Calculate Expectation Value & SD of N1, N2 in 0.1m^3 Vacuum Chambers

In summary: N-1}\left(1-\frac{10^6}{0.1}\right)}and\sigma_{N2} = \sqrt{N\left(\frac{10^6}{0.1}\right)\left(1-\frac{10^6}{0.1}\right)^{N-1}\left(1-\frac{10^6}{0.1}\right)}In summary, to calculate the expectation value and standard deviation for N1 and N2, we use the binomial distribution formula and the formula for standard deviation. The expectation value for both N1 and
  • #1
agentnerdo
13
0
The conc.of molecules in two ultra-high-vacuum chambers of volumes V1=V2=0.1m^3 is 10^6 molecules/m3

You count the # of molecules N1 and N2, in each chamber at some moment of time.

a.find the expectation value and the SD of N1 and N2 . calc [tex]\sigma[/tex][tex]_{N1}[/tex], sketch the prob distribution of N1.


I used SD= sqt(pq) x sqt(N)

=sqt(0.5x0.5) x sqt (2,000,000)
=707.1

I assume for Expectation value i use

W= [tex]\underline{N!}[/tex](v/V) (1-v/V)[tex]^{N-n}[/tex]
n!(N-n)!

I am just sure if N will be 2million, or even what my n will be...

any help be appriciated!
 
Physics news on Phys.org
  • #2



Hello, thank you for your question. Let me help you with the calculation of the expectation value and standard deviation for N1 and N2.

First, we need to define the variables:

N1 = number of molecules in chamber 1
N2 = number of molecules in chamber 2
V1 = volume of chamber 1 = V2 = volume of chamber 2 = 0.1 m^3
n = number of molecules in a given state
N = total number of molecules in the chamber (N = N1 + N2)

The expectation value for N1 can be calculated as:

E[N1] = \sum_{n=0}^{N} nP(n)

where P(n) is the probability of having n molecules in chamber 1, which can be calculated using the binomial distribution:

P(n) = \frac{N!}{n!(N-n)!} \left(\frac{v}{V}\right)^n \left(1-\frac{v}{V}\right)^{N-n}

Substituting the values, we get:

E[N1] = \sum_{n=0}^{N} n \frac{N!}{n!(N-n)!} \left(\frac{10^6}{0.1}\right)^n \left(1-\frac{10^6}{0.1}\right)^{N-n}

We can simplify this equation to:

E[N1] = N \left(\frac{10^6}{0.1}\right) \left(1-\frac{10^6}{0.1}\right)^{N-1}

Similarly, the expectation value for N2 can be calculated as:

E[N2] = N \left(\frac{10^6}{0.1}\right) \left(1-\frac{10^6}{0.1}\right)^{N-1}

The standard deviation for N1 and N2 can be calculated using the formula:

\sigma_{N1} = \sqrt{\sum_{n=0}^{N} (n-E[N1])^2 P(n)}

and

\sigma_{N2} = \sqrt{\sum_{n=0}^{N} (n-E[N2])^2 P(n)}

We can also simplify these equations to:

\sigma_{N1} = \sqrt{N\left(\frac{10^
 

1. What is the formula for calculating the expectation value and standard deviation of N1 and N2 in 0.1m^3 vacuum chambers?

The formula for calculating the expectation value of N1 and N2 in 0.1m^3 vacuum chambers is: E[N] = N1 + N2. The formula for calculating the standard deviation is: SD[N] = sqrt(N1^2 + N2^2).

2. How do you obtain the values for N1 and N2 in 0.1m^3 vacuum chambers?

The values for N1 and N2 can be obtained through experiments or simulations. In experiments, N1 and N2 can be measured using detectors or sensors placed inside the vacuum chamber. In simulations, N1 and N2 can be calculated based on the properties of the particles present in the vacuum chamber and their interactions.

3. Why is it important to calculate the expectation value and standard deviation of N1 and N2 in 0.1m^3 vacuum chambers?

Calculating the expectation value and standard deviation of N1 and N2 in 0.1m^3 vacuum chambers is important as it helps in understanding the behavior and characteristics of the particles present in the chamber. It also allows for the prediction of the number of particles that will interact with each other, which is crucial in many scientific experiments and processes.

4. What are some factors that can affect the values of N1 and N2 in 0.1m^3 vacuum chambers?

The values of N1 and N2 in 0.1m^3 vacuum chambers can be affected by various factors such as the type and properties of particles present in the chamber, temperature, pressure, and the presence of any external fields or forces. Additionally, any changes in the experimental or simulation setup can also impact the values of N1 and N2.

5. Can the expectation value and standard deviation of N1 and N2 in 0.1m^3 vacuum chambers be used to predict the behavior of particles in larger chambers?

Yes, the expectation value and standard deviation of N1 and N2 in 0.1m^3 vacuum chambers can be used as a basis for predicting the behavior of particles in larger chambers. However, it is important to note that other factors, such as the size and geometry of the chamber, may also play a significant role in particle behavior and should be considered when making predictions.

Back
Top