probability, ideal gas, and temperaturebiophysics/pc2267/lecture-03.pdf · tutorial outline...

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Probability reviewDimensions of physical quantities

Ideal gas and temperature

Probability, Ideal gas, and Temperature

Jie Yan

August 22, 2006

Jie Yan Probability, Ideal gas, and Temperature

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Time slots

I Tutorial starts from week 3 (28AUG-1SEP).

I We will have a class tutorial every week, on Friday 9 am - 10am.

I We will have a small group tutorial every even week (4,6,8,10),on Mon (12.00-12.50pm) or on Wed (5.00-5.50pm). Thesmall group tutorial classroom is S13-04-02. Your will bepartitioned into two groups. Group A takes week 4 and 8, andgroup B takes week 6 and 10. In the week 12, we will havetwo small tutorials: ”A” on Mon, and ”B” on Wed.

I Which time slot for the small group tutorial is the mostconvenient for you?


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Contents

I Review of probability theory introduced in the previous class.

I Dimensions of physical quantities we should know.

I The ideal gas and the temperature


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Sample space

Sample space: list of all possible outcomes of an event


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Probability properties

2.pdf

it takes over 40 minutes

it takes between 35 and 40 minutes

it takes between 30 and 35 minutes

it takes under 30 minutes

OUTCOME •Any probability assigned must be a nonnegative number. •The probability of the sample space (the collection of all possible outcomes) is equal to 1. •If you have two outcomes that can't happen at the same time, then the probability that either outcome occurs is the sum of the probabilities of the individual outcomes.

0 .1 .3 .4 it takes over 40 minutes.8 .1 .4 .4 it takes between 35 and 40 minutes.2 .4 .3 -.1 it takes between 30 and 35 minutes0 .4 .2 .3 it takes under 30 minutes Mary Sue Joe Max OUTCOME

Survey of likelihood: how long does it take you from your home to the school?

http://www-math.bgsu.edu/~albert/m115/probability/prob_rules.html


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Equal probability

3.pdf

00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

0,1,2,3,4,5,6,7,8,9 0,1,2,3,4,5,6,7,8,9


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Probability calculation: example

1/10 M1, M2 1/10 M2, W3 1/10 M2, W2 1/10 M2, W1 1/10 M1, W3 1/10 M1, W2 1/10 M1, W1 1/10 W2, W3 1/10 W1, W3 1/10 W1, W2

PROBABILITY COMMITTEE

1/10 0M1, M2 1/10 1M2, W3 1/10 1M2, W2 1/10 1 M2, W1 1/10 1M1, W3 1/10 1M1, W2 1/10 1M1, W1 1/10 2W2, W3 1/10 2W1, W3 1/10 2 W1, W2

PROBABILITY # OF WOMEN COMMITTEE

3/10 2 6/10 1 1/10 0

PROBABILITY # OF WOMEN

Candidates: woman 1, woman 2, woman 3, man 1, man 2Questions: P(no. of women in a committee) = ?Sample space={0,1,2}


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Probability calculation: example

12 T H T H H H T H H T H T H H H H T T TH

NUMBER OF HEADS COIN TOSSES


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Addition rule: example

6.pdf Jie Yan Probability, Ideal gas, and Temperature

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Mean/average/expectation): example of gambling

7.pdf

26/38 -10 12/38 20

PROBABILITY AMOUNT YOU WIN

-20/38 = -.53 SUM

(-10)(26/38) = -260/38 26/38 -10 (20)(12/38) = 240/38 12/38 20

PRODUCT PROBABILITY AMOUNT YOU WIN


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Gaussian distribution: Graphs


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Variance of Gaussian distribution


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1-d diffusion process


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Dimensions

I Length: dimension is [L], SI unit is ”meter”, abbrev ”m”.

I Mass: dimension is [M], SI unit is ”kilograms”, abbrev ”kg”.

I Time: dimension is [T], SI unit is ”second”, abbrev ”s”.

I Electric change: dimension is [Q], SI unit is ”Coulombs”,abbrev ”coul”.

I Temperature: its definition will be discussed later. Its SI unitis Kelvin, apprev ”K”. 0oC = 273K . Room temperature isaround 300K .


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Derivation of the dimensions of physical quantities

I Velocity and acceleration:[V]=[LT−1],[A]=[LT−2].

I Force and Energy: [F]=[MLT−2],[E]=[ML2T−2].

I Electric current: [I]=[QT−1].

I Coulomb law: f = 14πε0

q1q2

r2 , [ε0]=?


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The idea gas law

PV = NkBT , where P is the pressure, T is the temperature, N isthe number of the gas molecules, and kB is the proportion constantcalled ”Boltzmann constant”. kB = 1.38× 10−23 joules per Kelvin.

I Please derive [P].

I Please show [PV]=[Energy].

I Please derive [kB ].

I At T = 300K , please showkBT = 4.1× 10−21J = 4.1pN.nm(1pN = 10−12 N; 1nm =10−9 m).


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molecule movements lead to Pressure


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Temperature is a measure of the molecule movements

I Ideal gas law: PV = NkBT .

I Pressure is determined by gas molecule move:PV = m < v2

x > N = m<v2>N3 .

I We thus have: The average kinetic energy of a molecule in anideal gas is 3

2kBT . Each degree of freedom contribute 12kBT .


probability, ideal gas, and temperaturebiophysics/pc2267/lecture-03.pdf · tutorial outline...

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