By how much 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% at constant temperature?
To calculate the increase in the total partition function (excluding electronic) when Neon undergoes expansion at constant temperature, we need to first understand the concept of partition function and how it relates to the given conditions.
The partition function, usually denoted by Z, is a concept from statistical mechanics that describes the distribution of energy states in a system. It is a sum over all possible energy states, with each state weighted by the Boltzmann factor exp(-Ei/kT), where Ei is the energy of the state, k is the Boltzmann constant, and T is the temperature.
In this case, we are interested in the change in the total partition function due to expansion at constant temperature. The partition function can be written as the product of the translational, rotational, and vibrational partition functions, i.e.,
Z_total = Z_trans * Z_rot * Z_vib
Since we are excluding the electronic partition function, we can focus on the translational, rotational, and vibrational partition functions.
For the translational partition function, it is given by
Z_trans = (V / λ^3) * (2πmkT / h^2)^(3/2)
where V is the volume of the system, λ is the de Broglie wavelength, m is the mass of the particles, k is the Boltzmann constant, T is the temperature, and h is the Planck constant.
The rotational partition function is given by
Z_rot = [(8π^2IkT) / h^2]^(1/2)
where I is the moment of inertia of the molecule.
The vibrational partition function depends on the specific molecule and its vibrational modes, so we would need more information to calculate it.
Now, let's calculate the increase in Z_total due to the expansion of Neon at constant temperature.
Given:
Initial volume (V_initial) = 20 m^3
Final volume (V_final) = V_initial + ΔV
ΔV = 0.0010% of V_initial = (0.0010/100) * V_initial
Pressure (P) = 1.00 atm
Temperature (T) = 300 K
To calculate the new volume, we can use the ideal gas law:
P_initial * V_initial = P_final * V_final
Substituting the values and solving for V_final:
1.00 atm * 20 m^3 = 1.00 atm * (20 m^3 + ΔV)
20 = 20 + ΔV
ΔV = 0
Since the change in volume (ΔV) is negligible (0), the increase in the total partition function for this expansion is effectively zero.
Therefore, the total partition function (excluding electronic) does not increase when 20 m^3 of Neon at 1.00 atm and 300 K expands by 0.0010% at constant temperature.