The T dS equations and Sackur-Tetrode equation

[mathman44]

The "T dS" equations and Sackur-Tetrode equation

Homework Statement

This is a bit of a dumb question (I hope)

I'm having trouble with part (e).

The Attempt at a Solution

to show it for (a), dS is given, so $$T\times{NK_bln{\frac{V_f}{V_i}} = P\times{dV}$$ since dU = 0 for isothermal process. P can be written as $$\frac{NK_bT}{V}$$, but this is clearly not equal to what I got in part A... although it would be if I integrated the right side (?)

 

[LeonhardEuler]

The TdS equation has TdS on the left hand side, you have an expression for $\Delta S$ of a process with a macroscopic change. Do you know that
$$d(\ln{x})=\frac{d(\ln{x})}{dx}dx =\frac{dx}{x}$$
?

 

[mathman44]

Was I wrong to assume that $$\Delta{S} = dS$$?

 

[LeonhardEuler]

Yes, if you use Vf and Vi on the left hand side, you need to use them on the right hand side and take the limit
$$\lim_{V_{f}\rightarrow V_{i}}$$
on both sides and show that the ratio of the left hand side to the right hand side approaches 1, but it's much easier to just use the relation I mentioned.

 

[paranormal]

I can understand your confusion with this problem. The "T dS" equations and Sackur-Tetrode equation are both fundamental equations in thermodynamics. The "T dS" equations, also known as the Clausius equality, state that for a reversible process, the change in entropy (dS) is equal to the heat transfer (T) divided by the temperature (T) at which the process occurs. This equation is a fundamental tool in understanding the relationship between heat and entropy.

The Sackur-Tetrode equation, on the other hand, is a thermodynamic equation that relates the entropy of a system to its temperature, volume, and number of particles. It is commonly used to calculate the entropy of an ideal gas.

In regards to your attempt at solving part (e) of your homework problem, it seems that you have correctly identified the relationship between entropy and volume for an isothermal process (dU = 0). However, the expression for pressure (P) that you have used is incorrect. The correct expression for pressure in this case would be P = \frac{Nk_BT}{V}, where N is the number of particles in the system, k_B is the Boltzmann constant, and T is the temperature. By substituting this expression for pressure into your equation, you should be able to obtain the correct result.

I hope this helps clarify the concepts behind the "T dS" equations and the Sackur-Tetrode equation. Keep up the good work in your studies of thermodynamics!