Question

The relation between the gas pressure \(\mathrm{P}\) and average kinetic energy per unit volume \(E\) is (A) \(\mathrm{P}=(2 / 3) \mathrm{E}\) (B) \(P=(3 / 2) E\) (C) \(\mathrm{P}=\mathrm{E}\) (D) \(\mathrm{P}=(\mathrm{E} / 2)\)

Short answer

The correct relation between gas pressure P and average kinetic energy per unit volume E is given by the formula \(P = \frac{2}{3}E\). This is derived from the ideal gas law and the expression for average kinetic energy per unit volume. The correct answer is (A).

Step-by-step solution

Recall the ideal gas law

The ideal gas law is given by the equation: \[P V = n R T\] where P is the gas pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvins.

Define average kinetic energy per unit volume

The average kinetic energy per unit volume is given by the equation: \[E = \frac{(\frac{3}{2})nRT}{V}\] where E is the average kinetic energy per unit volume.

Find the relation between P and E

Now we want to find the relation between the gas pressure P and the average kinetic energy per unit volume E. To do this, we'll use the expressions we have derived in Step 1 and Step 2. From the ideal gas law (Step 1), \[P = \frac{nRT}{V}\] Also, we have the expression for E from Step 2, \[E = \frac{(\frac{3}{2})nRT}{V}\] Divide the expression for E by the expression for P, \[\frac{E}{P} = \frac{(\frac{3}{2})nRT}{V} \times \frac{V}{nRT}\] This simplifies to \[\frac{E}{P} = \frac{3}{2}\] Now, we can rearrange the equation to find the relation between P and E: \[P = \frac{2}{3}E\] So the correct answer is (A) \(\mathrm{P}=(2 / 3) \mathrm{E}\).

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry and physics, describing the behavior of ideal gases by relating pressure, volume, and temperature. Its equation is expressed as \[ P V = n R T \] where:

• P is the pressure of the gas
• V is the volume it occupies
• n is the number of moles of the gas
• R is the ideal gas constant
• T is the temperature in Kelvins

The equation shows how these quantities are interconnected. For instance, at constant volume and mole number, an increase in temperature will increase pressure. Similarly, under constant temperature, increasing the volume will exert a lesser pressure for the same mole of gas. This law assumes that gases behave ideally, meaning there are no interactions between the molecules other than collisions. In real-world conditions, this assumption may not hold perfectly, but the ideal gas law serves as an excellent approximation for many applications.

Average Kinetic Energy

The average kinetic energy per unit volume of a gas is a measure of how the particles in the gas move. It helps in understanding how the temperature of a gas relates to the motion of its particles. The expression is given by: \[ E = \frac{(\frac{3}{2})nRT}{V} \] where:

• E is the average kinetic energy per unit volume
• n is the number of moles
• R is the ideal gas constant
• T is the temperature in Kelvin
• V is the volume

This equation tells us that the average kinetic energy per unit volume is directly proportional to the temperature of the gas. As temperature increases, the average kinetic energy also rises, indicating faster-moving particles. This relationship helps explain why hot gases expand more than cold gases under similar conditions.

Gas Pressure Formulas

Gas pressure in relation to kinetic energy is a pivotal concept that links the macroscopic properties of gases to their molecular behavior. From the ideal gas law, we know \[ P = \frac{nRT}{V} \] And we derived earlier, the relation of pressure to average kinetic energy: \[ P = \frac{2}{3}E \] This indicates that pressure is directly proportional to average kinetic energy per unit volume (\( E \)) and displays how pressure in a gas results from collisions of particles with the walls of its container. When molecules move faster, they hit the walls more frequently and with more force, raising the pressure. This direct relation helps explain phenomena like how heating a gas in an enclosed space increases its pressure while understanding the basic operations of engines and refrigerators.