Question

Which of the following statements is(are) true? a. At constant temperature, the lighter the gas molecules, the faster the average velocity of the gas molecules. b. At constant temperature, the heavier the gas molecules, the larger the average kinetic energy of the gas molecules. c. A real gas behaves most ideally when the temperature is low and the pressure of the gas is high. d. At least two of the above \((\mathrm{a}-\mathrm{c})\) are true. e. None of the above is true.

Short answer

The correct answer is option a: At constant temperature, the lighter the gas molecules, the faster the average velocity of the gas molecules. This is true because the average velocity of gas molecules is given by \(v_{avg} = \sqrt{\frac{8k_BT}{\pi m}}\), which shows the inverse relationship between the mass of the gas molecule and its average velocity.

Step-by-step solution

Statement a: At constant temperature, the lighter the gas molecules, the faster the average velocity of the gas molecules.

This statement is true. According to the kinetic theory of gases, the average velocity of gas molecules is given by the equation: \(v_{avg} = \sqrt{\frac{8k_BT}{\pi m}}\) where \(v_{avg}\) is the average velocity, \(k_B\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of a gas molecule. At constant temperature, the average velocity of the gas molecules is inversely proportional to the square root of their mass. Therefore, lighter gas molecules have a higher average velocity, and heavier molecules have a lower average velocity.

Statement b: At constant temperature, the heavier the gas molecules, the larger the average kinetic energy of the gas molecules.

This statement is false. The average kinetic energy of gas molecules is given by the equation: \(KE_{avg} = \frac{3}{2}k_BT\) The average kinetic energy depends only on the temperature and is independent of the mass of the gas molecules. Therefore, at constant temperature, the average kinetic energy of gas molecules is the same for all gases, regardless of their mass.

Statement c: A real gas behaves most ideally when the temperature is low and the pressure of the gas is high.

This statement is false. A real gas behaves most ideally when the temperature is high and the pressure is low. When the temperature is high, the gas molecules have more kinetic energy, and the intermolecular forces between them become negligible, making the gas more similar to an ideal gas. On the other hand, when the pressure is low, the volume occupied by the gas is much larger, and the size of the molecules becomes negligible compared to the volume. This also makes the gas behave more like an ideal gas.

Options d and e: At least two of the above (a-c) are true, or none of the above is true.

Since statement a is true and statements b and c are false, only one of the statements (a-c) is true. Therefore, option d (at least two of the above are true) is false, and option e (none of the above is true) is also false. Based on our analysis, the correct answer is option a.

Key concepts

Average Velocity of Gas Molecules

Understanding the average velocity of gas molecules is key to grasping how gases behave under different conditions. The average velocity of gas molecules is determined by their mass and the temperature of the gas. According to the kinetic theory of gases, which explains the movement of gas particles:

• The average velocity \( v_{avg} \) is given by the formula: \( v_{avg} = \sqrt{\frac{8k_BT}{\pi m}} \).
• In this formula, \( k_B \) represents the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of the gas molecule.
• As you can see from the equation, the average velocity is inversely proportional to the square root of the molecular mass.

Therefore, lighter gas molecules will move at a faster average speed than heavier ones at a constant temperature. This relationship is crucial in many scientific and engineering calculations, where the motion of molecules can affect outcomes.

Real Gas Behavior

Real gas behavior differs from ideal gas behavior, which is explained by the discrepancies arising from the properties of real gases. Real gases may not conform strictly to the ideal gas law due to factors like intermolecular forces and the non-negligible volume of the gas molecules themselves.

• Real gases behave most ideally under conditions of high temperature and low pressure.
• At high temperatures, the increased kinetic energy of molecules overcomes any intermolecular attractions, allowing them to move more freely and mimic the behavior of an ideal gas.
• At low pressures, the effect of the molecular volume becomes negligible, which also helps in aligning real gas behavior more closely with ideal conditions.

Understanding these conditions helps us manipulate and predict the behavior of gases in various applications, from industrial processes to natural phenomena.

Ideal Gas Conditions

Ideal gas conditions refer to theoretical conditions under which a gas perfectly follows the ideal gas law: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature.

• In an ideal gas, intermolecular forces are assumed to be nonexistent.
• The molecules are also considered to occupy no volume, meaning they are point particles.
• Ideal gas behavior is primarily used as a simplification for calculations and can often adequately describe real gases under conditions of high temperature and low pressure.

By studying ideal gas conditions, scientists and engineers are able to create models that predict how gases will react and change in response to external variables, which is essential for fields like chemistry, physics, and engineering.