# Average Kinetic Energy of molecules calculation

• tahmidbro
In summary: The question talks about "air", which consists mostly of dinitrogen (and the rest mostly of dioxygen, which is quite similar in mass to dinitrogen).
tahmidbro
Homework Statement
a) what is the rms speed of four hydrogen molecules with the speeds of 890, 755, 902, 866m/s?

Ans : 855m/s

My problem is part b)

b) at what temperature would these hydrogen molecules be?

( Mass of H atom = 1.67 x 10^(-27) kg )

Now, in 1/2m<c^2> = (3/2)kt, should I use the mass for only 1 hydrogen atom, or mass of 2 hydrogen atoms, or the mass for all 8 atoms in 4 molecules?
Relevant Equations
1/2m<c^2> = 3/2kT
the answer in the solution book is 29K which only comes if I use mass for only one atom. ( They did not show any working )

My attempt:

1/2 x (1.67 x 10^(-27)) x (355)^(2) = 3/2 x 1.38 x 10^(-23) x T
T = 29.48820652 K

The confusion arises when I tried the following question:

Q. Estimate the rms speed of the molecules of air in this room.

The answer given in the solution book :

Estimate of 20 °C (T = 293 K); assume all molecules are dinitrogen, so m = 4.676 × 10^(–26) kg; gives rms = 509m/s.
which is only possible if I use mass of two nitrogen atoms.

The question says hydrogen molecules, so the mass is that of 2 H atoms. Looks like the book got it wrong.

DrClaude
tahmidbro said:
The answer given in the solution book :

Estimate of 20 °C (T = 293 K); assume all molecules are dinitrogen, so m = 4.676 × 10^(–26) kg; gives rms = 509m/s.
which is only possible if I use mass of two nitrogen atoms.

Hi. The solution-book is correct. When you use$$\text{average kinetic energy per particle = }\frac {3}{2}kT$$you have to consider what a 'particle' is.

'Particle' doesn't necessarily mean an atom. A 'particle' in this context is an object which freely moves around and doesn't exert a force on other particles (except during collisions). Not a rigorous definition though!

The question says "assume all molecules are dinitrogen". You are being told that each 'particle' is a nitrogen molecule (N₂).

You can't treat individual atoms as moving around independently because they are moving around in pairs.

DrClaude
Steve4Physics said:
Hi. The solution-book is correct. When you use$$\text{average kinetic energy per particle = }\frac {3}{2}kT$$you have to consider what a 'particle' is.

'Particle' doesn't necessarily mean an atom. A 'particle' in this context is an object which freely moves around and doesn't exert a force on other particles (except during collisions). Not a rigorous definition though!

The question says "assume all molecules are dinitrogen". You are being told that each 'particle' is a nitrogen molecule (N₂).

You can't treat individual atoms as moving around independently because they are moving around in pairs.

Actually, ''dinitrogen'' is not part of the second question. It was part of the answer from solution book.

Steve4Physics said:
Hi. The solution-book is correct. When you use$$\text{average kinetic energy per particle = }\frac {3}{2}kT$$you have to consider what a 'particle' is.

'Particle' doesn't necessarily mean an atom. A 'particle' in this context is an object which freely moves around and doesn't exert a force on other particles (except during collisions). Not a rigorous definition though!
The question concerns hydrogen molecules, therefore H2, so the mass of a particle is twice that of a hydrogen atom. The temperature thus obtained differs from that of the textbook.

I have some doubts. It happens I know the speed of just four molecules and, from those four values, I can say temperature is T? I understand the first part is about computing the RMS value but the second part leads the reader to believe it's OK to use a sample that small.

DrClaude said:
The question concerns hydrogen molecules, therefore H2, so the mass of a particle is twice that of a hydrogen atom. The temperature thus obtained differs from that of the textbook.

DrClaude said:
The question concerns hydrogen molecules, therefore H2, so the mass of a particle is twice that of a hydrogen atom. The temperature thus obtained differs from that of the textbook.
Apologies for the confusion. I was looking at the part of the post relating to air. not hydrogen.

tahmidbro said:
Actually, ''dinitrogen'' is not part of the second question. It was part of the answer from solution book.
The question talks about "air", which consists mostly of dinitrogen (and the rest mostly of dioxygen, which is quite similar in mass to dinitrogen).Even physicists are assumed to know that much chemistry. Assuming it's all N2 will give a good estimate.

## 1. What is the formula for calculating the average kinetic energy of molecules?

The formula for calculating the average kinetic energy of molecules is KE = (3/2) * k * T, where KE is the average kinetic energy, k is the Boltzmann constant, and T is the temperature in Kelvin.

## 2. How is the average kinetic energy of molecules related to temperature?

The average kinetic energy of molecules is directly proportional to the temperature. As the temperature increases, the average kinetic energy of molecules also increases.

## 3. What is the significance of calculating the average kinetic energy of molecules?

Calculating the average kinetic energy of molecules is important in understanding the behavior and properties of gases. It can also be used to determine the rate of diffusion and effusion of gases.

## 4. Can the average kinetic energy of molecules be negative?

No, the average kinetic energy of molecules cannot be negative. It is a measure of the average energy of molecules in motion, and therefore, it is always a positive value.

## 5. How does the mass of molecules affect the average kinetic energy?

The mass of molecules does not affect the average kinetic energy. According to the formula, the average kinetic energy is only dependent on temperature and the Boltzmann constant, not the mass of the molecules.

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