Calculating % of molecules at given temperature with a given velocity

In summary, the conversation discusses finding the percentage of molecules in water at 300K with a speed greater than 1 km/s, and the approach of integrating the distribution from 1000 to infinity. The speaker mentions using the equation for kinetic energy and taking into account the fact that they are only interested in the integral between 1000 and infinity. They also mention evaluating the answer numerically and finding an integral of 0 on their calculator.
  • #1
r34p3rex
2
0

Homework Statement



What percentage of molecules in water at 300K will have a speed greater than 1 km/s (evaluate numerically)?


Homework Equations


2}e^{-\frac{mv^{2}}{2KT}}dv.gif



The Attempt at a Solution


Based on what we did in class, I take it I'm supposed to integrate the distribution from 1000 to infinity.

After moving the constants out of the integral, I'm left with this:
gif.gif


I know that

gif.gif


But how do I account for the fact that I'm only interested in the integral between 1000 and infinity?
 
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  • #2
The instructions say to evaluate the answer numerically.
 
  • #3
SammyS said:
The instructions say to evaluate the answer numerically.

I punched it into my calculator and after a few minutes of calculating, it tells me the integral is 0.

Am I doing something wrong?
 
  • #4
r34p3rex said:
I punched it into my calculator and after a few minutes of calculating, it tells me the integral is 0.

Am I doing something wrong?
Probably something is being done wrong, or something is set wrong.
 
  • #5



As a scientist, it is important to first understand the context and background of the problem before attempting to solve it. In this case, we are dealing with a question about the percentage of molecules in water at a given temperature that have a speed greater than a certain value. To solve this problem, we can use the Maxwell-Boltzmann distribution, which describes the distribution of molecular speeds in a gas at a given temperature. The equation for this distribution is:

f(v) = 4π (m/2πkT)^(3/2) * v^2 * e^(-mv^2/2kT)

Where m is the mass of the molecule, k is the Boltzmann constant, T is the temperature, and v is the speed of the molecule.

To find the percentage of molecules with a speed greater than 1 km/s, we need to integrate this distribution from 1 km/s to infinity. This can be done numerically using a computer program or by using a table of integrals. Once the integral is solved, we can divide it by the total number of molecules in the sample to find the percentage.

It is important to note that the Maxwell-Boltzmann distribution assumes an ideal gas, so it may not be a perfect representation for water molecules. Additionally, the temperature of 300K may not accurately reflect the temperature of water. Therefore, the results obtained may not be exact, but they can provide a reasonable estimate.
 

1. How do you calculate the percentage of molecules at a given temperature with a given velocity?

The percentage of molecules at a given temperature with a given velocity can be calculated using the Maxwell-Boltzmann distribution. This distribution shows the relationship between the velocity of molecules and their probability of occurrence at a certain temperature.

2. What is the formula for calculating the percentage of molecules at a given temperature with a given velocity?

The formula for calculating the percentage of molecules at a given temperature with a given velocity is P(v) = 4π(v^2)(m/2πkT)^3/2 * e^(-mv^2/2kT), where P(v) is the probability of finding a molecule with a given velocity, m is the mass of the molecule, v is the velocity, k is the Boltzmann constant, and T is the temperature in Kelvin.

3. How does temperature affect the percentage of molecules at a given velocity?

As temperature increases, the percentage of molecules at a given velocity also increases. This is because at higher temperatures, molecules have more kinetic energy and therefore, a greater range of velocities. The Maxwell-Boltzmann distribution curve also shifts to the right with increasing temperature, indicating a higher percentage of molecules at higher velocities.

4. What happens to the percentage of molecules at a given velocity when the velocity increases?

When the velocity increases, the percentage of molecules at that given velocity decreases. This is due to the fact that as velocity increases, the probability of finding a molecule with that specific velocity decreases. The Maxwell-Boltzmann distribution curve also decreases as velocity increases.

5. Can the percentage of molecules at a given velocity ever be 100%?

No, the percentage of molecules at a given velocity can never be 100%. This is because the distribution of velocities follows a continuous curve and there will always be a small percentage of molecules with a different velocity. However, as the velocity approaches infinity, the percentage of molecules at that velocity approaches 0.

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