Find the Factor Increase of the Total Partition Function

[BelWonder]

Homework Statement

By what factor does the total partition function (excluding electronic) increase when 20 m^3 of Neon at 1.00 atm and 300 K is allowed to expand by 0.0010%?

Homework Equations

translational partition function qt= (V×[(2∏mkT)]^3/2])/ (h^3), vibrational partitition function qv = 1/(1-[e^-(ε/kT)]), rotational partition function qr = √∏/δ ×[(kT)^(3/2)]×[(1/Ba)^1/2]× [(1/Bb)^1/2] × [(1/Bc)^1/2)] Note: a, b, and c are subscripts and k = 0.695 cm^-1/K.

The Attempt at a Solution

Since volume is only in the translational partition function, I assumed that the factor would just be the difference in volume, 0.0010% of 20 m^3. Is this correct?

 

[nate808]

Thank you for your question. The factor by which the total partition function increases in this scenario is not simply the difference in volume (0.0010% of 20 m^3). To calculate the increase in the partition function, we need to take into account the change in the system's energy levels due to the expansion of the gas.

The total partition function is given by the product of the individual partition functions for translation, vibration, and rotation. In this case, we can assume that the electronic partition function remains constant since the problem specifies that we are excluding it.

To calculate the increase in the translational partition function, we can use the formula you provided: qt= (V×[(2∏mkT)]^3/2])/ (h^3). Here, V is the volume of the gas, m is the mass of a single neon atom, k is the Boltzmann constant, and T is the temperature. We can plug in the given values (V = 20 m^3, m = mass of neon atom, k = 1.381 x 10^-23 J/K, T = 300 K) to find the initial value of qt.

Next, we need to calculate the final value of qt after the expansion. Since the volume has increased by 0.0010% (0.000010), the final volume will be 20.002 m^3. We can plug this value into the same formula to find the final value of qt.

The factor by which the translational partition function increases is then given by the ratio of the final and initial values: qt(final)/qt(initial). This factor will also apply to the total partition function since the other two partition functions (vibration and rotation) do not depend on volume.

I hope this helps. If you have any further questions, please let me know.