What Is the vrms for This Gas Distribution?

• Qwurty2.0
In summary, the distribution of speeds for a gas consisting of 15,200 molecules, each with a mass of 2.00 x 10-26 kg, shows a weighted average of 654.32 m/s. To determine the rms speed, the velocities were squared, the weighted average of the squares was found, and then the square root was taken. The correct answer is 710 m/s.
Qwurty2.0

Homework Statement

A gas consisting of 15,200 molecules, each of mass 2.00 x 10-26 kg, has the following distribution of speeds, which crudely mimics the Maxwell distribution:

Number of Molecules - Speed (m/s)
1600 - 220
4100 - 440
4700 - 660
3100 - 880
1300 - 1100
400 - 1320

(a) Determine vrms for this distribution of speeds.

Homework Equations

vrms = √(2 * Ek/m)
Ek = (1/2) * n * M * v2

The Attempt at a Solution

Weighted Average
((1600 * 220) + (4100 * 440) + (4700 * 660) + (3100 * 880) + (1300 * 1100) + (400 * 1320)) / 15200
= (9944000 m/s) / 15200
= 654.32 m/s

vrms = √(2 * Ek/m)
Ek = (1/2) * n * M * v2
n = number of moles
M = molar mass
m = mass of single molecule

n = 15200 molecules / 6.02x1023 molecules/mole
= 2.525x10-20 mole

M = 2.00x10-26 kg / molecule * 6.02x1023 molecules / mole
= 0.01204 kg / mol

Ek = (1/2)(2.525x10-20moles)(0.01204 kg / mole)(654.21 m/s)2
= 6.506x10-17kg⋅m/s

vrms = √(2 * (6.506x10-17 kg⋅m/s) / 2.00x10-26 kg)
= 80659.78 m/s ...

The correct answer is 710 m/s so obviously I am either grossly overcomplicating this, or I am using the wrong equation.

RMS means "root mean square". It's the square root of the mean of the square, so take the velocities and square them, find the weighted average of the squares, and then take the square root of the average.

I did what you said: I squared the velocities, then I calculated the weighted average (same way as I did above), and then took the square root of the weighted average. I ended up getting 82914.03 m/s, which is close to the answer I originally got but not the correct answer.

What am I missing?

I got 707 m/s. I gather you're still messing around with calculating the kinetic energy, etc. That's unnecessary. The rms speed depends only on the velocity distribution of the molecules.

Your mistake seems to be that you're calculating the total internal energy of the gas, but ##E_k## is supposed to be the average kinetic energy of a single molecule.

My bad, I calculated it wrong. I tried again and got the right answer.

Thank you!

What is the Maxwell Distribution of speeds?

The Maxwell Distribution of speeds is a probability distribution that describes the distribution of speeds of particles in a gas at a given temperature. It was developed by physicist James Clerk Maxwell in the 19th century.

What factors affect the Maxwell Distribution of speeds?

The Maxwell Distribution of speeds is affected by temperature, mass, and the type of gas. As temperature increases, the distribution curve shifts towards higher speeds, while an increase in mass or a heavier gas will result in a distribution curve that is shifted towards lower speeds.

What is the most probable speed in the Maxwell Distribution?

The most probable speed in the Maxwell Distribution is the peak of the distribution curve. This speed is the speed at which the largest number of particles in the gas are moving.

How does the Maxwell Distribution relate to the Kinetic Theory of Gases?

The Maxwell Distribution is a key component of the Kinetic Theory of Gases. This theory states that the pressure, temperature, and volume of a gas are related to the average kinetic energy of its particles. The Maxwell Distribution illustrates the different speeds at which these particles are moving, and how they contribute to the overall kinetic energy of the gas.

Why is the Maxwell Distribution important in understanding gas behavior?

The Maxwell Distribution helps us understand how particles in a gas behave at a microscopic level. It allows us to make predictions about the average speed of particles, which is important in understanding properties such as temperature and pressure. This distribution also plays a crucial role in various fields of science, including chemistry, physics, and engineering.

• Introductory Physics Homework Help
Replies
4
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
4K
Replies
7
Views
8K
• Introductory Physics Homework Help
Replies
4
Views
5K
• Introductory Physics Homework Help
Replies
2
Views
3K
• Introductory Physics Homework Help
Replies
2
Views
6K
• Introductory Physics Homework Help
Replies
2
Views
2K
• MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
3K