Question

The kinetic theory of an ideal gas takes into account not only translational motion of atoms or molecules but also, for diatomic and polyatomic gases, vibration and rotation. Will the temperature increase from a given amount of energy being supplied to a monatomic gas differ from the temperature increase due to the same amount of energy being supplied to a diatomic gas? Explain.

Short answer

Answer: When the same amount of energy is supplied to both a monatomic gas and a diatomic gas, the temperature increase of the diatomic gas is 5/3 times less than the temperature increase of the monatomic gas. This is due to the diatomic gas having more degrees of freedom (rotational and translational) compared to the monatomic gas (only translational), resulting in a smaller temperature increase for the diatomic gas.

Step-by-step solution

Understand the degrees of freedom for monatomic and diatomic gases

Monatomic gases have 3 degrees of freedom, which are the three translational motions (motion in the x, y, and z directions). Diatomic gases have 5 degrees of freedom: 3 translational and 2 rotational degrees of freedom.

Recall the relation between internal energy, degrees of freedom, and temperature

For an ideal gas, the internal energy (U) can be calculated as follows: U = (f/2) nRT Where: - f is the degrees of freedom - n is the number of moles - R is the gas constant - T is the absolute temperature in Kelvin.

Calculate the change in temperature for the given energy supplied

Let's assume that the energy supplied to both gases is E and the number of moles is the same. We want to calculate the change in temperature (ΔT) after supplying energy E. For monatomic gas: ΔU_monatomic = E = (3/2)nRΔT_monatomic For diatomic gas: ΔU_diatomic = E = (5/2)nRΔT_diatomic

Compare the change in temperature for both gases

To compare the change in temperature for both the monatomic and diatomic gas, we can divide both equations: (ΔT_monatomic)/(ΔT_diatomic) = (5/3) From this, we can conclude that when the same amount of energy is supplied to both a monatomic gas and a diatomic gas, the temperature increase of the diatomic gas is 5/3 times less than the temperature increase of the monatomic gas. This is because the energy supplied to the diatomic gas is distributed among more degrees of freedom (rotational and translational) compared to the monatomic gas (only translational). Therefore, the temperature increase of the diatomic gas is less than the monatomic gas for the same amount of energy supplied.

Key concepts

Degrees of Freedom

When we discuss the kinetic theory of gases, the term 'degrees of freedom' is crucial. It refers to the ways in which a molecule can store energy through motion. Each degree of freedom equates to one independent way of moving in space, such as forward, backward, up, down, or rotational movements.

For monatomic gases, like helium or neon, there are three degrees of freedom, all of which are translational, meaning the atoms can move in three-dimensional space: left or right (x-axis), up or down (y-axis), and forward or backward (z-axis). However, diatomic or polyatomic gases, like oxygen or carbon dioxide, have additional degrees of freedom due to their ability to rotate and, for polyatomic gases, to vibrate.

Understanding degrees of freedom is vital because it directly influences how a gas absorbs and distributes energy. The more degrees of freedom a gas has, the more ways it can absorb energy without increasing its temperature as dramatically.

Internal Energy of an Ideal Gas

In the context of ideal gases, internal energy is the total energy contained by the gas molecules due to their movement. Specifically, this energy arises from the kinetic activity of the particles, which is directly related to the temperature of the gas.

The formula \( U = (f/2) nRT \) shows the relationship between internal energy (U), degrees of freedom (f), the amount of substance in moles (n), the ideal gas constant (R), and temperature (T). We can see that for a given amount of gas at a fixed temperature, the internal energy changes with the degrees of freedom. This is key in understanding why different gases behave in distinct ways when it comes to heat absorption and temperature changes. The degrees of freedom act as a kind of 'storage' for the energy, distributing it among the various available motions of the molecules.

Temperature Change in Gases

Temperature change in gases is a manifestation of how internal energy is shared among the available degrees of freedom. When heat energy is supplied to a gas, the temperature increase depends on how that energy is spread across the degrees of freedom.

As the kinetic theory suggests, a monatomic gas with only translational energy will have all the supplied energy go into enhancing the speed of the atoms, leading to a relatively larger increase in temperature. In contrast, a diatomic or polyatomic gas will distribute the same amount of energy among translational, rotational, and possibly vibrational degrees of freedom, resulting in a smaller temperature increase for the same amount of energy input.

This distribution is crucial when we apply heat to gases, as it affects not just temperature but also other thermodynamic processes such as work and entropy. The maxim 'one size does not fit all' applies aptly here—each gas has its unique behavior in response to energy thanks to its-specific degrees of freedom.