(I) What is the internal energy of 5.40 mol of an ideal diatomic gas at 2450 K, assuming all possible degrees of freedom are active?

Verified step by step guidance

1. Identify the type of gas and the degrees of freedom. For a diatomic gas with all degrees of freedom active (translational, rotational, and vibrational), there are 7 degrees of freedom.
2. Use the formula for the internal energy of an ideal gas, which is given by U = (f/2) * n * R * T, where f is the number of degrees of freedom, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
3. Substitute the values into the formula: f = 7 (for a diatomic gas with all degrees of freedom active), n = 5.40 mol, R = 8.314 J/(mol*K) (universal gas constant), and T = 2450 K.
4. Calculate the product of these values to find the internal energy U.
5. The result will give you the internal energy in Joules.

Key Concepts

Internal Energy

Internal energy is the total energy contained within a system, arising from the kinetic and potential energies of its particles. For an ideal gas, it primarily depends on the temperature and the number of moles of the gas. The internal energy can be calculated using the formula U = n * C_v * T, where n is the number of moles, C_v is the molar heat capacity at constant volume, and T is the temperature in Kelvin.

Degrees of Freedom

Degrees of freedom refer to the number of independent ways in which a system can possess energy. For a diatomic gas, there are translational, rotational, and vibrational degrees of freedom. At high temperatures, all these degrees of freedom are typically active, contributing to the internal energy of the gas. For diatomic gases, the molar heat capacity at constant volume (C_v) is often taken as (5/2)R when all degrees of freedom are active.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. While the question focuses on internal energy, understanding the Ideal Gas Law is essential for contextualizing the behavior of gases under various conditions.