Test bank 4 - Physics for Scientists and Engineers I | PH 141, Study notes of Physics

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**1. A. Fill in your student number, last name, and initials on the bubble sheet. Do these clearly and carefully. Errors will result in a rejected bubble sheet. B. Mark your exam version, A or B. C. Hand write in your section number (001 or 002) in the section block on the bubble sheet. D. Fill in your name and section number on the exam booklet.

2. While the exam is open book and open notes (no calculators or eletronic devices of any kind), you would be prudent to keep the use of these “open book” materials to a minimum.**

These are concept questions that involve basic understanding and a few key equations. There is essentially very little that you can “look up” from your book or notes. There are essentially no "equations to find." You should have the basic concept equations in your head.

**3. It is recommended that you work in this booklet, and wait until 10-20 minutes before the end of the period to mark your answer sheet. Mark up the booklet any way you like. It is recommended that you not waste time erasing. This is a workbook.

4. Make sure to use a soft #2 pencil to fill in the ovals on the bubble sheet. Note that erasures on the bubble sheet can result in misreads and machine grading errors. Please make sure of your answer before marking the bubble sheet.
5. Ten minutes before the end of the period you will be reminded to begin to wrap things up and mark answers on your bubble sheet.
6. Five minutes before the end of the period you will be reminded to wrap things up and turn in your booklet and answer sheet.
7. The exam has 15 questions. Each question is worth 7 points. The maximum score is
8. It is the raw score that counts. So you get a "spot" of 5 points! A perfect score will net you 105 points toward the in-term exam average. E.g., you can miss 2 problems and still get an A, miss 3 and still get B, miss 5 and still get a C, etc.**

Addendum on academic integrity: the academic integrity policy for this course is stated briefly in the syllabus and in detail in the separate document posted on RamCT. Failure to abide by these rules will be considered academic dishonesty and will be dealt with according to university policy. For exams: exams are to be your own work, and no consultation with other persons, directly or indirectly, is allowed. You are not allowed to use any materials unless specifically authorized to do by course policy or by the instructor. It is never acceptable to copy work from another student.

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Possibly useful physics equations and connections:

Wherever needed, take the gravitational acceleration parameter g as equal to 10 m/s 2.

Derivative connections between 1D position x ( ) t , velocity v (^) x ( ) t , and acceleration a (^) x ( ) t :

x ( )^ x ( );^ x ( )^ ( )

d d a t v t v t x t dt dt

= =. Use slopes instead of derivatives whenever possible.

Integral connections between 1D position x ( ) t , velocity v (^) x ( ) t , and acceleration a (^) x ( ) t :

( ) ( ) ; ( ) ( )

final final

initial initial

a v x x x a v

∆ v t = ∫ a t dt ∆ x t = ∫ v t dt.

Use areas instead of integration whenever possible.

Centripetal acceleration a (^) c and centripetal force Fc (both inward) for a point mass ( m ) going

around a circle of radius R with a speed v : v^2 a R

=.

2 c

mv F R

=

Newton’s First Law: An object in motion (constant velocity) tends to stay in motion (same velocity)

Newton’s Second Law:

2 2

dv d x F ma m m dt dt

= = =

Newton’s Third Law: For every force applied to/in a system, there is an equal and opposite force (action = reaction) Hooke’s law (spring force law): F = − kx

Work = force acting over a distance. " (in 1D) distance

end

start

end x x avg start x

W = ∫ F d r ⋅ = ∫ F dx = F ×

G G

Kinetic energy: 2

K = mv

Gravity work: W (^) grav = mgh

Spring work: 2

W spring = kx

Work – kinetic energy theorem (work on a “free mass”): 2 2

"

end end start start

∫ F d r ⋅^^ =^ mv^ − mv

G G

Maximum static friction force: F static friction max = μ s Fn (stationary mass)

Kinetic friction force: F kinetic friction = μ k Fn (moving mass)

Momentum defined (from " F = ma "):

(^2) ( ) 2 ;

d x dv d mv dp F m m p mv dt dt dt dt

= = = = =

" F = ma "and action-reaction basis for momentum conservation: (1)

dp F dp Fdt dt

= ⇒ =

means that a force F 1 acting over an incremental time dt on an object (#1) “causes” a change in momentum ( dp ) for object #1. (2) Action- reaction says that the object (#2) providing this force

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suffers an equal and opposite force, namely, F 2 (^) = − F 1 , and this force is applied for the same dt.

