Kinetic energy, Temperature, Pressure and · PDF filecontainers: Marco: You need to hold the opening of the box ... cigarettes fall at the same rate either way. They hold their boxes

Internal energy, Temperature, and Heating

Physics 116

Tues. 2/28, Thurs. 3/2

• So far we’ve dealt only with macroscopic gas properties, Pressure, Volume, and Temperature. How do we connect these to microscopic properties of the gas? In particular, how can we think about Pressure and Temperature at the atomic scale? •A real gas consists of a vast number of molecules, each moving randomly and undergoing millions of collisions every second.

• Despite the apparent chaos, averages, such as the average number of molecules in the speed range 600 to 700 m/s, have precise, predictable values. • The “micro/macro” connection is built on the idea that the macroscopic properties of a system, such as temperature or pressure, are related to the average behavior of the atoms and molecules.

Macro and Microscopic descriptions

Creating a mathematical model: How do the macroscopic quantities of pressure and temperature depend on the kinetic energies of the particles?

1) The pressure arises from the particles hitting the wall and transferring momentum.

2) Characterize gas by its density

3) How many particles hit a wall in a time Dt?

4) What is the pressure exerted? Recall that force is the rate of change of the momentum:

F = total momentum change /Dt = Dmv/Dt

And Pressure = Force/Area, so

P = Dmv/(Dt*A)

5) Now relax assumptions that all particles have same speed and move in the + direction:

vx2

½ vaverage(x)2

(two directions)

Now relax assumptions that all particles move in the x-direction:

vaverage(x)2

⅓ vrms2

(three dimensions)

Where vrms2 = vaverage(x)

2 + vaverage(y)2 + vaverage(z)

2

1) Momentum change for each hit = 2mvx

2) particle density = N/V

3) #hits in a time interval Dt:

#hits = (N/V)(vx*Dt)(Area of wall)

• Total momentum change in time Dt:

Dmv = (2mvx) [(N/V)(vx*Dt)(Area of wall)]

4) Pressure (P) = Force / (Area of wall)

= Dmv / Dt*(Area of wall)

P =(2mvx) (N/V)(vx*Dt)(Area of wall)/Dt*(Area of wall)

P = (2mvx) *(N/V)(vx)

P = (2mvx2) *(N/V)

5) Pressure = (⅓mvrms2) *(N/V)

P = (2/3)(½mvrms2) *(N/V)

P = (2/3)(KEave) *(N/V)

From the ideal gas law: PV=NkBT or P = kBT*(N/V)

• T=(2/3kB)(KEave)

Temperature measures average Kinetic Energy – how fast the particles move!

How does this relate to Pressure? Recall: P= NkT/V = nRT/V

Macro-micro connection • Assumptions for ideal gas:

– # of molecules N is large – They obey Newton’s laws – Short-range interactions with

elastic collisions – Elastic collisions with walls (an

impulse…pressure)

l What we call pressure P is a direct measure of the number density of molecules, and how fast they are moving (vrms)

l What we call temperature T is a direct measure of the average translational kinetic energy

m

Tkvv B

rms

3)(

avg

2

VTNkP B /

https://phet.colorado.edu/en/simulation/legacy/gas-properties

T =2

3kB(KE)avg/particle

leavg/partic)(3

2KE

V

NP





The distribution of molecular speeds in a sample of N2 gas

m

Tkvv B

rms

3)(

avg

2

Side note: What is “root mean square (rms)” and why do we use it? Why not just use “average”? A simple (not complete) reason: Velocities are vectors with sign and direction, some can be negative. Calculate the average of the following particle velocities: [-2, 5, -8, 9, -4] = 0 But we know the particles have velocity!

RMS process squares each number then takes the average, and then square roots that whole thing, so we get information about magnitudes.

Distribution of velocities for various temperatures of N2 gas – distribution broadens and the peak shifts

Flux

Flux is how much of something flows through a particular 2D slice of space in some amount of time.

Examples: water flux out of a faucet, wind (or rain) flux through your window, flux of rain on your umbrella.

• http://www.shutterstock.com/video/clip-13665980-stock-footage-animated-rain-of-cigarettes-against-transparent-background-in-k-low-angle-shot-alpha-channel.html?src=rel/13677485:2/3p

http://www.shutterstock.com/video/clip-13665980-stock-footage-animated-rain-of-cigarettes-against-transparent-background-in-k-low-angle-shot-alpha-channel.html?src=rel/13677485:2/3p



































Catching cigarettes

Marco and Donald are at a celebration sponsored by Phillip Morris, where cigarettes are just falling from the sky. They are discussing the quickest way to fill up their identical empty containers:

Marco: You need to hold the opening of the box horizontally so it fills fastest. Donald: It doesn’t matter if you tilt your box, the cigarettes fall at the same rate either way.

