# Neon Atom Speed Ratio at Different Temperatures

• brake4country
In summary, the equation relating average speed to temperature is v=sqrt. 3RT/m. Kinetic energy is proportional to temperature, but the relationship between speed and kinetic energy is not clear.

## Homework Statement

What is the ratio of the average speed of an atom of neon to another atom of neon at twice the temperature?
(A) 1:1
(B) 1:1.4
(C) 1:2
(D) 1:4

KE = 3/2RT

## The Attempt at a Solution

I used the above equation and substituted 2 for T for twice the temperature and 1 for half of that:

KE=3/2RT = 3R
KE=3/2RT = 3/2R

My ratio is 3:1.5. The correct answer is B. So kinetic energy is proportional to temperature, but what is the relationship between speed and kinetic energy?

brake4country said:
My ratio is 3:1.5. The correct answer is B. So kinetic energy is proportional to temperature, but what is the relationship between speed and kinetic energy?
Can't you figure it out? At least the proportionality between kinetic energy and speed? Write your ratio correctly (1 to something, 1:x), and then try to figure out the relation between x and 1.4.

brake4country said:
what is the relationship between speed and kinetic energy?

Is that really the first time you hear about kinetic energy? Of any object?

Your relevant equation is only one of two required to solve the problem. There is another one, one so basic you will feel ashamed once you realize what it is. Or if you don't know it, you should feel ashamed for not knowing it ;)

Ok. No, this is not the first time i heard about kinetic energy. For example, the average speed of an atom is defined in terms of KE, which is KE = 3/2RT. Also, another equation defines KE, which is KE = 1/2 mv^2. Putting these together I get v = sqrt. 3RT/m. Also what is known is that temperature is directly proportional to KE.

I can eliminate A because it cannot be 1:1 if the temperature is doubled on one of them.

When I use v = sqrt. 3RT/m, I get 0.024 (gas double temp) and 0.012 for the gas with standard temp. This still gives me a proportionality of 2:1.

brake4country said:
Ok. No, this is not the first time i heard about kinetic energy. For example, the average speed of an atom is defined in terms of KE, which is KE = 3/2RT. Also, another equation defines KE, which is KE = 1/2 mv^2. Putting these together I get v = sqrt. 3RT/m. Also what is known is that temperature is directly proportional to KE.

I can eliminate A because it cannot be 1:1 if the temperature is doubled on one of them.

When I use v = sqrt. 3RT/m, I get 0.024 (gas double temp) and 0.012 for the gas with standard temp. This still gives me a proportionality of 2:1.

You are soooo very close!

Do what Dr. Claude suggests (set this up as a ratio of speeds, using the equation for velocity that you have)

v(T_2)/v(T_1) = SQRT (3RT_2/m)/SQRT (3RT_1/m) ==> all of the constants on the right side (3, R, T, m) cancel, and you get ...

brake4country said:
When I use v = sqrt. 3RT/m, I get 0.024 (gas double temp) and 0.012 for the gas with standard temp. This still gives me a proportionality of 2:1.

Got it. So in order to relate T and avg. kinetic energy, we must set 1/2mv^2 = 3/2RT. We get sqrt. 3RT/m.

Then, using Graham's law: v1/v2 = sqrt. ms/sqrt. m1, we get (sqrt. 10)/(sqrt. 5) = sqrt. 2 = 1.41.

So, Graham's law only relates rms velocities to molar mass. In order to integrate temperature, we must set the above equations equal to each other and solve for v. Conceptually, we would not see a double in v since the relationship between v and KE is a sqrt. Am I finally understanding this now? Thanks in advance!

You don't need Graham's law here.

Let's start from the very beginning: please write formula relating average speed with the temperature (either use LaTeX or treat it with parentheses, to avoid any ambiguity).

(1/2mv2)=(3/2RT)
v = (sqrt. 3RT/m)

brake4country said:
v = (sqrt. 3RT/m)

(sqrt. 3RT/m) or sqrt(3RT/m)?

sqrt (3RT/m); sorry.

OK, so we have

$$V_1 = \sqrt{\frac{3RT_1}{m}}$$

Assume $T_2 = 2T_1$ (twice higher), plug it into identical equation for V2, calculate ratio of $\frac{V_1}{V_2}$, what do you get?

V1=sqrt (6R/20) and V2=sqrt. (3R/20). Since R is a constant, can I just use that value (0.08)?

If R is a constant, it cancels out.

brake4country
Ok. Now that temperature is accounted for already, v1/v2 =0.38/0.70. I see how the math is done but for some reason I don't see the proportions.

Why do you go for numbers instead of canceling everything out? For example

$$\frac{\sqrt{40}}{\sqrt{60}}=\frac{\sqrt{2\times 2\times 2\times 5}}{\sqrt{3\times 2\times 2 \times 5}} = \frac {\sqrt {2}} {\sqrt{3}}$$

brake4country
Yes, I see. But I get v1/v=sqrt. 6/sqrt.3 = sqrt. 2 = 1.4. I understand your example above; that was very helpful!

brake4country said:
sqrt. 6/sqrt.3 = sqrt. 2 = 1.4

You are looking for a ratio form, so it is best to leave it as a fraction:

$$\frac{\sqrt{3}}{\sqrt{6}}=\frac{\sqrt{1}}{\sqrt{2}}=\frac{1}{\sqrt{2}}$$

and then it is obvious why 1:1.4 is the correct answer. Pure algebra.

With some experience the answer is obvious. Energy is proportional to the temperature and to the velocity squared, 1.41... is a square root of 2 so it becomes a natural suspect ;)

brake4country

## 1. What is the significance of the speed ratio of neon atoms at different temperatures?

The speed ratio of neon atoms at different temperatures is important because it can provide insight into the kinetic energy and behavior of these atoms. It can also help us understand the relationship between temperature and atomic movement.

## 2. How does the speed ratio of neon atoms change as temperature increases?

As temperature increases, the speed ratio of neon atoms also increases. This is because higher temperatures provide more energy to the atoms, causing them to move faster and have a higher speed ratio.

## 3. Can the speed ratio of neon atoms be calculated or measured accurately?

Yes, the speed ratio of neon atoms can be calculated or measured accurately using various methods such as spectroscopy or particle velocity measurements. These methods can provide precise values for the speed ratio at different temperatures.

## 4. How does the speed ratio of neon atoms compare to other elements at the same temperature?

The speed ratio of neon atoms is unique to neon and cannot be directly compared to other elements. However, it follows a similar trend as other noble gases, with increasing speed ratio as temperature increases.

## 5. What are some practical applications of studying the speed ratio of neon atoms at different temperatures?

Studying the speed ratio of neon atoms at different temperatures can have various applications, such as understanding the properties of gases, improving gas sensing technologies, and developing new materials for high-temperature applications.