We have, therefore, dp (^) 2 = − dp 1.

Elastic collisions: momentum and kinetic energy are conserved; in 1D, the speed of approach for the oncoming masses in a two mass collision equals the speed separation after the collision.

Inelastic collisions: momentum is conserved but energy is not conserved.

Rotational dynamics (based on TM and “standard” notation):

S = R θ , dS / dt = vt = Rd θ/ dt = R ω , d^2 S / dt^2 = Rd^2 θ/ dt^2 = Rd ω / dt = R α

Torque (for forces perpendicular to the lever arm, pivot line, etc.): τ= RF = I α (rotational

analog to F = ma

Moment of inertia: Mass on the end of a massless rod: I = MR^2

Rimmed wheel (all mass uniformly distributed around circumference: I = MR^2 Rod of length L and mass M rotating about an axis at one end and perpendicular to the rod:

I =(1/ 3) ML^2

Uniform disk of radius R and mass M rotating about an axis through the center and perpendicular to the disk: I =(1/ 2) MR^2

Torque and angular momentum vector connections: τ= r × F = I α= dL / dt

G G G^ G G

, L = r × p = I ω

G G G G

.

The vector (cross) product: A × B = A B sin θ AB

G G G G

; direction of A × B

G G

by right hand rule.

Universal gravitation force law: F 12 (^) = Gm m 1 (^) 2 / r 122 (attractive) – this is a concept equation

Gravitational potential energy U r ( ) of a point mass m in the gravitational field of a uniform

spherical shaped object of mass M at a distance r away from the center, and with U taken as zero at r = ∞ : U r ( ) = − GMm r /.

Effective g -value at the surface of a planet: g = GM / R^2

Energy balance for the escape velocity vesc of a mass m launched from the surface of a uniform

spherical planet of mass M and radius R : 2

esc

GMm mv R

=

Solid angle rules for hollow or variable density spheres with spherical symmetry: All mass elements outside the radius ( ) r of the observer yield a net g − value of zero.

How do F = ma and Hooke’s Law, F = − kx , combine to give harmonic motion, that is, with

x = x 0 (^) sin ( ω 0 t ) ( ω 0 = k / m ), etc.

Frequency and period connections: ω 0 = 2 π f 0 , f 0 = 1/ T 0 , f 0 = ω 0 / 2π, etc. (do you see the

connections?)

Harmonic motion connections: x = x 0 sin(^ ω 0 t ), v x = x 0 ω 0 cos(^ ω 0 t ), a x = − x 0 ω 02 sin(^ ω 0 t ), etc.

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Harmonic motion and phase: How can x t ( ) = x 0 cos( ω 0 t + δ), x t ( ) = x 0 sin( ω 0 t + φ), and

x t ( ) = x C cos ( ω 0 t ) + x S sin( ω 0 t ), all be equivalent?

How do τ = I αand the restoring force dynamics for a pendulum made of a light string of length

L and a mass m , combined with the small angle approximation, sin θ ≈ θ, yield small angle

harmonic motion, θ = θ 0 sin (^ ω 0 t ) ( ω 0 = g / L ), etc.

How does pendulum motion in accelerated reference frames work? For linear accelerated motion? For circular motion? For accelerated motion down an incline? Answer; If there is an a

G

, the accelerated system “feels” an effective gravity, g (^) eff = − a

G G

(in addition to any “regular” g

G

that might be present.

Transverse harmonic wave: (^ )^ ( )

sin sin ( , ) cos or cos 2 2 etc. etc.

x t y x t A kx t T

ω π π λ

= − −

Velocity for a wave: v = λ f =ω/ k

Transverse wave equation:

2 2 2 2 2

d y x t ( , ) 1 d y x t ( , ) dx v dt

=

Doppler effect (concept equation – useless without understanding): (^) receiver air^ receiver source air source

v u f f v u

=

∓

∓

.

