Thermodynamics: PV diagram: High temperature adiabat

[msavg]

1. The problem statement, all variables and given/known

Hi, this is my first post on here.

Does an adiabat get steeper at higher temperatures? I think it does, but I wanted to make sure I'm thinking about it in the right way. Also, what is the constant in the PV relations?

Homework Equations

PV=const, for constant temperature.

PV^γ=const, for zero heat transfer.

γ=(f+2)/f, where f=degrees of freedom.

The Attempt at a Solution

Using the equations for total thermal energy, the first law, and the ideal gas law, we can derive relationships between pressure and volume that show us what the PV diagram will look like. Since an isotherm comes from
PV=const
and an adiabat comes from
PV^γ=const, where γ=(f+2)/f,
we can readily see that the adiabat will be steeper than the isotherm.

However, as temperatue increases, so do the degrees of freedom for a multi-atom molecule. As degrees of freedom increase, so too does the exponent on the V for the adiabatic case. This would lead to a steeper adiabat for non-monatomic molecules at higher temperatures. Furthermore, because the maximum total degrees of freedom possible=3N, where N=number of atoms comprising the molecule, our adiabat would get steeper and steeper for larger and larger molecules at higher temperatures.

Is that correct?

As for the constant, I've looked all over my textbook, but it doesn't describe it at all. Is it some kind of characteristic constant dependent on the kind of gas we're talking about? Are the constants in the isothermal case and adiabatic case the same?

 

[Andrew Mason]

Using the equations for total thermal energy, the first law, and the ideal gas law, we can derive relationships between pressure and volume that show us what the PV diagram will look like. Since an isotherm comes from
PV=const
and an adiabat comes from
PV^γ=const, where γ=(f+2)/f,
we can readily see that the adiabat will be steeper than the isotherm.

What do you mean by "steeper"? Is this a P-V diagram (P on the vertical axis, V on the horizontal)? What part of the curve are you talking about?

However, as temperature increases, so do the degrees of freedom for a multi-atom molecule.

This is not necessarily true. It depends on the temperature. The number of degrees of freedom are limited by geometry. So when all modes are activated, there is no increase in the number of degrees of freedom with further temperature increases.

Furthermore, because the maximum total degrees of freedom possible=3N, where N=number of atoms comprising the molecule, our adiabat would get steeper and steeper for larger and larger molecules at higher temperatures.

Is that correct?

The adiabatic condition assumes an ideal gas obeying PV=nRT. As you increase the number of atoms in the molecule, the atoms tend to lose their ideal gas behaviour.

AM