Kinetic Energy of Gases: Videos & Practice Problems

Kinetic energy, denoted as KE, is the energy an object possesses due to its motion. It can be calculated using two different formulas depending on the available information. When the mass (in kilograms) and velocity (in meters per second) of a gas are known, KE is determined by the equation:

$\mathrm{KE}=\frac{1}{2}m{v}^2$

with the resulting energy in joules (kg·m²/s²). Alternatively, when dealing with the amount of gas in moles and its temperature in Kelvin, the kinetic energy can be calculated using:

$\mathrm{KE}=\frac{3}{2}nRT$

where 'n' is the number of moles, 'R' is the ideal gas constant (8.314 J/mol·K), and 'T' is the temperature in Kelvin. The units of this calculation are in liters·atmospheres, which can be converted to joules using the conversion factor 1 L·atm = 101.325 J. It's important to select the appropriate formula based on whether mass and velocity or moles and temperature are the known variables.

Kinetic Energy Formulas

Kinetic energy is the energy an object has due to its motion, and when studying gases, we can apply kinetic energy formulas to understand their behavior. There are two primary equations for calculating kinetic energy, depending on the information available.

The first equation is used when the mass and velocity of a gas are known. It is expressed as:

\( KE = \frac{1}{2} m v^2 \)

In this formula, \( KE \) represents kinetic energy measured in joules (J), \( m \) is the mass of the gas in kilograms (kg), and \( v \) is the velocity of the gas in meters per second (m/s). The units of kinetic energy can be derived as follows: when mass (kg) is multiplied by the square of velocity (m²/s²), the resulting unit is kg·m²/s², which is equivalent to joules.

The second equation is applicable when dealing with the moles and temperature of a gas. It is given by:

\( KE = \frac{3}{2} n R T \)

In this case, \( n \) denotes the amount of gas in moles, \( R \) is the ideal gas constant (8.314 J/(mol·K)), and \( T \) is the temperature in Kelvin (K). When calculating kinetic energy using this formula, the units can be expressed in liters times atmospheres, which can be converted to joules using the conversion factor of 101.325 J.

It is important to note that these two kinetic energy formulas are not typically required to be memorized, as they are often provided in questions or on formula sheets. When faced with a problem, identify whether you are given mass and velocity or moles and temperature to determine which formula to use.

Kinetic Energy Example

To determine the kinetic energy of a gaseous particle, we use the formula for kinetic energy, which is given by:

\( KE = \frac{1}{2} m v^2 \)

In this equation, \( KE \) represents kinetic energy, \( m \) is the mass in kilograms, and \( v \) is the velocity in meters per second. In this example, we have a particle with a mass of \( 1.56 \times 10^{13} \) picograms and a velocity of \( 6.21 \) meters per second.

First, we need to convert the mass from picograms to kilograms. Since \( 1 \) pico is \( 10^{-12} \) and \( 1 \) kilogram is \( 10^{3} \) grams, we can perform the conversion as follows:

\( 1.56 \times 10^{13} \, \text{pg} = 1.56 \times 10^{13} \times 10^{-12} \, \text{kg} = 0.0156 \, \text{kg} \)

Now that we have the mass in kilograms, we can substitute it into the kinetic energy formula:

\( KE = \frac{1}{2} (0.0156 \, \text{kg}) (6.21 \, \text{m/s})^2 \)

Calculating the velocity squared:

\( (6.21 \, \text{m/s})^2 = 38.6241 \, \text{m}^2/\text{s}^2 \)

Now substituting this value back into the kinetic energy equation:

\( KE = \frac{1}{2} (0.0156 \, \text{kg}) (38.6241 \, \text{m}^2/\text{s}^2) \)

Calculating this gives:

\( KE = 0.0003 \, \text{kg} \cdot \text{m}^2/\text{s}^2 = 1.203 \, \text{J} \)

Thus, the kinetic energy of the particle is \( 1.203 \) joules. If we want to express this in kilojoules, we can convert joules to kilojoules by dividing by \( 1000 \):

\( 1.203 \, \text{J} = 0.001203 \, \text{kJ} = 1.20 \times 10^{-3} \, \text{kJ} \)

Both \( 1.203 \, \text{J} \) and \( 0.001203 \, \text{kJ} \) are acceptable answers for the kinetic energy, depending on the required units.

A baseball with a mass of 503 g possesses a kinetic energy of 0.815 kJ. Calculate its velocity in m/s.

• A. 18.5 m/s
• B. 28.2 m/s
• C. 38.6 m/s
• D. 56.9 m/s
• E. 69.1 m/s

Answer: (D)

A 10.0 L flask contains a mixture of neon and argon gases at a pressure of 2.38 atm. Calculate the total kinetic energy of the gaseous mixture.

• A. 2815 J
• B. 3620 J
• C. 3850 J
• D. 4750 J
• E. 5250 J

Answer: (B)

Here’s what students ask on this topic:

Why does gas have the most kinetic energy?

Gases have the most kinetic energy among the states of matter (solid, liquid, and gas) due to the nature of their particles. In a gas, the particles are much farther apart compared to solids and liquids, and they move freely and rapidly in all directions. This high-speed movement of particles is a manifestation of kinetic energy.

Since there are no significant forces holding the gas particles close together, they collide with each other and the walls of their container in a random and continuous manner. These collisions are perfectly elastic, meaning the gas particles do not lose energy when they collide; instead, they transfer kinetic energy to each other and their surroundings.

The temperature of a gas is directly related to the average kinetic energy of its particles. As the temperature increases, the particles move faster, thus increasing the kinetic energy of the gas. This is why, at the same temperature, gas particles have more kinetic energy than the particles in solids or liquids, where the movement is more restricted.