Perfect gas law: (^2) av

(temperature = kinetic energy) 3

PV = nRT = Nk T B = m v

Absolute zero temperature: T 0 (^) = 0 K ≈ −273.15 ° C Thermodynamic internal energy - point atom with three (translational) degrees of freedom:

( ) ( ) ( )

1 2 3 3 3 E int (^) = N (^) 2 m v = N (^) 2 k TB = nN (^) A (^) 2 k TB = n (^) 2 RT

Gas constant - Boltzmann constant connection: R = N kA B

Normalized Boltzmann speed distribution:

2 ( ) 4 2 e (^2) B (^2) B

m mv f v v π k T^ k T

⎛ ⎞ ⎛^ ⎞

= ⎜ ⎟ ⎜ − ⎟

⎝ ⎠ ⎝ ⎠

Degrees of freedom – energy connection: For any energy form such as 12 mv^2 x , 12 mv^2 y , or 12 mv^2 z

(translational kinetic energy), 12 Ix ω x^2 or 12 Iy ω 2 x (rotational kinetic energy), 12 kx^2 and 12 μ u^2 x

(spring potential and kinetic energy), etc., Boltzmann averaging gives 12 mv^2 x = 12 k TB , etc.

Gas constant value: R = 0.083 L atm / mol⋅ ⋅ K = 8.314 J / mol K⋅ (when to use which one)?

Specific heats and thermodynamic internal energy: E int (^) = C Tv (in K)= mc Tv = nc Tv ' (understand

the notation too). First law of thermodynamics: One version: ∆ E (^) int (^) = Qin + Won ; Another version: Qin = ∆ E (^) int + Wby.

Why do these say the same thing? Internal energy of an ideal gas: E int (^) = C Tv (why?) Why does the specific heat at constant volume always apply to the internal energy? Specific heat at constant pressure:

( )

int '^ '^ ; '^ ' Qin = ∆ E + P V ∆ = ncv ∆ T + nR T ∆ = n cv + R ∆ T c (^) p = cv + R

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Adiabatic expansion ( Qin = 0 ): PV = nRT differential analysis and integration gives

PV^ γ^ = const. γ = C p / Cv. For a monatomic ideal gas, we have cv '^ = 32 R and c ' p^ = 52 R ,

so we also have γ = 5 / 3.

Thermodynamic work:

2

1

W by = (^) ∫ PdV.

2

1

W on = − (^) ∫ PdV

Work done for single legs in simple perfect gas thermodynamic PV cycles:

Constant volume: W = 0 (why?)

Constant pressure: Wby = P V ( final − Vinitial )(why?)

Isothermal ( T = const .):

final

initial

V by V

dV W nRT V

= (^) ∫ (why?)

Adiabatic ( Qin = 0 ): Wby = − C (^) v ∆ T (why?)

Efficiency of a reversible thermodynamic engine: ε = 1 − Qout / Qin

Efficiency of a Carnot engine: ε = 1 − TC / TH

Coefficient of performance for a refrigerator: in (from cold reservoir) on

Q

COP

W

=

Possibly useful math connections: opposite adjacent opposite sin ; cos ; tan. hypotenuse hypotenuse adjacent

= = =

sin 30 ° = 1/ 2; cos 30 ° = 3 / 2. sin 30 ° = cos 60 ;° sin 60 ° = cos 30° (rule for complimentary angles) sin 45 ° = 2 / 2 = 1/ 2; cos 45 ° = 2 / 2 =1/ 2.

tan θ =sin θ / cos θ.

sin 2 θ + cos 2 θ= 1.

2sin θ cos θ =sin 2θ

sin ( A + B )= sin A cos B + cos A sin B (get sin( A − B )from symmetry)

1 2 (^1 2 )^ (^1 2 )

sin sin 2 cos sin 2 2

θ + θ = ⎡^ θ − θ ⎤^ ⎡^ θ +θ ⎤

⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦

Derivative connection for x n : n^ n^1 d x nx dx

= −.

