To calculate the partition function for a system of 10 molecules of a monoatomic gas confined to a 1-dimensional box of length 1 nm at a temperature of 300 K, we will first use the classical method and then the quantum mechanical method.1. Classical Method:In the classical method, we use the classical partition function, which is given by:q = 2mkT/h^2 ^1/2 * Lwhere q is the classical partition function, m is the mass of the molecule, k is the Boltzmann constant, T is the temperature, h is the Planck constant, and L is the length of the box.For a system of N molecules, the total partition function is given by:Q = q^N / N!Assuming the gas is an ideal gas, we can use the mass of an argon atom as an example m = 6.63 x 10^-26 kg . The Boltzmann constant k is 1.38 x 10^-23 J/K, and the Planck constant h is 6.63 x 10^-34 Js.q = 2 6.63 x 10^-26 kg 1.38 x 10^-23 J/K 300 K / 6.63 x 10^-34 Js ^2 ^1/2 * 1 x 10^-9 m q 1.21 x 10^10Now, we calculate the total partition function for 10 molecules:Q_classical = 1.21 x 10^10 ^10 / 10!Q_classical 2.76 x 10^982. Quantum Mechanical Method:In the quantum mechanical method, we use the quantum partition function, which is given by:q = exp -E_i/kT where E_i is the energy of the ith energy level.For a 1-dimensional box, the energy levels are given by:E_i = i^2 * h^2 / 8 * m * L^2 The quantum partition function is the sum of the Boltzmann factors for each energy level. Since there are infinite energy levels, we will sum up to a large number e.g., 1000 to approximate the partition function.q_quantum = exp - i^2 * 6.63 x 10^-34 Js ^2 / 8 * 6.63 x 10^-26 kg * 1 x 10^-9 m ^2 * 1.38 x 10^-23 J/K * 300 K q_quantum 1.00 x 10^10 approximated by summing up to i=1000 Now, we calculate the total partition function for 10 molecules:Q_quantum = 1.00 x 10^10 ^10 / 10!Q_quantum 1.02 x 10^98Comparison:The partition functions obtained from the classical and quantum mechanical methods are:Q_classical 2.76 x 10^98Q_quantum 1.02 x 10^98The discrepancy between the classical and quantum mechanical partition functions arises due to the different treatment of energy levels in both methods. The classical method assumes a continuous distribution of energy levels, while the quantum mechanical method considers discrete energy levels. For a system of particles in a 1-dimensional box, the quantum mechanical method provides a more accurate representation of the energy levels, as the energy levels are indeed quantized. However, as the temperature increases or the size of the box increases, the difference between the classical and quantum mechanical partition functions becomes smaller, and the classical method becomes a reasonable approximation.