📂Thermal Physics

Gas Flux

Definition¹

In physics, flux is the number of particles (or a physical quantity such as energy, momentum, etc.) that pass through (collide with) a unit area per unit time. Flux is often denoted by the capital letter Phi (Φ) $\Phi$.

$$ \Phi = \dfrac{\text{physical quantity}}{\text{area} \times \text{time}} $$

Explanation

According to the definition, the flux of a gas refers to the number of gas molecules passing through a unit area per unit time, and heat flux refers to the amount of thermal energy passing through a unit area per unit time.

The number of gas molecules with velocity $v$ hitting a unit area per unit time at an angle of $\theta$ is given in the process of deriving the ideal gas law as follows.

$$ \left( n f(v) \right) \left( \dfrac{1}{2} \sin\theta\right) \left( v \cos \theta \right) $$

Therefore, integrating this with respect to $r \in [0,\infty)$ and $\theta=[0,\pi / 2]$ gives the number of molecules hitting a unit area per unit time.

$$ \begin{align*} \Phi &= \int_{0}^{\infty} \int_{0}^{\pi/2} \left( n f(v) \right) \left( \dfrac{1}{2} \sin\theta\right) \left( v \cos \theta \right) dv d\theta \\ &= \dfrac{n}{2} \int_{0}^{\infty} f(v) v dv \int_{0}^{\pi/2} \sin \theta \cos \theta d\theta \end{align*} $$

The integration mentioned above for $v$ is the expected value of velocity.

$$ \int_{0}^{\infty} f(v) v dv = \left\langle v \right\rangle $$

The integral value for $\theta$ is $\dfrac{1}{2}$, so the flux of a gas is expressed as the number of molecules hitting a unit area per unit time as follows.

$$ \Phi = \dfrac{1}{4}n\left\langle v \right\rangle $$

Let’s calculate this value. The expected value of velocity is as follows.

$$ \left\langle v \right\rangle = \sqrt{\dfrac{8 k_{B} T}{\pi m }} $$

And, given the ideal gas law, $p = n k_{B}T$,

$$ \Phi = \dfrac{1}{4} \dfrac{p}{k_{B}T} \sqrt{\dfrac{8 k_{B} T}{\pi m }} = \dfrac{p}{\sqrt{2 \pi m k_{B} T}} $$

From the above equation, it can be seen that the flux of a gas is inversely proportional to the square root of the mass. This is the same result experimentally discovered by Graham, known as Graham’s law of effusion.