Circumference of a circle: C = 2 π r

Area of a circle: A = π r^2

Area of a sphere: A = 4 π r^2

Volume of a sphere: 3

V = π r

∫ sin^ θ^ d θ^ =cosθ,^ ∫cos^ θ^ d θ^ = −sinθ

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ln ( )

b

a

dx b x a

∫^ =

ln ( ) ln( )

b a a b

= −

Disclaimer: This test bank is provided to the student as a guide only. The problems below have not been totally vetted and may contain errors and ambiguities. They are intended only to give the student an idea of the genre of problems they may expect on the exam. Final exam problems will be drawn from this test bank and all previous test banks (1-3) for the course. Once this test bank is posted, specific questions on these problems will not be entertained. Feedback on any possible errors and ambiguities, of course, is welcoe.

Good luck on the final exam.

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TB

On the Réaumur temperature scale, the melting point of ice is 0°R and the boiling point of water is 80°R. What is the expression for converting temperatures on the Réaumur scale ( t (^) R) to the

Celsius scale ( t (^) C)?

TB

On the Réaumur temperature scale, the melting point of ice is 0°R and the boiling point of water is 80°R. What is the expression for converting temperatures on the Réaumur scale ( t (^) R) to the

Fahrenheit scale ( t (^) F)?

TB

A thermistor is a solid-state device widely used in a variety of engineering applications. Its primary characteristic is that its electrical resistance R varies greatly with temperature T. Its

temperature dependence is given approximately by R = 100 e 200 / T. Which of the following statements is true?

TB The fractional distribution function of molecular speeds in a gas is f ( ) v. Which of the

following integrals gives the average speed of the molecules?

TB The fractional distribution function of molecular speeds in a gas is f ( ) v. Which of the

following integrals gives the mean square speed of the molecules?

TB Equal amounts of heat are transferred to two objects A and B. If the mass of object A is twice that of object B, and the specific heat of object A is one third of that of object B, how do the subsequent changes in their temperatures compare?

TB Object A has a mass that is twice the mass of object B, and object A has a specific heat that is four times of the specific heat of object B. If the two objects have the same amount of increases in temperature, how do the amounts of heat transferred to the objects compare?

TB Object A has a mass that is twice the mass of object B, and object A has a specific heat that is three times of the specific heat of object B. If equal amounts of heat are transferred to these two objects , how do the subsequent changes in their temperatures compare?

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TB

A gas may change its state according to different quasi- static paths or combinations of paths shown in the figure. Which of the following statements is WRONG?

TB

A gas may change its state according to different quasi- static paths or combinations of paths shown in the figure. For which path is the work done on the gas by the external world positive?

TB

A gas may change its state according to different quasi- static paths or combinations of paths shown in the figure. The work done on the gas by the external world is:

TB

For purposes of these numerics, take the specific heat of ice as 2 kJ/kg K, the heat of fusion for ice as 300 kJ/kg, and the specific heat of water as 4 kJ/kg K. How much heat energy ( Qin ) is

required (extracted or added) to take 100 g of water at 50 °C, freeze it, and lower the temperature of the ice to -100 °C?

TB For purposes of these numerics, take the specific heat of ice as 2 kJ/kg K, the heat of fusion for ice as 300 kJ/kg, and the specific heat of water as 4 kJ/kg K. How much heat energy ( Qin ) is

required (extracted or added) to take 100 g of ice at -100 °C, warm it up, melt it, and raise the temperature of the water to 50 °C?

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TB If a person is (accidentally) exposed to a blast of water vapor (steam) at 100 °C, he/she can be burned severely. A blast of hot air, on the other hand, will have much less of an effect. Why?

TB

If 100 J of heat is extracted from a system at the same time that 100 J of work is done on the system, what is the energy change of the system ( ∆ E (^) int)?

TB

If 100 J of heat is added to a system at the same time that 100 J of work is done on the system, what is the energy change of the system ( ∆ E (^) int)?

