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

• animboy
In summary: If you want to say specifically about diatomic gas, you might say "specific heat capacity at constant volume for diatomic gas is 5/2R per mole of 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

This is value known as specific heat capacity at constant volume, or CV. 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=CP/CV known as heat capacity ratio. You can look it up for gas of interest, and use the fact that CP=CV+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.

Last edited:
K^2 said:
This is value known as specific heat capacity at constant volume, or CV. 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=CP/CV known as heat capacity ratio. You can look it up for gas of interest, and use the fact that CP=CV+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?

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, CP is higher than CV.

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 = nCVΔT, or for diatomic gas, ΔQ = 5/2 nRΔT

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

,

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.

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

The (5/2) in the calculation of thermal energy of diatomic gas comes from the equipartition theorem, which states that each degree of freedom in a molecule contributes an average energy of (1/2)kT, where k is the Boltzmann constant and T is the temperature in Kelvin. For a diatomic gas, there are 5 degrees of freedom, resulting in a total average energy of (5/2)kT.

## 2. Why is the equipartition theorem used in the calculation of thermal energy?

The equipartition theorem is used because it provides a way to determine the average energy of a molecule based on its temperature and the number of degrees of freedom it has. This is important in understanding the behavior of gases and other systems at the molecular level.

## 3. Are all diatomic gases assumed to have (5/2) degrees of freedom?

No, not all diatomic gases have (5/2) degrees of freedom. The (5/2) value is an average for diatomic gases at room temperature. Depending on the specific molecule, some diatomic gases may have more or less than 5 degrees of freedom, which can affect the value of the thermal energy calculation.

## 4. How does the value of (5/2) affect the thermal energy calculation?

The value of (5/2) is a constant in the calculation of thermal energy for diatomic gases. It represents the average energy contributed by each degree of freedom in the molecule. Changing the value of (5/2) would result in a different average energy for the gas at a given temperature.

## 5. Can the (5/2) value be used for other types of gases?

No, the (5/2) value is specific to diatomic gases. Other types of gases have different numbers of degrees of freedom, and therefore, the value used in the thermal energy calculation would be different. For example, monatomic gases have (3/2) degrees of freedom, while polyatomic gases have varying numbers of degrees of freedom depending on their molecular structure.

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