<p>There are two things going on. (1) When you add heat to a gas that is on the threshold of ionisation or partially ionised, some of that heat goes into ionisation. This means that it takes a much larger amount of energy to produce a rise in temperature. (2) However, as the gas becomes ionised the number of particles per unit mass also increases and so the pressure increases at a given temperature. The net effect is that <span class="math-container">$C_V$</span> (heat capacity for a fixed volume) increases by more than <span class="math-container">$C_P$</span> (heat capacity for a fixed pressure) and so the adiabatic index decreases.</p>

<p>It is a tricky and detailed computation to work out how <span class="math-container">$C_V$</span> and <span class="math-container">$C_P$</span> change as ionisation proceeds. An example calculation for pure hydrogen gas is give on pp 123-126 of "Principles of stellar evolution and nucleosynthesis" (1983, D. Clayton). Both <span class="math-container">$C_V$</span> and <span class="math-container">$C_P$</span> increase drastically (by factor of 30-40) as ionisation increases, both peaking at around 50% ionisation before falling back to exactly <em>twice</em> their unionised values once the gas is completely ionised (when it just behaves like a monatomic perfect gas but with twice the number of particles per unit mass). However, the behaviour of <span class="math-container">$C_V$</span> and <span class="math-container">$C_P$</span> are slightly different, <span class="math-container">$C_P$</span> increases by less than <span class="math-container">$C_V$</span>, and so the ratio of <span class="math-container">$C_P/C_V$</span> changes and goes through a minimum at around 50% ionisation before regaining the standard monatomic gas value once completely ionised.</p>

CC BY-SA 4.0
2018-09-28T17:00:56.137Z
2018-09-29T10:29:09.883Z
2018-09-29T10:29:09.883Z
ProfRob
<p>The context is the ionization zones in stellar atmospheres or interiors, the Sun, for instance.</p>

<p>The adiabatic exponent is the heat capacity ratio:</p>

<p><span class="math-container">$$\gamma = \frac{c_P}{c_V} = \frac{C_P}{C_V}$$</span></p>

<p>And, for an ideal gas, it can be shown that  <span class="math-container">$\gamma=(f+2)/f$</span> where <span class="math-container">$f$</span> is the degrees of freedom. Resulting in <span class="math-container">$\gamma\approx1.66$</span>. </p>

<p>In most of the extent of the solar interior, <span class="math-container">$\gamma\sim1.66$</span> but it presents dips when ionization is present, like the H, HeI and HeII ionization zones. <strong>My question is why does this happen?</strong></p>

<p>I thought it could be because when ionization, the heat is used to ionize atoms instead of increasing the temperature and, therefore, the heat capacity would be larger in those regions. But, because the <span class="math-container">$\gamma$</span> is the ratio between the heat capacities, the argument does not work.</p>

<p>Any suggestion?</p>

CC BY-SA 4.0
2018-09-25T20:25:13.860Z
Not sure I understand what you're asking. Yes, $\gamma$ is in general not a constant when ionization processes are involved.
AtmosphericPrisonEscape
CC BY-SA 4.0
2018-09-26T16:57:03.323Z
@AtmosphericPrisonEscape - My question is why the ionization processes affect the value of $\gamma$, in particular, why do they suppress it? I clarified it in the question too.
Stefano
CC BY-SA 4.0
2018-09-28T13:39:22.650Z
I believe it is because of EM interaction. You might want to check that.
Kornpob Bhirombhakdi
<star><the-sun><astrophysics><stellar-astrophysics><stellar-atmospheres>
Why does the adiabatic exponent decrease at the ionization zones?
Stefano
# Why does the adiabatic exponent decrease at the ionization zones?

(Asked by: username_0 on 2018-09-25)

The context is the ionization zones in stellar atmospheres or interiors, the Sun, for instance.

The adiabatic exponent is the heat capacity ratio:

$$\gamma = \frac{c\_P}{c\_V} = \frac{C\_P}{C\_V}$$

And, for an ideal gas, it can be shown that $\gamma=(f+2)/f$ where $f$ is the degrees of freedom. Resulting in $\gamma\approx1.66$.

In most of the extent of the solar interior, $\gamma\sim1.66$ but it presents dips when ionization is present, like the H, HeI and HeII ionization zones. **My question is why does this happen?**

I thought it could be because when ionization, the heat is used to ionize atoms instead of increasing the temperature and, therefore, the heat capacity would be larger in those regions. But, because the $\gamma$ is the ratio between the heat capacities, the argument does not work.

Any suggestion?

**Tags:** star, the-sun, astrophysics, stellar-astrophysics, stellar-atmospheres

> **Comments:**
> **username_1**: Not sure I understand what you're asking. Yes, $\gamma$ is in general not a constant when ionization processes are involved.  
> **username_0**: @username_1 - My question is why the ionization processes affect the value of $\gamma$, in particular, why do they suppress it? I clarified it in the question too.  
> **username_2**: I believe it is because of EM interaction. You might want to check that.  

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## 1 answers

### Answer by username_3 on 2018-09-28. Score: 3 | Accepted answer

There are two things going on. (1) When you add heat to a gas that is on the threshold of ionisation or partially ionised, some of that heat goes into ionisation. This means that it takes a much larger amount of energy to produce a rise in temperature. (2) However, as the gas becomes ionised the number of particles per unit mass also increases and so the pressure increases at a given temperature. The net effect is that $C\_V$ (heat capacity for a fixed volume) increases by more than $C\_P$ (heat capacity for a fixed pressure) and so the adiabatic index decreases.

It is a tricky and detailed computation to work out how $C\_V$ and $C\_P$ change as ionisation proceeds. An example calculation for pure hydrogen gas is give on pp 123-126 of "Principles of stellar evolution and nucleosynthesis" (1983, D. Clayton). Both $C\_V$ and $C\_P$ increase drastically (by factor of 30-40) as ionisation increases, both peaking at around 50% ionisation before falling back to exactly *twice* their unionised values once the gas is completely ionised (when it just behaves like a monatomic perfect gas but with twice the number of particles per unit mass). However, the behaviour of $C\_V$ and $C\_P$ are slightly different, $C\_P$ increases by less than $C\_V$, and so the ratio of $C\_P/C\_V$ changes and goes through a minimum at around 50% ionisation before regaining the standard monatomic gas value once completely ionised.


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