Degrees of Freedom: Gases vs Solids/Liquids

[User Name]

The homework assignment that I've been given in Physics deals with degrees of freedom (which my assignment refers to as "modes").

We're given the equation E$$_{thermal}$$= (# of modes) $$\times$$ $$\frac{1}{2}$$ $$\times$$ $$k$$ $$\times$$ T.

(k being the Boltzmann's Constant).

We are also told that gases have 3 modes of energy while solids/liquids have 6 modes of energy.

Then we're given a a list of two things to compare and to determine which has the higher thermal energy; most of them I have been fine with, but there has been one that is confusing me to no end.

I'm comparing 1 mole of a monatomic gas at 290 K versus 1 mole of a monatomic solid at 290 K.

If I use the equation given, then the solid would have a higher thermal energy (since both have 1 mole of the substance, which is $$N_{A}\ =\ 6.02214199(47)\ \times\ 10^{23} mol^{-1}$$ and the number of modes are dependent on how many particles there are.. so a gas would have 3 $$\times$$ Avagadro's number while a solid would have 6 $$\times$$ Avagadro's number, which results in solids having more thermal energy).

However, this doesn't really make sense to me. Shouldn't a gas have a higher thermal energy than a solid?

The subsequent question also raised this issue with me, because I had to explain how to use that particular equation I had to determine whether gases or liquids (specifically H$$\_{2}\$$O liquid and O$$\_{2}$$ gas) move faster at thermal equilibrium. If I similarly use the equation like I did in the previous problem, then I would again have that the liquid would have a higher thermal energy than the gas since it has 6 modes of energy versus 3 modes for gases.

However, I believe that thermal energy is directly related to kinetic energy at the microscopic level, so a higher thermal energy would then mean higher kinetic energy, which would then mean the liquid is moving faster than the gas.

And yet again, this doesn't make sense to me, because gases most definitely move much quicker than liquids.

Any help would be much appreciated!

 

[anathema]

Hello,

Thank you for bringing up this confusion regarding the comparison of thermal energy between gases and solids. It is important to understand that the equation given, E_{thermal}= (# of modes) \times \frac{1}{2} \times k \times T, is a simplified version and does not take into account the complexities of the molecules in a substance.

Firstly, let's clarify the concept of modes of energy. In physics, a mode refers to a specific way in which energy can be distributed among the particles of a substance. In the case of gases, there are three modes of energy: translational, rotational, and vibrational. This means that the energy of a gas molecule can be distributed among these three modes. On the other hand, in solids and liquids, there are six modes of energy: translational, rotational, vibrational, and three additional modes known as phonons, which are related to the vibrations of the atomic lattice.

Now, let's consider the comparison between 1 mole of a monatomic gas at 290 K and 1 mole of a monatomic solid at 290 K. As you correctly stated, the solid would have a higher thermal energy according to the given equation. However, this does not necessarily mean that the solid is moving faster. In fact, the molecules in a solid are not moving freely like in a gas. They are bound to their lattice structure and can only vibrate. This means that, even though the solid may have more modes of energy, the molecules are not able to move as freely as in a gas, resulting in a lower kinetic energy.

Similarly, when comparing a liquid and a gas, the liquid may have more modes of energy, but the gas molecules are moving much faster due to their higher kinetic energy. This is because in a gas, the molecules are not bound to a lattice structure and can move freely in any direction.

In conclusion, the number of modes of energy is not the only factor determining thermal energy and kinetic energy. The complexities of the molecules in a substance also play a significant role. I hope this helps to clarify your confusion. If you have any further questions, please feel free to ask. Good luck with your assignment!

 

[kerimek]

First of all, it is important to understand the concept of degrees of freedom. In physics, degrees of freedom refer to the number of independent ways in which a system can store energy. In the case of gases, there are 3 modes of energy: translational, rotational, and vibrational. This means that the molecules in a gas can move in 3 different ways and store energy in those modes. On the other hand, in solids and liquids, there are 6 modes of energy: 3 translational and 3 vibrational. This is because the molecules in a solid or liquid are confined to a fixed position and can only vibrate around that position.

Now, let's look at the equation E_{thermal}= (# of modes) \times \frac{1}{2} \times k \times T. This equation tells us that the thermal energy of a system is directly proportional to the number of modes and the temperature. So, if we compare 1 mole of a monatomic gas and 1 mole of a monatomic solid at the same temperature, the solid will have a higher thermal energy because it has more modes (6 compared to 3 for the gas).

However, as you correctly pointed out, this does not necessarily mean that the solid will have a higher kinetic energy. This is because thermal energy and kinetic energy are not the same thing. Thermal energy is the total energy of a system due to the random motion of its particles, while kinetic energy is the energy that results from the motion of particles in a particular direction. In other words, thermal energy is the average of all the different kinetic energies of the particles in a system.

In the case of gases and liquids, even though a gas may have a lower thermal energy, its particles are moving much faster than the particles in a liquid. This is because gases have a higher average kinetic energy due to their high degree of freedom. So, while the liquid may have a higher thermal energy, the gas is still moving faster.

In conclusion, the number of modes of energy in a system does not determine its kinetic energy. It is important to distinguish between thermal energy and kinetic energy, and to understand that they are not the same thing. I hope this helps clarify any confusion you may have had.