They hold their boxes out, the opening to Marco’s box is horizontal, and Donald’s is tilted at q degrees to the horizontal. Whose box fills faster? A. Donald B. Marco C. They fill at the same rate.

Catching cigarettes




Catching cigarettes




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Let’s break this problem down step by step

Limiting cases

What would be a “limiting case” argument that supports the outcome of this experiment?

How many cigarettes go through each unit of area of the opening in each unit

of time? • It depends on the rate at which the cigarettes

fall – higher rate will result in more cigarettes

• It depends on the size of the opening – a bigger opening will fill faster

• It depends on the orientation – horizontal will fill fastest and vertical won’t fill at all – how to quantify?











Flux: How many cigarettes go through each unit of area of the opening in

each unit of time?

Number that falls in each time

interval

Area of plane it

falls through

Cosine of “tilting”

angle

X X

The flux characterizes the flow by combining the rate and the crowdedness


each unit of time?


interval

Area of plane it

falls through


angle (selects the fraction of Area that faces the

falling stuff)

X X



each unit of time?


interval

Area of plane it

falls through

X X



angle (selects the fraction of Area that faces the

falling stuff)

How might you characterize rainfall onto an umbrella using the idea of

flux? • http://www.shutterstock.com/video/clip-

9063377-stock-footage-animated-very-heavy-rainfall-on-transparent-background-alpha-embedded-with-hd-png-file.html?src=rel/9063365:2

http://www.shutterstock.com/video/clip-9063377-stock-footage-animated-very-heavy-rainfall-on-transparent-background-alpha-embedded-with-hd-png-file.html?src=rel/9063365:2

































Consider Pressure to be a “momentum change” flux as the particles hit a wall

• Total momentum change in a time interval =(2mvx (N/V)(vx*Dt)(wall Area), so

• Pressure = total momentum change per unit area per unit of time

• Pressure = (2mvx) *(N/V)(vx)

= (2mvx2) *(N/V)

• Pressure = (⅓mvrms2) *(N/V) (considering two directions and 3

dimensions)

= (2/3)(½mvrms2) *(N/V)

= (2/3)(KEave) *(N/V)

From the ideal gas law: PV=NkT

• T=(2/3kB) (KEave) (kB=1.38 × 10-23 J/K) Boltzmann constant

Internal energy and the 1st law of thermodynamics

• The average kinetic energy of each particle is:

• Summing up the energy of the particles is the same as multiplying the average energy per particle times the total number of particles (N). This total energy is called the internal energy of the system:

• Energy can be added to/removed from the system either by doing work on/by the system or heating/cooling the system. This is the first law of thermodynamics:

TkKE B2

3)( leavg/partic

TNkU Binternal2

3

QWU systemoninternal D _

Last semester: ΔKEtotal = W. If Q=0 above, all of the positive work done on the gas increases the gas temperature, and so increases the total internal energy, ΔUinternal

Changing the system energy

DUint =

The algebraic sign (+) or (-) of W and Q indicate if energy is gained or lost by the system, respectively.

Heating and Temperature

Q: What happens to matter when it is heated or cooled?

“heating” and “cooling” are processes which add or take away energy from the system, so they increase or decrease the kinetic energy of the particles in an ideal gas at a constant volume. What about other

matter, like solids and liquids? There is electrical potential energy involved

between particles, and so the response to heating is more complicated. Back to the macroscopic world…

In real matter, it depends…

Heating is a way of adding energy to a system, so it can:

• cause the particles to become more energetic.

• cause them to change their average positions relative to each other, which will change the potential energy of the system and result in a phase change.

Phases (states) of Matter

Phase Transitions

Hea

tin

g: A

dd

en

erg

y to

sys

tem

Co

olin

g: Rem

ove en

ergy from

system

Heating Matter

• When matter is heated, its temperature changes OR its phase changes, but they don’t happen simultaneously.

• For example: a solid is heated, its temperature changes until it reaches the melting temperature for that substance. Continued heating melts the substance and the temperature stays constant at the melting temperature until the entire solid is melted. Once the matter is in a liquid form entirely then continued heating results in an increase in temperature, until the liquid reaches the vaporization temperature, etc.

