Chemistry - Physical chemistry Assignment 1

Chem 240 One Hour Exam 1

1. For this question you are going to solve this identity that is very frequently

encountered in PChem II:

∫ x2e−ax 2 ∙ ∂x

c

0

= √π

4 ∙ a3 2⁄ ⋅ erf(√a ∙ c) −

c

2a e−a∙c

2

where “erf” is the “error function”. To do so, please use the following steps:

a) Start with substituting y2 = ax2, and don’t forget to change the upper and lower limits!

Here is the formula for variable substitution for x → y, where g(y) = x:

∫ f(x) ∙ ∂x

c

0

→ ∫ f(g(y)) ∙ | ∂g(y)

∂y | ∂y

g(y)=c

g(y)=0

b) Starting with 1

√a ∫ y2 ∙ e−y

2 ∙ ∂y

√a∙c

0 use integration by parts:

∫f(y) ∙ ∂g(y)

c

0

= f(y) ∙ g(y)|0 c −∫g(y) ∙ ∂f(y)

c

0

and don’t forget to apply limits on the 1st term on the right. The integral in the 2nd term

on the right will be solved using this identity:

∫ e−x 2 ∙ ∂x

c

0

= 1

2 √π ∙ erf⁡(c)

where “erf” is just a function like sine or tangent etc.

2. a. Try to fit the data provided here

(sig3.txt). The function should be:

f(x) = m

𝑥 + b

Please provide the optimized variables and a

figure of the data and fit.

b. Report the error in m and b using the

variance-covariance method.

3. You have yet to turn in your HW1 makup. You might want to do so, since this

question is predicated on it. In that problem set, in question 1, you were asked to fit data

to an exponential decay:

Amplitude = A ∙ e−t τ⁄

and to calculate the errors thereof to report: A ± σA and τ ± στ.

For this question, use the results from #1 to calculate the following function:

f = π ∙ A + τ

A

and calculate the associated error as to report f ± 𝜎𝑓.

Here are the standard error equations:

For f(x) = a·x 𝜎𝑓= |a|·𝜎𝑥 where a has no error.

For f(x,y) = x ± y, 𝜎𝑓 = √(𝜎𝑥)2 + (𝜎𝑦) 2

For f(x,y) = x·y, 𝜎𝑓 = |𝑓|√( 𝜎𝑥

|𝑥| ) 2

+ ( 𝜎𝑦

|𝑦| ) 2

For f(x,y) = x

y , 𝜎𝑓 = |𝑓|√(

𝜎𝑥

|𝑥| ) 2

+ ( 𝜎𝑦

|𝑦| ) 2

For f(x) = xn 𝜎𝑓 = |𝑓||𝑛| 𝛥𝑥

|𝑥|