TB

If 100 J of heat is added to a system at the same time that 200 J of work is done by the system, what is the energy change of the system ( ∆ E (^) int)?

TB

Two identical blocks, each of mass m (^) blockare rubbed together to produce heat. The specific

heat of the blocks is specified as c block (^) (in J/kg K)⋅. Take the normal force pressing the blocks

together as Fn (in N), the coefficient of kinetic friction as μ k , and the average relative speed of

the rubbing motion as v (^) rub. Which equation below gives the heating power ( P heating ) generated

from the rubbing process, assuming that all of the non-conservative friction work goes into heat. Note: Not all of the parameters specified above may be needed to obtain P heating.

TB

Two identical blocks, each of mass m (^) blockare rubbed together to produce heat. The specific

heat of the blocks is specified as c block (^) (in J/kg K)⋅. Take the normal force pressing the two

blocks together as Fn (in N), the coefficient of kinetic friction as μ k , and the average relative

speed of the rubbing motion as v (^) rub. Which equation below gives the rate of increase in the

block temperatures (assumed the same), taken as dT (^) block / dt generated from the rubbing process,

assuming that all of the non-conservative friction work goes into heat. Note: Not all of the parameters specified above may be needed to obtain dT (^) block / dt.

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TB

The figure to the right shows a PV diagram for a reversible, quasi-static two leg gas compression process. The gas starts out at point ( P Vi , (^) i ). The gas is first heated

at constant volume ( Vi ), so that the pressure is increased

from some Pi to some (higher) value P (^) f. Call this

intermediate point ( P (^) f , Vi ). The gas is then compressed

at constant pressure ( P (^) f ) to a final volume V (^) f. The

final state is at ( P (^) f , V (^) f ). At what point or points is the

temperature the highest?

TB

The figure to the right shows a PV diagram for a reversible, quasi-static two leg gas compression process. The gas starts out at point ( P Vi , (^) i ). The gas is first heated

at constant volume ( Vi ), so that the pressure is increased

from some Pi to some (higher) value P (^) f. Call this

intermediate point ( P (^) f , Vi ). The gas is then compressed

at constant pressure ( P (^) f ) to a final volume V (^) f. The

final state is at ( P (^) f , V (^) f ). At what point or points is the

temperature the lowest?

TB The figure to the right shows a PV diagram for a reversible, quasi-static two leg gas compression process. The gas starts out at point ( P Vi , (^) i ). First leg: The gas is

first heated at constant volume ( V (^) i ), so that the pressure

is increased from some P (^) i to some (higher) value P (^) f.

Call this intermediate point ( Pf , Vi ). Second leg: The

gas is then compressed at constant pressure ( P (^) f )to a

final volume V (^) f. The final state is at ( Pf , V (^) f ). For

which leg does heat flow into the gas?

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TB The figure to the right shows a PV diagram for a reversible, quasi-static two leg gas compression process. There are n moles of gas. Take the specific heat of the

gas as cv '^ (in J/mol K or L atm/mol K)⋅ ⋅ ⋅. The gas

constant is denoted as R (in J/mol K or L atm/mol K)⋅ ⋅ ⋅.

The gas starts out at point ( P Vi , (^) i ). First leg: The gas is

first heated at constant volume ( V (^) i ), so that the pressure

is increased from some P (^) i to some (higher) value P (^) f.

Call this intermediate point ( Pf , Vi ). Second leg: The

gas is then compressed at constant pressure ( P (^) f )to a final volume V (^) f. The final state is at

( Pf , V (^) f ). What is the total change in the internal energy ( ∆ E (^) int) of the gas?

TB

The figure to the right shows a PV diagram for a reversible, quasi-static isothermal gas expansion process. There are n two moles of gas. Take the specific heat of the gas as

c ' v^ (in J/mol K or L atm/mol K)⋅ ⋅ ⋅. The gas constant is

denoted as R (in J/mol K or L atm/mol K)⋅ ⋅ ⋅. The gas

starts out at point ( P Vi , (^) i )and ends up at point ( P (^) f , Vi ).

