physics 207: lecture 23, pg 1 lecture 23 goals: chapter 16 chapter 16 use the ideal-gas law. use...

Physics 207: Lecture 23, Pg 1

Lecture 23Goals:Goals:

• Chapter 16Chapter 16 Use the ideal-gas law. Use pV diagrams for ideal-gas processes.

• Chapter 17Chapter 17 Employ energy conservation in terms of 1st law of TD Understand the concept of heat. Relate heat to temperature change Apply heat and energy transfer processes in real situations Recognize adiabatic processes.

• AssignmentAssignment HW9, Due Wednesday, Apr. 14th HW10, Due Wednesday, Apr. 21st (9 AM)

Exam 3 on Wednesday, Apr. 21 at 7:15 PM (Statics, Ang. Mom., Ch.14 through 17)


Thermodynamics: A macroscopic description of matter

Recall “3” Phases of matter: Solid, liquid & gas All 3 phases exist at different p,T conditions Phase diagram

Triple point of water:

p = 0.06 atm

T = 0.01°C

Triple point of CO2:

p = 5 atm

T = -56°C


Modern Definition of Kelvin Scale

Water’s triple point on the Kelvin scale is 273.16 K One degrees Kelvin is defined to be 1/273.16 of the

temperature at the triple point of water

Triple point

Accurate water phase diagram


Changes due to Heat or Thermal Energy Transfer

Change the temperature (it gets hotter) Change the state of matter (solidliquid, liquidgas)

Accurate water phase diagram

Path onheating


Energy Transfer to a solid (ice)1. Temperature increase or

2. State Change

If a gas, then V, p and T are interrelated….equation of state


Defining a temperature scale

Three main scales

212

Farenheit

100

Celsius

32 0 273.15

373.15

Kelvin

Water boils

Water freezes

0-273.15-459.67Absolute Zero


Some interesting facts In 1724, Gabriel Fahrenheit made thermometers

using mercury. The zero point of his scale is attained by mixing equal parts of water, ice, and salt. A second point was obtained when pure water froze (originally set at 30oF), and a third (set at 96°F) “when placing the thermometer in the mouth of a healthy man”. On that scale, water boiled at 212. Later, Fahrenheit moved the freezing point of

water to 32 (so that the scale had 180 increments).

In 1745, Carolus Linnaeus of Upsula, Sweden, described a scale in which the freezing point of water was zero, and the boiling point 100, making it a centigrade (one hundred steps) scale. Anders Celsius (1701-1744) used the reverse scale in which 100 represented the freezing point and zero the boiling point of water, still, of course, with 100 degrees between the two defining points.

T (K)

108

107

106

105

104

103

100

10

1

0.1

Hydrogen bomb

Sun’s interior

Solar corona

Sun’s surface

Copper melts

Water freezes

Liquid nitrogen

Liquid hydrogenLiquid helium

Lowest T~ 10-9K


Ideal gas: Macroscopic description Consider a gas in a container of volume V, at

pressure p, and at temperature T Equation of state

Links these quantities Generally very complicated: but not for ideal gas


Ideal gas: Macroscopic description

pV = nRT R is called the universal gas constant

In SI units, R =8.315 J / mol·K

n ≡ number of moles

Equation of state for an “ideal gas” Collection of atoms/molecules moving randomly No long-range forces

Their size (volume) is negligible Density is low Temperature is well above the condensation point


Boltzmann’s constant

In terms of the total number of particles N

p, V, and T are thermodynamic variables

pV = nRT = (N/NA ) RT

kB is called the Boltzmann’s constant

kB = R/NA = 1.38 X 10-23 J/K

pV = N kB T

Number of moles: n = m/M

One mole contains NA=6.022 X 1023 particles :

Avogadro’s number = number of carbon atoms in 12 g of carbon

m = mass (kg)M= mass of one mole (kg/mol)


What is the volume of 1 mol of gas at STP ?T = 0 °C = 273 Kp = 1 atm = 1.01 x 105 Pa

nRTpV

4.22m0224.0

Pa1001.1K 273Kmol/J31.8

3

5

p

nRTV

The Ideal Gas Law


There are four things that can vary: p, V, n & T

Typically, two of these are held constant and the relationship between the remaining two is studied

nRTpV

The Ideal Gas Law


Example

A spray can containing a propellant gas at twice atmospheric pressure (202 kPa) and having a volume of 125.00 cm3 is at 27oC. It is then tossed into an open fire. When the temperature of the gas in the can reaches 327oC, what is the pressure inside the can?

Assume any change in the volume of the can is negligible.

