Kinetic energy is the energy an object possesses due to its motion. This energy is present in everything from a moving car to the constant, random movement of microscopic particles like atoms and molecules. In a large collection of particles, such as a sample of gas or liquid, not every particle moves at the same speed, exhibiting a wide range of velocities.
Because of this variation in speed, scientists rely on the concept of average kinetic energy (\(\text{KE}_{avg}\)). This value represents the mean energy of motion across all particles within the sample. Calculating this average characterizes the substance’s motion, providing a single, representative figure for its internal energy state. This calculation is a fundamental tool in thermodynamics and physical chemistry.
Kinetic Energy and Temperature: The Core Relationship
The foundation for calculating average kinetic energy is rooted in the kinetic molecular theory (KMT) of gases. This theory describes gases as collections of particles in constant, random motion that frequently collide. The average kinetic energy of these particles is directly proportional to the absolute temperature of the gas.
Temperature, specifically measured on the Kelvin scale, is a macroscopic manifestation of the microscopic average kinetic energy. Raising the temperature causes atoms and molecules to move faster, increasing their average kinetic energy. Conversely, lowering the temperature slows the average motion, decreasing the average kinetic energy. This direct relationship means temperature is a precise measure of this average energy.
A consequence of the kinetic molecular theory is that all gases at the same absolute temperature will have the same average kinetic energy, regardless of their chemical identity or mass. For example, light helium gas and heavy xenon gas share the same average kinetic energy if both are at \(300 \text{ Kelvin}\). This means the lighter helium atoms must move at a greater average speed than the heavier xenon atoms to achieve this equal energy value. Temperature is thus the sole determinant of average kinetic energy in an ideal gas system.
The Formula for Average Kinetic Energy
To move from the conceptual relationship to a quantitative value, a specific formula derived from statistical mechanics is used. The expression for the average kinetic energy of a single particle (atom or molecule) is \(\text{KE}_{avg} = \frac{3}{2} kT\). This equation demonstrates the direct proportionality between the average energy and the absolute temperature (\(T\)). The factor \(\frac{3}{2}\) arises from considering the three independent spatial directions (degrees of freedom) in which a particle can move.
Scientists often work with macroscopic quantities of matter, typically measured in moles, making the single-particle equation impractical for bulk calculations. For these larger samples, a modified formula is used: \(\text{KE}_{avg} = \frac{3}{2} RT\). This second formula calculates the total average kinetic energy per mole of gas.
The structure of both equations highlights that average kinetic energy depends only on the sample’s temperature. Both forms include the \(\frac{3}{2}\) factor and temperature \(T\). The constants \(k\) and \(R\) adjust the calculation for the specific quantity of matter: \(kT\) is for a single particle, and \(RT\) is for a standard mole of a substance.
Understanding the Constants
The two formulas rely on distinct proportionality constants to scale the result correctly. The Boltzmann Constant (\(k\)) is used for calculating the average kinetic energy per single particle. Its numerical value is \(1.380649 \times 10^{-23} \text{ J/K}\). This value translates the temperature in Kelvin directly into an energy value in Joules for one molecule.
When calculating the average kinetic energy for a mole of substance, the Ideal Gas Constant (\(R\)) is used instead. This constant is approximately \(8.314 \text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}\). The difference in magnitude between \(k\) and \(R\) is due to the scale of the measurement. The Ideal Gas Constant \(R\) is the product of the Boltzmann Constant \(k\) and Avogadro’s number (\(N_A\)).
Avogadro’s number (\(N_A\)) is the number of particles in one mole of any substance, defined as \(6.02214076 \times 10^{23} \text{ particles per mole}\). Multiplying the energy per particle (\(k\)) by \(N_A\) yields the energy per mole (\(R\)). This relationship ensures both formulas are mathematically consistent for calculations at both the microscopic and macroscopic levels.
Applying the Formula: A Worked Example
Calculating the average kinetic energy requires applying the appropriate formula and ensuring all units are correctly converted. Consider the calculation for one mole of nitrogen gas at \(25^\circ \text{C}\). The \(\text{KE}_{avg} = \frac{3}{2} RT\) formula is selected because the problem specifies one mole of gas.
The first step is converting the temperature from Celsius to the absolute Kelvin scale, as the formula requires this unit. The conversion is \(25^\circ \text{C} + 273.15 = 298.15 \text{ K}\). Failing to convert the temperature to Kelvin will result in an incorrect calculation of the average kinetic energy.
Next, substitute the known values into the formula. Using \(R = 8.314 \text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}\), the equation becomes \(\text{KE}_{avg} = \frac{3}{2} \times (8.314 \text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}) \times (298.15 \text{ K})\). The Kelvin units cancel out during multiplication, leaving the final result in Joules per mole.
The calculation proceeds by multiplying the constant and the temperature: \(8.314 \times 298.15 \approx 2480.9 \text{ J}\cdot\text{mol}^{-1}\). The final step is to multiply this result by the \(\frac{3}{2}\) factor (1.5). This yields an average kinetic energy of \(3721.4 \text{ Joules per mole}\) for the nitrogen gas.