# Thermal Energy Calculator

Created by Luciano Mino
Last updated: Sep 14, 2022

Our thermal energy calculator can find the thermal energy of an ideal gas based on a few parameters from kinetic molecular theory.

We've paired this calculator with a short text covering:

• What is thermal energy;
• How to calculate thermal energy for an ideal gas;
• Relationship between kinetic energy and thermal energy for an ideal gas; and

## Kinetic molecular theory postulates

We can derive all the definitions used in this text from the kinetic molecular theory, which states:

• Gases consist of a large number of particles (atoms or molecules) in constant random motion;
• The size of these particles is negligible compared to the average distance between them;
• The collisions between particles themselves or with the walls of the container are perfectly elastic; and
• The particles exert no force on one another except during collisions.

Let's now take a look at the thermal energy definition.

## What is thermal energy?

Thermal energy is the part of the internal energy of a system associated with the random motion of molecules.

For an ideal gas, its thermal energy is proportional to the average kinetic energy of its molecules. Therefore, gas at higher temperatures will possess greater internal energy.

## Thermal energy equation

For a monoatomic gas with three degrees of freedom, the average kinetic energy of its atoms will be:

$\lang \text{KE} \rang = \frac{3}{2}k_{B}T$

where:

• $k_{B}$ is Boltzmann's constant, $k_{B} = 1.38064852 * 10^{-23}\ \text{J/K}$;
• $T$ is the temperature;
• $KE$ is the average kinetic energy of its atoms.

🙋 You can change the degrees of freedom number in the advanced-mode of this thermal energy calculator

And we can relate this formula to thermal energy in the thermal energy equation:

$U = n\cdot N_{A}\cdot \lang \text{KE} \rang$

where:

• $n$ is the number of moles;
• $N_{A}$ is Avogadro constant, $N_{A} = 6.02214076×10^{23}\ \text{mol}^{−1}$; and
• $U$ is the thermal energy of the gas.

### Average velocity of particles

Our thermal energy calculator also allows you to input the average velocity of particles derived from this equation:

$\lang \text{v} \rang = \sqrt{\frac{2\cdot \lang \text{KE} \rang\cdot N_{A}}{M}}$

where $M$ is the molar mass of the gas.

Luciano Mino
Molar mass
g/mol
Temperature
K
Moles of gas
mol
Average kinetic energy
meV
Average speed
ft/s
Total thermal energy
J
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