Question

Which of the following statements is(are) true? For the false statements, correct them. 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 container volume is relatively large and the gas molecules are moving relatively quickly. d. As temperature increases, the effect of interparticle interactions on gas behavior is increased. e. At constant \(V\) and \(T\), as gas molecules are added into a container, the number of collisions per unit area increases resulting in a higher pressure. f. The kinetic molecular theory predicts that pressure is inversely proportional to temperature at constant volume and moles of gas.

Short answer

True statements: a, c, e False statements and corrections: - b: At constant temperature, the average kinetic energy of gas molecules is the same regardless of their mass. - d: As temperature increases, the effect of interparticle interactions on gas behavior is decreased. - f: The kinetic molecular theory predicts that pressure is directly proportional to temperature at constant volume and moles of gas.

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 molecular theory, the velocity of gas molecules is inversely proportional to their mass. Therefore, lighter gas molecules will have a higher average velocity compared to heavier gas molecules at a constant temperature.

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 any gas molecules is determined by the temperature and not their mass. In fact, the average kinetic energy is given by \( \frac{3}{2}kT \), where \(k\) is the Boltzmann constant and \(T\) is the temperature. To correct the statement: "At constant temperature, the heavier the gas molecules, the larger the average kinetic energy of the gas molecules." should be "At constant temperature, the average kinetic energy of gas molecules is the same regardless of their mass."

Statement c

A real gas behaves most ideally when the container volume is relatively large and the gas molecules are moving relatively quickly. This statement is true. A real gas tends to behave most like an ideal gas under conditions of low pressure (large container volume) and high temperature (fast-moving gas molecules). Under these conditions, the gas molecules are far apart, and the intermolecular interactions between them are minimized.

Statement d

As temperature increases, the effect of interparticle interactions on gas behavior is increased. This statement is false. As the temperature increases, the gas molecules move faster, and the effect of intermolecular interactions is actually decreased. To correct the statement: "As temperature increases, the effect of interparticle interactions on gas behavior is increased." should be "As temperature increases, the effect of interparticle interactions on gas behavior is decreased."

Statement e

At constant \(V\) and \(T\), as gas molecules are added into a container, the number of collisions per unit area increases resulting in a higher pressure. This statement is true. According to the Ideal Gas Law (PV = nRT), if the volume and temperature of a gas are held constant while the number of gas molecules (moles, n) increases, the pressure exerted by the gas will increase due to the increased number of collisions with the container walls.

Statement f

The kinetic molecular theory predicts that pressure is inversely proportional to temperature at constant volume and moles of gas. This statement is false. According to the Ideal Gas Law (PV = nRT), if the volume and moles of a gas are held constant, the pressure is directly proportional to the temperature. To correct the statement: "The kinetic molecular theory predicts that pressure is inversely proportional to temperature at constant volume and moles of gas." should be "The kinetic molecular theory predicts that pressure is directly proportional to temperature at constant volume and moles of gas."

Key concepts

Kinetic Energy of Gases

Understanding the concept of kinetic energy in gases is essential when studying the behavior of these tiny particles. Kinetic energy is basically the energy of motion, and in the realm of gases, each molecule zips around with its own share of this energy. It's fascinating to know that this energy is directly related to temperature. In fact, at a given temperature, all gas molecules, regardless of their size, have the same average kinetic energy. An intriguing point brought up in the exercise is the misconception that heavier gas molecules have more kinetic energy, when in reality, the kinetic energy is a temperature-dependent affair.

So, how do we calculate this kinetic energy? It's actually given by the equation \( \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature. Imagine billions of gas particles bouncing around - their collective motion is what we measure as temperature. When they move faster, temperature rises, indicating an increase in kinetic energy. But despite varying speeds, the energy average stays consistent across the board, painting a dynamic and heated picture of a gas at any given temperature.

Therefore, it's key to understand that while individual speeds vary, the average kinetic energy remains a constant at a specific temperature, reflecting the democratic nature of energy distribution among gas particles.

Ideal Gas Behavior

When we talk about ideal gas behavior, we're entering a simplified world where gases obey certain theoretical laws without fail. An ideal gas is an imaginary gas where the particles don't interact with each other and take up no space. Of course, in reality, no gas is truly 'ideal', but many gases behave closely enough to this model under certain conditions, making it a useful representation for many scientific calculations.

As highlighted in the exercise, when a gas is in a large volume container and the particles are moving quickly (which is usually the case at higher temperatures), the behavior of the gas aligns more closely with our ideal model. This is because the increased space and speed mean that the particles are less likely to interact with each other, and even if they do, these interactions are brief and don't significantly affect the overall behavior of the gas.

In the classroom, students are often taught the 'high temperature and low pressure' mantra when memorizing the conditions for ideal gas behavior. This mantra syncs with the concept of minimizing particle interactions and maximizing the kinetic energy so that the gas laws we study can be applied with minimal deviations from the observed behavior of real gases.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of thermodynamics and provides a clear lens through which to view the behavior of gases under various conditions. Its formula, \( PV = nRT \), where \( P \) represents pressure, \( V \) stands for volume, \( n \) indicates the number of moles of gas, \( R \) is the gas constant, and \( T \) signifies temperature, is elegant in its simplicity and powerful in its application.

The beauty of the Ideal Gas Law lies in its ability to relate all the primary variables that define a gas's condition. For instance, if you know the amount of gas (in moles), the volume it occupies, and the temperature it's at, you can calculate the pressure it exerts. Conversely, change the pressure or temperature, and you'll observe a predictable shift in volume, assuming you keep other factors constant.

The exercise points to a fundamental relationship captured by the law: pressure and temperature are directly proportional when volume and moles remain unchanged. This means that heating a confined gas will increase its pressure, as the molecules collide more energetically against the container walls.

All in all, while real gases may not obey the Ideal Gas Law perfectly, many come close enough that this simple equation can be used for a wide range of practical calculations, from designing engines and balloons to understanding the atmosphere's behavior.