Question

Calculate the internal energy of He, Ne, and Ar under standard thermodynamic conditions. Do you need to redo the entire calculation for each species?

Short answer

The internal energy of helium (He), neon (Ne), and argon (Ar) under standard thermodynamic conditions is approximately 6.128 x 10^{-21} J. The calculation did not need to be redone for each species because they are all ideal monoatomic gases with the same degrees of freedom.

Step-by-step solution

Determine the degrees of freedom for each gas

Since helium (He), neon (Ne) and argon (Ar) are all monoatomic gases, they will have the same degrees of freedom. Monoatomic gases have 3 translational degrees of freedom, as each particle can move in the 3-dimensional coordinate system (x, y, and z axis).

Calculate the internal energy using the equipartition theorem

The equipartition theorem states that for a system at a given temperature, the average energy contributed by each degree of freedom can be calculated as \(\dfrac{1}{2}kT\), where \(k\) is the Boltzmann constant (1.38 x 10^{-23} J/K) and \(T\) is the temperature in Kelvin. Under standard thermodynamic conditions, the temperature is 298 K. Since all the three gases have the same degrees of freedom, their internal energy can be calculated as: \(U = f \cdot \dfrac{1}{2} k T\) Where \(U\) is the internal energy, \(f\) is the number of degrees of freedom (3 for these monoatomic gases), \(k\) is the Boltzmann constant, and \(T\) is the temperature in Kelvin.

Compute the internal energy for each gas

Now, plug in the values for \(f\), \(k\), and \(T\) to calculate the internal energy: \(U_{He} = U_{Ne} = U_{Ar} = 3 \cdot \dfrac{1}{2} (1.38 \times 10^{-23} \, \text{J/K}) (298 \, \text{K})\) \(U_{He} = U_{Ne} = U_{Ar} = 6.128 \times 10^{-21} \, \text{J}\) The internal energy of helium, neon, and argon under standard thermodynamic conditions is approximately 6.128 x 10^{-21} J. The calculation for each species did not need to be redone because the degrees of freedom are the same for these ideal monoatomic gases.

Key concepts

Equipartition Theorem

The equipartition theorem is a fundamental principle in statistical mechanics that helps us understand how energy is distributed among the various degrees of freedom of a system at thermal equilibrium. According to this theorem, each degree of freedom contributes an equal amount of energy to the total internal energy of the system. In mathematical terms, each independent degree of freedom has an average energy of \(\frac{1}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) is the absolute temperature measured in Kelvin.

In the context of our gas molecules, such as helium, neon, and argon, the equipartition theorem tells us that for each translational degree of freedom, a molecule will have an average energy of \(\frac{1}{2}kT\). Since these are monoatomic gases and their atoms only possess translational motion, we know they each have three degrees of freedom - one for each spatial dimension. Therefore, when calculating the internal energy of these gases, the theorem simplifies the process, avoiding the need for complex calculations and making it straightforward to understand the energy contained within a sample of gas at a given temperature.

Degrees of Freedom

In physics, the term 'degrees of freedom' refers to the number of independent ways in which a system can move without violating any constraints. For a single particle, it generally correlates with the directions of movement allowed in three-dimensional space. A monoatomic gas particle, like He, Ne, or Ar, has three degrees of freedom that correspond to motion along the x, y, and z axes, known as translational movement. These degrees of freedom are essential for computing the internal energy of gases because they relate to the ways energy can be stored within the system.

For monoatomic gases under standard conditions, since only translational energy is considered, the three degrees of freedom result from the three possible directions a gas atom can move about freely. It's noteworthy that when more complex molecules are considered, with additional rotational or vibrational modes, the concept of degrees of freedom becomes more intricate, leading to a higher count of degrees, and consequently, influencing the calculation of internal energy.

Boltzmann Constant

The Boltzmann constant (\(k\)) is a fundamental physical constant that plays a vital role in the field of thermodynamics and statistical mechanics. It connects the macroscopic world to the microscopic world - relating the temperature, an averaged macroscopic quantity, to the kinetic energy of particles, a microscopic quantity. The constant is named after the Austrian physicist Ludwig Boltzmann and has a value of approximately \(1.38\times 10^{-23}\) joules per Kelvin. This tiny number quantifies the amount of energy that is associated with each degree of freedom per unit temperature.

The Boltzmann constant is featured prominently in the equipartition theorem and is used to calculate the average kinetic energy of particles within a gas. This constant is a bridge between the energy of molecules bouncing around at the atomic level and the temperature we feel and measure in everyday life, thus serving as a key to unlocking the behavior of gases at the atomic scale.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In essence, it describes how thermal energy is converted to and from other types of energy and how it affects matter. The four laws of thermodynamics lay the groundwork for energy exchanges within a physical system, including systems that involve ideal gases.

Specifically, the first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. This principle applies directly to our discussion of internal energy in gases. When we calculate the internal energy of a gas using the equipartition theorem, we're effectively working with the first law by quantifying the energy present in the molecules of the gas. It provides a foundation for understanding how energy is spread out within the system and why certain properties, like temperature and internal energy, are directly related, shaping the way we predict how gases will behave under different thermodynamic conditions.