The Saha equation (degree of ionization in plasma)

[Zarude22]

Homework Statement

We were asked to calculate the degree of ionization which is described by the Saha equation (below). We were given T=0.3 eV, dominating ion species of O+ with density of 10^11 / m^3 and ionization energy of oxygen of 13.62 eV.

Relevant Equations

The equation was given to us in a form of n_i/n_n=3*10^27*T^(3/2)*n_i^(-1)*e^(-U/T) (some approximations used and constants bunched together)

I tried to understand the equation and plug in the numbers, but I just don't get how that is supposed to give us a ratio (with no units!), when it only has the temperature/energy to the power of 3/2 and that multiplied by m^3. Other units (in the exponent of e) cancel each other out. Thank you.

 

[TSny]

Did you check to see if the numerical factor of $3 \times 10^{27}$ has units?

 

[Zarude22]

Did you check to see if the numerical factor of $3 \times 10^{27}$ has units?

Hi! It didnt, which I also thought was kinda of weird, since one of the course books has another constant, 2.4*10^21, with the only difference being KT instead of T in the e^U/T term.
Also in that equation not sure how they are supposed to cancel out.

 

[TSny]

The numerical factor does have units. The Saha equation as given to you must not have indicated the units for the numerical factor. A quick search will show that the numerical factor is expressible in terms of certain fundamental constants such as Planck's constant and the mass of the electron. See here for example. It's a good exercise to show that for the units that you are using, where temperature $T$ is in energy units of eV and length is in meters, the numerical factor has units of (m³ eV^3/2)^-1.

 

[Zarude22]

The numerical factor does have units. The Saha equation as given to you must not have indicated the units for the numerical factor. A quick search will show that the numerical factor is expressible in terms of certain fundamental constants such as Planck's constant and the mass of the electron. See here for example. It's a good exercise to show that for the units that you are using, where temperature $T$ is in energy units of eV and length is in meters, the numerical factor has units of (m³ eV^3/2)^-1.

Yes, I figured it must have OR I have understood the equation and assignment wrong and it's more complex. But if it a simple plug in the value to the equation, then it must have units for it to cancel out. I was just wondering if there was something else that I didn't realize. But thank you!