Calculate the translational partition function for Ar confined to a volume of 1000 cm^3 at 298 K. At what temperature will the translational partition function of Ne be identical to that of Ar at 298 K confined to the same volume?
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To calculate the translational partition function for a gas molecule, we use the formula:
Z_trans = V / λ^3
Where:
Z_trans is the translational partition function,
V is the volume of the container,
λ is the thermal de Broglie wavelength.
For argon (Ar) confined to a volume of 1000 cm^3 at 298 K, we can calculate the translational partition function as follows:
1. Convert the volume to cubic meters:
V = 1000 cm^3 = 1000 * (10^-6) m^3 = 0.001 m^3
2. Calculate the thermal de Broglie wavelength using the formula:
λ = h / √(2 * π * m * k * T)
Where:
h is the Planck's constant (6.62607015 × 10^-34 J·s),
m is the mass of the molecule (for Ar it is 6.6335209 × 10^-26 kg),
k is the Boltzmann constant (1.380649 × 10^-23 J/K),
T is the temperature in Kelvin (298 K).
Plugging in the values:
λ = (6.62607015 × 10^-34 J·s) / √(2 * π * (6.6335209 × 10^-26 kg) * (1.380649 × 10^-23 J/K) * (298 K))
Calculate λ and then square it to get λ^2.
3. Calculate the translational partition function:
Z_trans = V / λ^3
Now, to find the temperature at which the translational partition function of neon (Ne) will be identical to that of argon (Ar) at 298 K confined to the same volume, we need to equate their partition functions:
Z_trans_Ar = Z_trans_Ne
Therefore, we can rearrange the equation for Ne:
λ_Ne = (V / Z_trans_Ar)^(1/3)
Now, we can calculate the thermal de Broglie wavelength for Ne using the volume and the value of Z_trans_Ar. Once we have the value of λ_Ne, we can use the equation to calculate the temperature (T_Ne) at which the translational partition function of Ne is identical to that of Ar.
Note: In this calculation, we are assuming an ideal gas behavior and neglecting any intermolecular interactions.
Hope this helps you in calculating the translational partition function and finding the temperature at which Ne's partition function is identical to Ar's.