Heating Matter

• Matter’s resistance to temperature change is characterized by an index, called the specific heat capacity. The symbol lower case “c” is commonly used and its value gives the energy required for every 1 kg of mass to increase in temperature by 1 K. (Units: J / Kkg )

Q = M c ΔT

• Side note – there’s another quantity you developed in recitation which describes how much heat it takes to change the temperature of a substance by 1 Kelvin, the units were J/K. This is known as the heat capacity (sometimes written as upper-case C).

• Heat capacity differs from specific heat capacity because it changes based on how much of the substance you have – it will take more heat to heat up a larger quantity of substance, whereas specific heat capacity is a material property and doesn’t depend on how much you have.

The specific heat capacity of aluminum is about twice that of iron. Consider two blocks of equal mass, one made of aluminum and the other one made of iron, initially in thermal equilibrium. Both blocks are heated at the same constant rate until reaching a temperature of 500 K. Which of the following statements is true? a) The iron takes less time than the aluminum to

reach 500 K b) The aluminum takes less time than the iron to

reach 500 K c) The two blocks take the same amount of time to

reach 500 K

• Matter’s resistance to temperature change is characterized by an index, called the specific heat . The symbol lower case “c” is commonly used and its value gives the energy required for every 1 kg of mass to increase in temperature by 1 K. (Units: J / Kkg )

Q = M c ΔT • When matter changes phase (e.g., between solid and liquid

or between liquid and gas) that phase change has energy associated with it. The phase change index, or latent heat of transformation is the energy required to complete the phase change for every kg of the substance. The symbol upper case “L” is commonly used and its units are (J / kg)

Q = ±ML

cAl = 0.897 J/Kg or 897 J/Kkg

LAl = 398 J/g or 398,000 J/kg

cFe = 0.412 J/Kg or 412 J/Kkg

LFe = 272 J/g or 272,000 J/kg

• When matter changes phase (e.g., between solid and liquid or between liquid and gas) that phase change has energy associated with it. The phase change index, or latent heat of transformation is the energy required to complete the phase change for every kg of the substance. The symbol upper case “L” is commonly used and its units are (J / kg)

Q = ±ML

LFe = 272 J/g or 272,000 J/kg

LAl = 398 J/g or 398,000 J/kg

https://www.youtube.com/watch?v=If5h3CMzdrw

Since LAl is so large (it takes a lot of energy to melt Al) it’s cheaper to ship Al as a liquid and keep it hot than ship it as a room temp solid.

https://www.youtube.com/watch?v=If5h3CMzdrw

40 g of water at 100 °C and 60 g of water at 0 °C are mixed together. When the mixture reaches thermal equilibrium, the final temperature will be:

a) greater than 50 °C .

b) equal to 50 °C .

c) less than 50 °C .

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Can reason what the answer must be without considering equations?

Let’s work through the physics together AFTER you’ve thought through the problem.

50 g of steam at 100 °C and 50 g of liquid water at 80 °C are mixed together in an insulated container. When the combination reaches thermal equilibrium, the final temperature will be:




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60 g of ice at 0 °C and 40 g of liquid water at 20 °C are mixed together in an insulated container. When the combination reaches thermal equilibrium, the final temperature will be:




Does the water melt the ice? How much energy does it take to melt the ice? How much energy can the liquid water supply as it cools to 0°C?

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60 g of ice at 0 °C and 40 g of liquid water at 20°C are mixed together in an insulated container. When the combination reaches thermal equilibrium, the final temperature will be…

Does the water melt the ice? Let’s see: How much energy does it take to melt the ice? It takes Q = mLf energy to melt a mass “m” of ice, in this case. So we have Q=(0.06 kg)(334000 J/kg) = 20,040 J to melt the ice entirely. How much energy can the liquid water supply as it cools from 20°C to 0°C? If the 40g of liquid water starts at 20C, the most energy it can give to the ice is if it cools to 0C as well, so Qwater,max = mwcw(273K – 293K), where we’ve converted the final temp of 0C to 273K, and the initial temp of 20C to 293K. Qwater,max = (0.04 kg)(4180 J/kgK)(-20 K) = 3344J Since the warm water only supplies 3344 J max, it will only melt a bit of the ice, not nearly all of it. The amount of ice melted can be calculated using Q=mL where Q is 3344J, then solving for m gives m=Q/L = 3344 J / (334000 J/kg) = 0.01 kg, or 10g. The warm water melts just 10g of the ice, so in the final state we end up having a nice glass of ice-water with 50g of ice and 50g of water.

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Kinetic energy, Temperature, Pressure and · PDF filecontainers: Marco: You need to hold the opening of the box ... cigarettes fall at the same rate either way. They hold their boxes