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The average translational kinetic energy is an important concept in understanding how particles behave in an ideal gas. It provides insight into the energy associated with the motion of individual gas molecules.
To calculate this energy for a molecule in an ideal gas, you can use the formula:

• KE = \(\frac{3}{2} kT\)

Here.
- KE stands for the average translational kinetic energy.
- \(k\) is Boltzmann's constant, \(1.38 \times 10^{-23} \text{J/K}\). This constant essentially acts as a bridge between macroscopic and microscopic worlds.
- \(T\) is the absolute temperature in Kelvin.
This relationship tells us that as the temperature of the gas increases, so does the average kinetic energy of its molecules.
It's crucial in understanding that energy and temperature are directly linked at the molecular level.

Knowing the temperature is vital in many physics calculations involving gases. In our problem, we determine it using the Ideal Gas Law, which is stated as \(PV = nRT\).
Here's a breakdown of the terms:

• \(P\) is pressure, given as \(4.5 \times 10^5 \, \text{Pa}\), which means Pascals.
• \(V\) is volume, provided as \(8.5 \times 10^{-3} \, \text{m}^3\).
• \(n\) is the number of moles, \(2\) here.
• \(R\) is the ideal gas constant, \(8.314 \, \text{J/mol K}\), which we will discuss in detail below.

To find the temperature \(T\), rearrange the formula to: \(T = \frac{PV}{nR}\). Substitute in the values to get \(T \approx 230 \, \text{K}\).
This temperature calculation helps in determining the kinetic energy of particles, showing the strong connection between these concepts.

The ideal gas constant \(R\) is pivotal in linking many properties of gases. It appears in the Ideal Gas Law \(PV = nRT\) and has a value of \(8.314 \, \text{J/mol K}\). This constant enables physical relationships between pressure, volume, temperature, and number of moles of a gas.

• Unit Explanation: \(\text{J/mol K}\) represents Joules per mole per Kelvin. These units are crucial because they directly relate energy (Joules) to temperature (Kelvin) and amount of substance (moles).
• Application: \(R\) is used in situations involving thermodynamic processes and calculations, highlighting its importance in fields like chemistry and physics.

Grasping \(R\) helps understand how energy changes within a system as gases undergo different transformations. This concept is not only theoretical but also applicable in practical scenarios involving gas behaviors.