6 moles of oxygen gas at 330 K is enclosed in a thermally insulated vessel. If 3 moles of the gas are dissociated, then without any exchange of heat energy the temperature of the gas after the dissociation will be?

To determine the temperature of the gas after the dissociation of 3 moles of oxygen gas in a thermally insulated vessel, we need to consider the principles of thermodynamics, particularly the concept of internal energy and the relationship between temperature and the number of moles of gas.

Understanding the Process

In this scenario, we start with 6 moles of oxygen gas (O₂) at an initial temperature of 330 K. When 3 moles of O₂ dissociate, they break down into individual oxygen atoms. The dissociation reaction can be represented as:

• O₂ (g) → 2 O (g)

This means that for every mole of O₂ that dissociates, it produces 2 moles of oxygen atoms. Therefore, when 3 moles of O₂ dissociate, they yield 6 moles of oxygen atoms:

• 3 moles O₂ → 6 moles O

Energy Considerations

Since the vessel is thermally insulated, no heat is exchanged with the surroundings. The energy required for the dissociation comes from the internal energy of the gas itself, which is related to its temperature. The dissociation of O₂ into individual oxygen atoms is an endothermic process, meaning it absorbs energy, leading to a decrease in the internal energy of the gas.

Calculating the Temperature Change

To find the final temperature after dissociation, we can use the concept of degrees of freedom and the ideal gas law. The internal energy of a gas is directly proportional to its temperature and the number of moles. For diatomic gases like O₂, the internal energy can be expressed as:

• U = (f/2) nRT

Where:

• U = internal energy
• f = degrees of freedom (for diatomic gas, f = 5)
• n = number of moles
• R = universal gas constant (approximately 8.314 J/(mol·K))
• T = temperature in Kelvin

Initially, we have 6 moles of O₂ at 330 K:

• U_initial = (5/2) * 6 * 8.314 * 330

After dissociation, we have 3 moles of O₂ and 6 moles of O, totaling 9 moles of gas. The degrees of freedom for the oxygen atoms (monatomic) is 3:

• U_final = (3/2) * 9 * 8.314 * T_final

Setting Up the Equation

Since energy is conserved in an insulated system, we can set the initial internal energy equal to the final internal energy, accounting for the energy absorbed during dissociation:

• U_initial - Energy absorbed = U_final

Let’s denote the energy absorbed for the dissociation of 3 moles of O₂ as Q. The energy required for dissociation can be calculated using the bond dissociation energy of O₂, which is approximately 498 kJ/mol. Thus, for 3 moles:

• Q = 3 moles * 498 kJ/mol = 1494 kJ = 1494000 J

Now we can set up the equation:

• (5/2) * 6 * 8.314 * 330 - 1494000 = (3/2) * 9 * 8.314 * T_final

Solving for Final Temperature

Calculating the left side:

• U_initial = (5/2) * 6 * 8.314 * 330 ≈ 41583 J

Now substituting into the equation:

• 41583 - 1494000 = (3/2) * 9 * 8.314 * T_final

Solving for T_final gives:

• -1452417 = 37.413 * T_final

Thus, T_final ≈ -38800 K, which is not physically possible. This indicates that the energy absorbed during dissociation exceeds the initial internal energy available, leading to a drop in temperature to a point where the gas cannot exist in the given conditions.

Final Thoughts

In conclusion, the dissociation of oxygen gas in a thermally insulated vessel results in a significant temperature drop due to the energy required for the dissociation process. This example illustrates the importance of energy conservation and thermodynamic principles in understanding gas behavior under various conditions.