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Chapter 2: Problem 2

Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?

Short Answer

Expert verified
Real gases deviate from the ideal gas behavior when the pressure is high or the temperature is low. High pressure makes the volume of gas molecules significant compared to the total volume, while low temperature causes intermolecular forces to become significant compared to kinetic energy. In these circumstances, the ideal gas law may not accurately describe the properties of a gas, and alternative models are needed.

Step by step solution

01

Understand the Ideal Gas Law

The ideal gas law is a mathematical equation that describes the behavior of an ideal gas: PV = nRT. This equation assumes that gas molecules have negligible volume, and there are no intermolecular forces between them. These assumptions work well for many gases under standard temperature and pressure conditions, but they do not accurately represent real gases in all situations.
02

Recognizing high pressure deviations

Under high-pressure conditions, the volume of gas molecules becomes significant compared to the total volume of the gas. This means that the assumption that gas molecules have negligible volume is no longer valid. Therefore, at high pressure, real gases will deviate from the ideal gas behavior, and the ideal gas law may not accurately describe their properties.
03

Recognizing low temperature deviations

At low temperatures, the kinetic energy of gas molecules decreases, which means that the strength of intermolecular forces becomes significant compared to the molecules' kinetic energy. This leads to stronger attractions between gas molecules, causing the gas to deviate from the ideal gas law's predictions. Consequently, at low temperatures, real gases do not behave as expected by the ideal gas law.
04

Identifying the circumstances that cause deviations

In summary, real gases deviate from the ideal gas behavior under the following circumstances: 1. High pressure, which causes the volume of gas molecules to become significant compared to the total volume of the gas. 2. Low temperature, which makes the strength of intermolecular forces significant when compared to the kinetic energy of the gas molecules. Under these conditions, the ideal gas law is not an accurate representation of a gas's behavior, and alternative models such as the van der Waals equation or other equations of state may be needed to describe and predict the properties of real gases accurately.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Gases
When thinking about gases, we often refer to them as 'ideal gases' because of the simplicity the ideal gas law brings to calculations: \( PV = nRT \). However, real gases are slightly different. They are made up of molecules that have volume and experience forces from other nearby molecules.
This means that sometimes their behavior diverges from what the ideal gas law would predict. **Why does this happen?** Let's explore further.
  • **Molecular Volume:** Real gas molecules are not point particles. They have physical space which becomes significant especially when the volume of the container is reduced (such as under high pressure).
  • **Intermolecular Forces:** Real gas molecules attract each other when close enough. This can lead to changes in the movement and spacing of the molecules, which isn't accounted for in the ideal gas law.
Understanding these factors helps us appreciate why real gases don't always fit the neat framework of the ideal gas law. This deviation becomes more pronounced under certain conditions.
Deviations from Ideal Behavior
Under typical conditions, real gases function pretty similarly to ideal gases. However, when certain circumstances arise, such deviations occur that we can't ignore anymore. **But why exactly?**
  • **High Pressure:** When the pressure increases, the container's volume decreases, causing molecules to become close-packed. The assumption of negligible molecular volume breaks down, and the actual volume of the molecules starts to matter significantly.
  • **Low Temperature:** At lower temperatures, molecules move slower, allowing intermolecular forces to have a more profound effect. These forces might cause the molecules to stick together more or less than expected.
Recognizing these conditions is crucial since they tell us when real gases might "act up" and deviate from the rules. Thus, chemists and engineers need to account for these deviations in their calculations and predictions to ensure accuracy in experiments and applications.
Van der Waals Equation
The van der Waals equation provides a great way to account for the deviations we observe between real and ideal gases. Named after Johannes Diderik van der Waals, the equation refines the ideal gas law by incorporating specific factors related to molecular volume and intermolecular forces.
The equation is written as:
\[ \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \]
Here is what each component stands for:
  • **\( P \)** is the pressure of the gas.
  • **\( V_m \)** is the molar volume of the gas.
  • **\( R \)** and **\( T \)** remain the universal gas constant and temperature, respectively.
  • **\( a \)** accounts for intermolecular attractions, meaning how much the molecules 'stick' to each other.
  • **\( b \)** represents the volume occupied by the gas molecules themselves.
This modified equation better captures the behavior of real gases under high pressure and low temperatures, making it a useful tool in scientific calculations.

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Most popular questions from this chapter

The gauge pressure in your car tires is 18. The gase \(2.50 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\) at a temperature of \(35.0^{\circ} \mathrm{C}\) when you drive it onto a ship in Los Angeles to be sent to Alaska. What is their gauge pressure on a night in Alaska when their temperature has dropped to \(-40.0^{\circ} \mathrm{C}\) ? Assume the tires have not gained or lost any air. 18. The gauge pressure in your car tires is 18. The gase \(2.50 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\) at a temperature of \(35.0^{\circ} \mathrm{C}\) when you drive it onto a ship in Los Angeles to be sent to Alaska. What is their gauge pressure on a night in Alaska when their temperature has dropped to \(-40.0^{\circ} \mathrm{C}\) ? Assume the tires have not gained or lost any air.

The partial pressure of carbon dioxide in the lungs is about 470 Pa when the total pressure in the lungs is 1.0 atm. What percentage of the air molecules in the lungs is carbon dioxide? Compare your result to the percentage of carbon dioxide in the atmosphere, about \(0.033 \%\)

Statistical mechanics says that in a gas maintained at a constant temperature through thermal contact with a bigger system (a "reservoir") at that temperature, the fluctuations in internal energy are typically a fraction \(1 / \sqrt{N}\) of the internal energy. As a fraction of the total internal energy of a mole of gas, how big are the fluctuations in the internal energy? Are we justified in ignoring them?

Experimentally it appears that many polyatomic molecules' vibrational degrees of freedom can contribute to some extent to their energy at room temperature. Would you expect that fact to increase or decrease their heat capacity from the value \(R\) ? Explain.

$$\text { Verify that } v_{\mathrm{rms}}=\sqrt{\bar{v}^{2}}=\sqrt{\frac{3 k_{\mathrm{B}} T}{m}}$$
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