Question

At room temperature, identical gas cylinders contain 10 moles of nitrogen gas and argon gas, respectively. Determine the ratio of energies stored in the two systems. Assume ideal gas behavior.

Short answer

Answer: The ratio of energies stored in the nitrogen gas to the argon gas is 5/3.

Step-by-step solution

Understand the Equipartition Theorem

Equipartition Theorem states that the average energy per particle in a system is given by (1/2)kT for each degree of freedom, where k is the Boltzmann constant and T represents the temperature. We will use this theorem to calculate the energy of each gas.

Determine the degrees of freedom for nitrogen and argon gases

Nitrogen gas (N2) is a diatomic molecule, which has 5 degrees of freedom at room temperature (3 translational and 2 rotational). Argon gas (Ar) is a monatomic gas with 3 degrees of freedom (3 translational).

Write the formulas for energies of both gases

We will use the Equipartition Theorem to write the formulas for the energy of both nitrogen and argon gases. For nitrogen gas (N2), the energy is E_N2 = (5/2) * n * k * T, where n represents the number of moles (10 moles). For argon gas (Ar), the energy is E_Ar = (3/2) * n * k * T, where n represents the number of moles (10 moles).

Find the ratio of energies of the two gases

To find the ratio of energies of nitrogen and argon gases, we will divide their respective energy formulas: Ratio = E_N2 / E_Ar = ((5/2) * n * k * T) / ((3/2) * n * k * T) Notice that n, k, and T are the same for both gases, and they can be cancelled out. The final ratio will be: Ratio = (5/2) / (3/2) = 5/3

Interpret the result

The ratio of energies stored in the nitrogen gas to the argon gas is 5/3, which means that the nitrogen gas has more energy stored in its identical gas cylinder compared to the argon gas at room temperature, given the same number of moles.