In terms of the parameters given (if needed), what is the total change in the internal energy ( ∆ E (^) int) of the gas?

TB

The figure to the right shows a PV diagram for a reversible, quasi-static isothermal gas expansion process. There are n moles of gas. The gas constant is denoted as R (in J/mol K or L atm/mol K)⋅ ⋅ ⋅. The gas

starts out at point ( P Vi , (^) i )and ends up at point ( P (^) f , Vi ).

In terms of the parameters given (if needed), what is total work done on the gas ( W on gas ) during this

compression?

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TB The figure to the right shows a PV diagram for a reversible, quasi-static isothermal gas expansion process. There are n moles of gas. The gas constant is denoted as R (in J/mol K or L atm/mol K)⋅ ⋅ ⋅. The gas

starts out at point ( P Vi , (^) i )and ends up at point ( P (^) f , Vi ).

In terms of the parameters given (if needed), what is total heat absorbed by the gas ( Q (^) in) during this

compression?

TB

The figure to the right shows a PV diagram for a reversible, quasi-static gas cycle. The cycle goes A → B → C → D → A. The A → B and the C → D legs are adiabatic legs. As evident from the diagram, the B → C and D → A legs are at constant volume. For which of the four legs does heat flow into the gas?

TB

The figure to the right shows a PV diagram for a reversible, quasi-static gas cycle. The cycle goes A → B → C → D → A. The A → B and the C → D legs are isothermal at temperatures T hot and T cold , respectively.

As evident from the diagram, the B → C and D → A legs are at constant volume. We will take the volumes for DA and BC as at V high and V low , respectively. There are n moles of

gas. The gas constant is denoted as R (in J/mol K or L atm/mol K)⋅ ⋅ ⋅. For which of the four legs

does the internal energy of the gas ( E (^) int) increase?

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TB

The figure to the right shows a PV diagram for a reversible, quasi-static gas cycle. The cycle goes A → B → C → D → A. The A → B and the C → D legs are isothermal at temperatures T hot and T cold , respectively.

As evident from the diagram, the B → C and D → A legs are at constant volume. For which of the four legs is positive work done by the gas on the outside world?

TB

The figure to the right shows a PV diagram for a reversible, quasi-static gas cycle. The cycle goes A → B → C → D → A. The A → B and the C → D legs are isothermal at temperatures T hot and T cold , respectively.

As evident from the diagram, the B → C and D → A legs are at constant volume. We will take the volumes for DA and BC as at V high and V low , respectively. There are n moles of

gas. The gas constant is denoted as R (in J/mol K or L atm/mol K)⋅ ⋅ ⋅. In terms of the parameters

given (as needed), what is the total work done by the gas on the outside world for one complete cycle?

TB

The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. If the system is brought to point C, the change in the internal energy of the gas is:

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TB

The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. If the system is brought to point C, the work done by the gas is:

TB

The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the pressures, volumes, and temperatures at the labeled points be denoted as P (^) A, P (^) B , etc., and

V (^) A, V (^) B , etc., and T (^) A, T (^) B , etc.,

respectively. CV and C (^) P denote the

specific heats ( in J or L atm per K⋅ ) at

constant volume or constant pressure, respectively. The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→E→C, which expression gives the work done by the gas on the external world?

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TB The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc., respectively. CV and C (^) P denote the

specific heats ( in J or L atm per K⋅ in J) at

constant volume or constant pressure, respectively. The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→E→C, what is the change in the internal energy of the gas?

TB

The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc., respectively. The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→E→C, what is the change in the internal energy of the gas?

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TB The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the pressures, volumes, and temperatures at the labeled points be denoted as P (^) A, P (^) B , etc., and V (^) A,

V (^) B , etc., and T (^) A, T (^) B , etc., respectively. If

the system is brought to point C along the path A→E→C, what is the heat absorbed by the gas?

