Problem 134 - Master Resource Book in JEE Main Physics

Question

Under which of the following conditions, the law \(p V=R T\) is not obeyed by a real gas? (a) High pressure and high temperature (b) Low pressure and low temperature (c) Low pressure and high temperature (d) High pressure and low temperature

Step-by-Step Solution

Answer

Real gases deviate most under condition (d) high pressure and low temperature.

Step 1: Understanding Real Gas Behavior

Real gases deviate from ideal gas behavior due to intermolecular forces and the finite volume of gas particles. Under ideal conditions, the equation \( pV = RT \) is perfectly obeyed, but real gases differ from this behavior.

Step 2: Conditions Affecting Real Gases

Ideal gas behavior is more closely followed at low pressures where intermolecular forces are negligible and at high temperatures where kinetic energy overcomes these forces. Thus, deviations are prominent at high pressures (where particle volume and forces become significant) and low temperatures (where forces have a greater effect).

Step 3: Evaluating Provided Options

From the previous step, real gases deviate notably under high pressure and low temperature conditions because these maximize intermolecular forces and the volume occupied by gas molecules relative to the container's volume.

Step 4: Selecting the Correct Condition

Comparing the effect of conditions on gases:- (a) High pressure and high temperature: less deviation due to high temperature.- (b) Low pressure and low temperature: some deviation, but not maximized.- (c) Low pressure and high temperature: minimal deviation.- (d) High pressure and low temperature: maximum deviation from ideal behavior.Thus, condition (d) is where the law \( pV = RT \) is not obeyed by a real gas.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the behavior of an ideal gas. It is expressed as \( pV = RT \), where \( p \) represents pressure, \( V \) represents volume, and \( T \) denotes temperature. \( R \) is the universal gas constant, which serves as a bridge between the units used for pressure, volume, and temperature.
The ideal gas law assumes that gas particles are point masses with no attraction or repulsion between them. This means that the particles move in random motion and collide elastically without energy loss. This hypothetical model works well under certain conditions.

• It assumes high temperature and low pressure, where intermolecular forces are minimal.
• At such conditions, the gas particles are so far apart that their volume is negligible compared to the container they are in.

Understanding these assumptions is crucial for grasping why real gases sometimes do not follow this law. Real-world conditions often diverge from this ideal behavior, leading to notable deviations under certain circumstances.

Deviation from Ideal Gas

Real gases do not always behave as ideal gases; this is referred to as deviation from the ideal gas law. Such deviation occurs due to two main reasons:

• Intermolecular Forces: In real gases, particles do experience attractions and repulsions between them. These forces become significant at high pressures because the particles are closer together.
  
• Finite Volume of Particles: Unlike the assumption in the ideal gas law, real gas particles have a finite size. This volume becomes important relative to the volume of the container when the gas is compressed under high pressure.
  

Under conditions where real gases do follow the ideal gas law more closely, such as low pressure and high temperature, these factors have less impact because:

• At high temperatures, kinetic energy of gas particles is high enough to overcome intermolecular attractions.
• At low pressures, the volume of individual gas particles can be ignored compared to the overall volume.

Recognizing these deviations helps us understand where and when the ideal gas law is applicable and why adjustments or different models are sometimes necessary for real gases.

Conditions Affecting Real Gases

The behavior of real gases and their deviation from the ideal gas law is influenced significantly by certain conditions. Let's explore these conditions that impact real gas behavior:

• High Pressure: Under high pressure, real gases exhibit more pronounced deviation from ideal behavior. This is because the volume occupied by gas molecules is no longer negligible compared to the container's volume, and intermolecular forces become much more significant.
  
• Low Temperature: At low temperatures, the kinetic energy of gas particles is reduced, making intermolecular forces more noticeable. When the temperature is low, gas particles move more slowly, allowing attractions or repulsions to have a bigger impact.
  

To minimize these deviations, gases should ideally be kept at:

• Low Pressure: Allows molecules to be spaced far apart, reducing interactions.
• High Temperature: Increases particle energy, overcoming intermolecular forces.
  

In our study scenario (option d), high pressure and low temperature increase both the effect of particle volume and the influence of intermolecular forces, causing maximum deviation from the ideal gas law. Understanding these conditions helps in predicting gas behavior in various environments and applications.