm{C}\). Then perform the same calculation for the pressure exerted by \(\mathrm{CO}_{2}\) at this new temperature, using both the ideal gas law and the van der Waals equation. Again, what is the percent error when using the ideal gas equation?

Short answer

At 298 K, the percent error using the ideal gas law is 1.82%. At -70°C, the percent error is 10.55%.

Step-by-step solution

Ideal Gas Pressure at 298 K

To find the pressure exerted by the gas using the ideal gas law, use the formula: \[ P = \frac{nRT}{V} \]Where:- \( R = 0.0821 \ \text{L atm/mol K} \) (ideal gas constant)- \( V = 0.500 \ \text{L} \) (volume)- \( T = 298 \ \text{K} \) (temperature)- \( n \) is the number of moles, \( n = \frac{12.0 \ \text{g}}{44.01 \ \text{g/mol}} = 0.2727 \ \text{mol} \)Substitute these into the equation:\[ P = \frac{0.2727 \times 0.0821 \times 298}{0.500} = 13.38 \ \text{atm} \]

van der Waals Pressure at 298 K

The van der Waals equation is given by:\[ \left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT \]Where for \( \text{CO}_2 \):- \( a = 3.59 \ \text{L}^2 \text{atm/mol}^2 \)- \( b = 0.0427 \ \text{L/mol} \)\( V - nb = 0.500 - 0.2727 \times 0.0427 = 0.4884 \ \text{L} \)\( \frac{an^2}{V^2} = \frac{3.59 \times (0.2727)^2}{(0.500)^2} = 1.072 \ \text{atm} \)Substitute \( a, b, n, R, T \) into the equation:\[ P = \frac{nRT}{V-nb} - \frac{an^2}{V^2} = \frac{0.2727 \times 0.0821 \times 298}{0.4884} - 1.072 = 13.14 \ \text{atm}\]

Percent Error at 298 K

The percent error is calculated using the formula:\[ \text{Percent Error} = \left( \frac{| P_{\text{ideal}} - P_{\text{v.d.W.}} |}{P_{\text{v.d.W.}}} \right) \times 100 \% \]\[ \text{Percent Error} = \frac{|13.38 - 13.14|}{13.14} \times 100\% = 1.82\% \]

Ideal Gas Pressure at -70°C

First convert \(-70^{\circ} \mathrm{C}\) to Kelvin: \( T = 273 - 70 = 203 \text{K} \).The ideal gas law gives:\[ P = \frac{nRT}{V} = \frac{0.2727 \times 0.0821 \times 203}{0.500} = 9.11 \ \text{atm} \]

van der Waals Pressure at -70°C

Using the van der Waals equation with \( T = 203 \text{K} \), we recalculate:\( V - nb = 0.4884 \ \text{L} \) (unchanged as calculations are same for volume adjustment)\( \frac{an^2}{V^2} = 1.072 \ \text{atm} \) (also unchanged)Substitute into the equation:\[ P = \frac{0.2727 \times 0.0821 \times 203}{0.4884} - 1.072 = 8.24 \ \text{atm} \]

Percent Error at -70°C

Calculate the percent error:\[ \text{Percent Error} = \frac{| 9.11 - 8.24 |}{8.24} \times 100 \% = 10.55 \% \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry used to describe the behavior of ideal gases. It is expressed as: \ \[ PV = nRT \]Here,

• \( P \) represents pressure,
• \( V \) is the volume,
• \( n \) is the number of moles of the gas (a measure of quantity),
• \( R \) represents the ideal gas constant, \( 0.0821 \ \text{L atm/mol K} \),
• and \( T \) is the temperature in Kelvin.

To apply the Ideal Gas Law, recall that it assumes gases behave ideally, meaning the gas particles do not attract or repel each other, and their volume is negligible compared to the container. As pressure increases or temperature decreases, gases deviate from ideality. Hence, under extreme conditions like high pressure or low temperature, more advanced models such as the van der Waals equation are recommended. Understanding when to use the Ideal Gas Law is crucial in calculations and scientific predictions.
By calculating the pressure exerted by \( 12.0 \, \text{g} \) of \( \text{CO}_2 \) in a \( 500-\text{mL} \) vessel at \( 298 \, \text{K} \), we apply these principles and derive a pressure estimate that's further verified against the van der Waals model to measure accuracy.

Percent Error Calculation

Percent error is an important concept when comparing experimental results to theoretical predictions. It tells us how far off a measured value is from a known or accepted value and is calculated using the formula:

• \[ \text{Percent Error} = \left( \frac{| \text{Value}_{\text{ideal}} - \text{Value}_{\text{exact}} |}{\text{Value}_{\text{exact}}} \right) \times 100\% \]

The formula subtracts the exact measurement from the estimated measurement, using their absolute difference to eliminate negative errors. Then it divides this difference by the exact value and multiplies by 100 to convert it to a percentage.
In our exercise, we calculated the pressure of carbon dioxide first using the Ideal Gas Law, considered our estimate, and compared it to the more accurate van der Waals pressure. The percent error indicated how much the Ideal Gas Law estimate deviated from the more sophisticated equation. Smaller errors suggest closer agreement with the exact pressure, while larger values highlight the inaccuracies introduced by simplifying assumptions.
At room temperature, we found a percent error of \( 1.82\% \). However, under cooler conditions, the simplifications present in the Ideal Gas Law led to a larger \( 10.55\% \) deviation.

Carbon Dioxide Pressure

In a contained system, carbon dioxide exerts pressure on its surroundings, which can be measured using the laws of gases. Calculating its pressure in different conditions provides insights into how gases behave across various environments.
Using the Ideal Gas Law and the van der Waals equation, we calculated the pressure exerted by \( 12.0 \, \text{g} \) of \( \text{CO}_2 \) within given volumes and temperatures.

• At \( 298 \, \text{K} \), the Ideal Gas Law estimated a pressure of \( 13.38 \, \text{atm} \), while the van der Waals equation provided a value of \( 13.14 \, \text{atm} \).
• When the temperature was reduced to \( -70^\circ \text{C} \) (203 K), the pressures were estimated as \( 9.11 \, \text{atm} \) and \( 8.24 \, \text{atm} \), respectively.

These calculations showcased the effect of temperature on the gas pressure: lower temperatures resulted in decreased pressures, consistent with gas behavior trends. Differences in results from the two equations highlight the gas's ideality limits and the importance of choosing appropriate models based on conditions. Understanding pressure dynamics in gases like carbon dioxide helps in designing systems involving air conditioning, refrigeration, and even fermentation processes.