TB

The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point C. The paths DA and CB represent isothermal changes. If the system is brought to point A, the change in the internal energy of the gas is:

TB

The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point C. The paths DA and CB represent isothermal changes. If the system is brought to point A, the work done on the gas is:

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TB The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the pressures, volumes, and temperatures at the labeled points be denoted as P (^) A, P (^) B , etc., and V (^) A,

V (^) B , etc., and T (^) A, T (^) B , etc., respectively.

The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→B→C, what is the work done by the gas?

TB

The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc. CV^ and C (^) P denote the specific heats

( in J or L atm per K⋅ in J) at constant

volume or constant pressure, respectively. If the system is brought to point C along the path A→B→C, what is the change in the internal energy of the gas?

TB

The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc. The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→B→C, what is the change in the internal energy of the gas?

Test version: X mark this version on your answer sheet!

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TB The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the pressures and the volumes at the labeled points be denoted as P A , PB , etc., and V (^) A, V (^) B , etc.,

respectively. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B , etc.

The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→B→C, what is the heat absorbed by the gas?

TB

The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc. Let the pressures and the volumes at the labeled points be denoted as P A , P (^) B , etc.,

and V (^) A, V (^) B , etc., respectively. The

numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→D→C, what is the work done by the gas on the external world?

Test version: X mark this version on your answer sheet!

YOU MUST RETURN THIS EXAM BOOKLET WITH YOUR ANSWER SHEET!

TB The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc. Let the pressures and the volumes at the labeled points be denoted as P A , P (^) B , etc.,

and V (^) A, V (^) B , etc., respectively. The

numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→D→C, what is the change in the internal energy of the gas?

TB

The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc. The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→D→C, what is the change in the internal energy of the gas?

Test version: X mark this version on your answer sheet!

YOU MUST RETURN THIS EXAM BOOKLET WITH YOUR ANSWER SHEET!

TB The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent isothermal changes. R is the universal gas constant. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B , etc. Let the pressures

and the volumes at the labeled points be denoted as P A , PB , etc., and V (^) A, V (^) B ,

etc., respectively. The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→D→C, what is the heat absorbed by the gas?

TB

The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent adiabatic processes. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B , etc. Let the pressures

and the volumes at the labeled points be denoted as P A , PB , etc., and V (^) A, V (^) B ,

etc., respectively. The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→B→C, what is the work done by the gas? \

Test version: X mark this version on your answer sheet!

YOU MUST RETURN THIS EXAM BOOKLET WITH YOUR ANSWER SHEET!

TB4 The PV diagram in the figure is for an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent adiabatic processes. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc. CV and C (^) P denote the specific heats

( in J or L atm per K⋅ in J) at constant

volume or constant pressure, respectively. Let the pressures and the volumes at the labeled points be denoted as P A , P (^) B , etc.,

and V (^) A, V (^) B , etc., respectively. The

numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→B→C, what is the heat absorbed by the gas?

TB4

The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent adiabatic processes. R is the universal gas constant. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc. Let the pressures and the volumes at the labeled points be denoted as P A , P (^) B , etc.,

and V (^) A, V (^) B , etc., respectively. The

numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→B→C, what is the heat absorbed by the gas?

Test version: X mark this version on your answer sheet!

YOU MUST RETURN THIS EXAM BOOKLET WITH YOUR ANSWER SHEET!

TB4 The PV diagram in the figure is for n moles of an ideal monatomic gas. The gas is initially at point A. The paths AD and BC represent adiabatic processes. R is the universal gas constant. Let the temperatures at the labeled points be denoted as T (^) A, T (^) B ,

etc. The numbers on the axes of the graph are not relevant to this problem. If the system is brought to point C along the path A→B→C, what is the change in the internal energy of the gas?

TB4

When an ideal gas undergoes a temperature change at constant volume, its internal energy change is given by ∆ E (^) int = C (^) V ∆ T. However, this equation also correctly gives the change in

internal energy whether the volume remains constant or not. Which statement below explains why this is so?

TB4 When an ideal gas undergoes a temperature change at constant volume, the heat input is connected to the change in temperature through Q in (^) = CV ∆ T , where CV is the specific heat at

constant pressure. When the temperature change occurs at constant pressure, one has Q in (^) = C (^) P ∆ T , where C (^) P = CV + nR is the specific heat at constant pressure. What is the origin

of the extra " nR "term when one goes from a constant volume to a constant pressure process?