Steps

1. Convert to Kelvin (From 300 K to 600 K)

2. Use P/T = nR/V = constant P1/T1 = P2/T2

3. Solve for final pressure P2 = P1 T2/T1

WD40 foolishness


Example problem: Air bubble rising A diver produces an air bubble underwater, where the absolute

pressure is p1 = 3.5 atm. The bubble rises to the surface, where the pressure is p2 = 1.0 atm. The water temperatures at the bottom and the surface are, respectively, T1 = 4°C, T2 = 23°C

What is the ratio of the volume,V2 , of the bubble just as it reaches the surface to its volume at the bottom, V1?

Is it safe for the diver to ascend while holding his breath?

No! Air in the lungs would expand, and the lung could rupture.


Example problem: Air bubble rising

A diver produces an air bubble underwater, where the absolute pressure is p1 = 3.5 atm. The bubble rises to the surface, where the pressure is p2 = 1 atm. The water temperatures at the bottom and the surface are, respectively, T1 = 4°C, T2 = 23°C

What is the ratio of the volume of the bubble as it reaches the surface,V2, to its volume at the bottom, V1? (Ans.V2/V1 = 3.74)

pV=nRT pV/T = const so p1V1/T1 = p2V2/T2

V2/V1 = p1T2/ (T1 p2)

= 3.5 296 / (277 1)

If thermal transfer is efficient.

[More than likely the expansion will be “adiabatic” and, for a diatomic gas, PV = const. where = 7/5, see Ch. 17 & 18]


Buoyancy and the Ideal Gas Law

A typical 5 passenger hot air balloon has approximately 700 kg of total mass and the balloon itself can be thought as spherical with a radius of 10.0 m. If the balloon is launched on a day with conditions of 1.0 atm and 273 K, how hot would you have to heat the air inside (assuming the density of the surrounding air is 1.2 kg/m3 and the air behaves and as an ideal gas) in order to keep the balloon at a constant altitude?

Hint: Remember the weight of the air inside the balloon.

p, V and R do not change!

Balloon weight = Buoyant force – Weight of hot air

Ideal gas law: pV = nRT nT= pV/R = const.

or T = const. = 1.2 x 273 kg K/m3


Buoyancy and the Ideal Gas Law

mballoon g = air at 273 K V g – air at T V g

mballoon = air at 273 K V – air at T V

mballoon = (1.2 – 330 / T) V

700 / 4200 = 1.2 – 330 / T

330 / T = (1.2 - 0.2)

T = 330 K 57 C


pV diagrams: Important processes

Isochoric process: V = const (aka isovolumetric) Isobaric process: p = const Isothermal process: T = const constant

TpV

Volume

Pre

ssur

e

2

2

1

1 T

p

T

p

1

2Isochoric

Volume

Pre

ssur

e

2

2

1

1 T

V

T

V

1 2

Isobaric

Volume

Pre

ssur

e

2211 VpVp 1

2

Isothermal


pV vs. Fx diagrams

Work (on the system) remains the area under the curve

dW = F dx

or

dW = F/A (A dx) = P dVworld = -p dVsystem

constant TpV

Volume

Pre

ssur

e

2211 VpVp 1

2

Isothermal

Position

For

ce

1

2


Work and Energy Transfer (Ch. 17)

K reflects the kinetic energy of the system

ΔK =Wconservative + Wdissipative + Wexternal

Wconservative = - ΔU (e.g., gravity)

Wdissipative = - ΔEThermal

Wexternal Typically, work done by contact forces

ΔK + ΔU + ΔETh = Wexternal= ΔEsys


Work and Energy Transfer

ΔK + ΔU + ΔETh = Wexternal= ΔEsys

But we can transfer energy without doing work

Q ≡ thermal energy transfer

ΔK + ΔU + ΔETh = W + Q = ΔEsys

If ΔK + ΔU = ΔEMech = 0 ΔETh = W + Q


1st Law of Thermodynamics

Thermal energy Eth : Microscopic energy of moving molecules and stretching molecular bonds. ΔEth depends on the initial and final states but is independent of the process.

Work W : Energy transferred to the system by forces in a mechanical interaction.

Heat Q : Energy transferred to the system via atomic-level collisions when there is a temperature difference.

ΔEth =W + Q

W & Q with respect to the system


Lecture 23

• AssignmentAssignment HW9, Due Wednesday, Apr. 14th HW10, Due Wednesday, Apr. 21st (9 AM)

Exam 3 on Wednesday, Apr. 21 at 7:15 PM (Statics, Ang. Mom., Ch.14 through 17)

physics 207: lecture 23, pg 1 lecture 23 goals: chapter 16 chapter 16 use the ideal-gas law. use...

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