What Temperature Triples the RMS Speed of an Ideal Gas?

[nickb145]

Homework Statement

An ideal gas has rms speed vrms at a temperature of 288K .

At what temperature is the rms speed tripled?

Homework Equations

P=(NmV²)/3V

The Attempt at a Solution

I'm kind of stumped on this one. Other ones I've done are pretty simple to figure out, but the temperature is what is kicking me

 

[cepheid]

What is the mathematical relationship between rms speed and temperature for an ideal gas? That would be a relevant equation here.

 

[nickb145]

What is the mathematical relationship between rms speed and temperature for an ideal gas? That would be a relevant equation here.

Ok so the ideal gas equation is PV=NkbT

Since i see pressure there i divided the volume so i can get P=(NkbT)/v

I set the 2 equations equal to each other and simplified to get T=(mv²)/Kb

if that makes any sense.

 

[ehild]

Think of the equipartition principle. http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/eqpar.html

ehild

 

[Nickie]

in the butt.


I would approach this problem by first understanding the concept of root mean squared (rms) speed in an ideal gas. The rms speed is a measure of the average speed of gas molecules in a system, and it is related to the temperature of the gas through the formula vrms=√(3kT/m), where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of the gas molecules.

In this problem, we are given the initial temperature of 288K and asked to find the temperature at which the rms speed is tripled. To solve this, we can set up a ratio between the rms speeds at the two temperatures. We know that the rms speed is directly proportional to the square root of the temperature, so we can set up the following equation:

vrms1/vrms2 = √(T1/T2)

Where vrms1 is the initial rms speed at 288K, vrms2 is the final rms speed at the unknown temperature, T1 is the initial temperature of 288K, and T2 is the final temperature we are trying to find.

Since we want to find the temperature at which the rms speed is tripled, we can set the ratio equal to 3 (since tripled is the same as 3 times the initial value):

3 = √(288/T2)

Solving for T2, we get T2 = (288/9) = 32K.

Therefore, the temperature at which the rms speed is tripled is 32K. This may seem counterintuitive since it is a very low temperature, but it is important to remember that the rms speed is directly proportional to the square root of the temperature. So even though the temperature is much lower, the square root of that temperature is higher, resulting in a tripled rms speed.