Final answer:
The translational partition function of a helium atom at 298 K and in a 1.00 m³ volume is calculated using the mass of the helium atom, the Boltzmann constant, the temperature, and the Planck constant. The mass of helium is derived by dividing the molar mass by Avogadro's number. Substituting all these values into the appropriate equation yields the partition function.
Step-by-step explanation:
To calculate the translational partition function (qtrans) of a helium atom at 298 K in a container of volume 1.00 m³, we can use the following formula:
qtrans = (((2πmkT) / (h²)))⁴ˡ V
Where:
- m is the mass of the helium atom.
- k is the Boltzmann constant (1.38 × 10-23 J/K).
- T is the temperature (298 K).
- h is the Planck constant (6.626 × 10-34 J·s).
- V is the volume of the container (1 m³).
First, we calculate the mass of one helium atom as m = M / NA, with M being the molar mass of helium (4.0026 g/mol) and NA being Avogadro's number (6.022 × 1023 mol-1).
m = (4.0026 × 10-3 kg/mol) / (6.022 × 1023 mol-1) = 6.646 × 10-27 kg
Then we substitute the known values into the equation to get:
qtrans = ((2π((6.646 × 10-27 kg)(1.38 × 10-23 J/K)(298 K)) / (6.626 × 10-34 J·s)²)⁴ˡ ( 1 m³ )
Calculating the expression above will give the value of the translational partition function for a helium atom at the given conditions.