Where does the (5/2) come from in calculating thermal energy of diatomic gas

[animboy]

So I am doing a second year thermodynamics course and would like to know. Do we just have to remember (5/2)PV for a diatomic gas, why is it 5/2 and also what is it for a monatomic gas. Also would we have to remember more complex ratios for exams?

Cheers

 

[K^2]

This is value known as specific heat capacity at constant volume, or C_V. You obtain said quantity by counting degrees of freedom and dividing it by 2. For diatomic gas, there are 7 total, 3 translational, 2 vibrational, and 2 rotational, but the 2 vibrational degrees of freedom are typically "frozen out". So you only count the 5. That gives you the 5/2. For monatomic gas, there are only the translational, so you get 3/2.

Realistically, the value will be off by a bit. There is an associated quantity γ=C_P/C_V known as heat capacity ratio. You can look it up for gas of interest, and use the fact that C_P=C_V+1 to compute the actual ratio.

Note that all these values are in units of R. So the actual specific heat capacity of diatomic gas at constant volume will be 5/2R per mole of gas.

 

[animboy]

This is value known as specific heat capacity at constant volume, or C_V. You obtain said quantity by counting degrees of freedom and dividing it by 2. For diatomic gas, there are 7 total, 3 translational, 2 vibrational, and 2 rotational, but the 2 vibrational degrees of freedom are typically "frozen out". So you only count the 5. That gives you the 5/2. For monatomic gas, there are only the translational, so you get 3/2.

Realistically, the value will be off by a bit. There is an associated quantity γ=C_P/C_V known as heat capacity ratio. You can look it up for gas of interest, and use the fact that C_P=C_V+1 to compute the actual ratio.

Note that all these values are in units of R. So the actual specific heat capacity of diatomic gas at constant volume will be 5/2R per mole of gas.

Thanks, that makes sense now. Except, my book doesn't go through the derivation, it doesn't give the unit R just the ratio. What does it physically represent?

 

[K^2]

Same as any heat capacity. How much energy you need to change the temperature. With gases, however, you can either hold the cylinder closed, and then the volume remain constant, but pressure changes with temperature, or you can have a piston in the cylinder, which keeps pressure constant, but let's volume vary. Because moving piston takes work, you need more heat to increase temperature when you keep pressure constant. Hence, C_P is higher than C_V.

So suppose you want to change the temperature by ΔT in a closed cylinder, id est, constant volume. The amount of heat will be ΔQ = nC_VΔT, or for diatomic gas, ΔQ = 5/2 nRΔT

Keep in mind that the symbol C_V may be used for specific heat capacity, as I have been doing, or for total heat capacity.

 

[sardar]

,

The value of (5/2) in the calculation of thermal energy for a diatomic gas comes from the equipartition theorem, which states that at thermal equilibrium, each degree of freedom in a molecule will have an average energy of (1/2)kT, where k is the Boltzmann constant and T is the temperature.

For a diatomic gas, there are five degrees of freedom - three translational and two rotational. This means that the average energy per molecule will be (5/2)kT, which is why we use (5/2) in the calculation of thermal energy for a diatomic gas.

For a monatomic gas, there are only three degrees of freedom - three translational. Therefore, the average energy per molecule is (3/2)kT.

In terms of remembering these values for exams, it is important to understand the underlying concepts and how they relate to the calculation of thermal energy. However, it is also important to familiarize yourself with common values and ratios for different types of gases, as they may be needed for problem-solving in exams. It is always best to check with your professor or study materials to determine which values you will need to remember for your specific course.