6.2129
×10^-21 J
3,741.5082
J
6,235.847
J
421.99
m/s
476.17
m/s
516.83
m/s
20.786
J/(mol·K)
29.101
J/(mol·K)
1.4
28.014
g
1.137984
kg/m^3
6.2129
×10^-21 J
3,741.5082
J
6,235.847
J
421.99
m/s
476.17
m/s
516.83
m/s
20.786
J/(mol·K)
29.101
J/(mol·K)
1.4
28.014
g
1.137984
kg/m^3
The Kinetic Theory of Gases Calculator provides a comprehensive analysis of gas behavior based on the molecular kinetic theory. This foundational framework connects macroscopic thermodynamic properties (temperature, pressure, energy) to the microscopic motion of gas molecules. The theory rests on key assumptions: molecules are in constant random motion, collisions are perfectly elastic, and molecular volume is negligible compared to the container.
The central result of kinetic theory connects temperature to molecular kinetic energy through the equipartition theorem:
$$\langle KE \rangle = \frac{f}{2}k_BT$$
where f is the number of degrees of freedom and $$k_B = 1.381 \times 10^{-23}$$ J/K. For translational motion alone (f = 3), the average kinetic energy per molecule is $$\frac{3}{2}k_BT$$. For n moles of gas, the total translational kinetic energy is $$KE_{total} = \frac{3}{2}nRT$$. The internal energy includes all degrees of freedom: translational, rotational, and vibrational, making it $$U = \frac{f}{2}nRT$$.
The calculator uses the equipartition theorem and Maxwell-Boltzmann statistics to compute all key properties. Select the degrees of freedom based on molecular structure:
The internal energy is: $$U = \frac{f}{2}nRT$$
The molar heat capacities follow from the equipartition theorem: $$C_v = \frac{f}{2}R$$ and $$C_p = C_v + R = \frac{f+2}{2}R$$. The heat capacity ratio $$\gamma = C_p/C_v = (f+2)/f$$ determines the speed of sound and adiabatic processes. For a monatomic gas, γ = 5/3 = 1.667; for diatomic, γ = 7/5 = 1.400; for polyatomic, γ = 8/6 = 1.333.
The three characteristic velocities from the Maxwell-Boltzmann distribution are also computed: $$v_{rms} = \sqrt{3RT/M}$$, $$v_{avg} = \sqrt{8RT/(\pi M)}$$, and $$v_{mp} = \sqrt{2RT/M}$$.
The internal energy represents the total thermal energy stored in the gas. It increases linearly with temperature and with the number of degrees of freedom. A diatomic gas stores 5/3 more energy than a monatomic gas at the same temperature because rotational modes contribute additional energy storage. The heat capacity ratio γ determines how gases respond to adiabatic compression: higher γ means more temperature change per unit compression, which is why monatomic gases heat up more during compression than polyatomic gases.
Inputs
Results
1 mol of N₂ at 300 K has 6235.5 J of internal energy (5 DOF) and γ = 1.4, consistent with experimental values.
Inputs
Results
For monatomic Ar, internal energy equals translational KE (no rotational modes). γ = 5/3 is the highest possible for an ideal gas.
Degrees of freedom (DOF) are independent modes of energy storage. A monatomic gas has 3 translational DOF (x, y, z motion). A diatomic gas adds 2 rotational DOF (rotation about two axes perpendicular to the bond). Polyatomic gases have 3 rotational DOF. At very high temperatures, vibrational modes become active, adding 2 more DOF per vibrational mode.
Temperature is defined through the average translational kinetic energy: $$\frac{3}{2}k_BT = \frac{1}{2}m\langle v^2 \rangle$$. While internal energy includes rotational and vibrational contributions, temperature directly measures the translational component. Two gases at the same temperature have the same average translational KE regardless of molecular complexity.
The equipartition theorem states that each quadratic degree of freedom contributes $$\frac{1}{2}k_BT$$ of average energy per molecule (or $$\frac{1}{2}RT$$ per mole). A "quadratic" DOF appears as a squared term in the energy expression, such as $$\frac{1}{2}mv_x^2$$ for translation or $$\frac{1}{2}I\omega^2$$ for rotation.
Quantum mechanics requires a minimum energy (quantum) to excite each mode. At low temperatures, $$k_BT$$ may be too small to activate rotational modes (below ~80 K for H₂) or vibrational modes (below ~1000-3000 K for most diatomics). The equipartition theorem is a classical approximation that breaks down when $$k_BT \ll h\nu$$ for a given mode frequency ν.
The molar heat capacity at constant volume is $$C_v = \frac{\partial U}{\partial T}\bigg|_V = \frac{f}{2}R$$. Since U is linear in T for an ideal gas, Cv is constant. For a monatomic gas, Cv = 12.47 J/(mol·K). For diatomic, Cv = 20.79 J/(mol·K).
The speed of sound is $$v_s = \sqrt{\gamma RT/M}$$, where γ = Cp/Cv is the heat capacity ratio. Higher γ and lower molar mass give faster sound speed. Helium (γ = 5/3, M = 4) carries sound at ~1007 m/s versus 343 m/s in air at room temperature.
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