TB4 When an ideal gas undergoes an isothermal expansion, the pressure volume connection P = nRT / V follows directly from the perfect gas law. When we expand a gas adiabatically, we

obtain a very different looking result, namely,

constant P V^ γ

= , where γ = C P / CV = 1 + nR C / V is

always greater than unity. This means that the pressure falls off more rapidly with increasing volume for an adiabatic expansion than for an isothermal expansion. What is the physical reason for this more rapid fall off?

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TB4

During an isothermal expansion of an ideal gas, the pressure starts out at some pressure P 0 and

volume V 0. As the expansion proceeds, the pressure drops by a factor of two (to P 0 (^) / 2) and the

volume doubles to 2 V 0. The gas is then compressed adiabatically from this

( P V , ) = ( P 0 (^) / 2, 2 V 0 )point back to its original volume ( V (^) 0 ) but the pressure is now equal to f P 0 , where f is some number that is greater than 1. What is the correct connection between

f and the thermodynamic parameter γ = C P / CV (where C P and CV are the specific heats at

constant pressure and volume respectively).

TB4

The figure to the right shows a PV diagram for a standard Carnot gas cycle (reversible, quasi-static). The cycle goes

1 → 2 (isothermal at temperature T h - hot bath), 2 → 3

(adiabatic from T h down to T c - cold bath), 3 → 4 (isothermal at temperature T c ), and finally (for one cycle)

4 → 1 (adiabatic from T c back to T h ). For which leg is the change in internal energy ( ∆ E (^) int) positive?

TB4

The figure to the right shows a PV diagram for a standard Carnot gas cycle (reversible, quasi-static). The cycle goes

1 → 2 (isothermal at temperature T h - hot bath), 2 → 3

(adiabatic from T h down to T c - cold bath), 3 → 4

(isothermal at temperature T c ), and finally (for one cycle)

4 → 1 (adiabatic from T c back to T h ). For which leg is the change in internal energy ( ∆ E (^) int) negative?

Test version: X mark this version on your answer sheet!

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TB4 The figure to the right shows a PV diagram for a standard Carnot gas cycle (reversible, quasi-static). The cycle goes

1 → 2 (isothermal at temperature T h - hot bath), 2 → 3

(adiabatic from T h down to T c - cold bath), 3 → 4 (isothermal at temperature T c ), and finally (for one cycle)

4 → 1 (adiabatic from T c back to T h ). Which legs contribute to the work done either by the gas or on the gas?

TB4

The figure to the right shows a PV diagram for a standard Carnot gas cycle (reversible, quasi-static). The cycle goes

1 → 2 (isothermal at temperature T h - hot bath), 2 → 3

(adiabatic from T h down to T c - cold bath), 3 → 4

(isothermal at temperature T c ), and finally (for one cycle)

4 → 1 (adiabatic from T c back to T h ). The figure shows heat going into the gas on isothermal leg 1 → 2 , and out of the gas for isothermal leg 3 → 4 , and no heat in or out for the 2 → 3 and 4 → 1 adiabatic legs. How is this “no heat flow in or out” for these adiabatic legs accomplished?

TB4

The figure to the right shows a PV diagram for a standard Carnot gas cycle (reversible, quasi-static). The cycle goes

1 → 2 (isothermal at temperature T h - hot bath), 2 → 3

(adiabatic from T h down to T c - cold bath), 3 → 4

(isothermal at temperature T c ), and finally (for one cycle)

4 → 1 (adiabatic from T c back to T h ). The figure shows (correctly) that heat Qin flows into the gas from the hot bath

for leg 1 → 2 and heat Qout flows out of the gas into the

cold bath for leg 3 → 4. Take the total work by the gas for

the cycle as Wby total. In terms of these parameters (if

needed), what is the total change in the internal energy of

the gas ( ∆ E (^) int^ total ) as one goes around one complete cycle?

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TB4

For a Carnot refrigerator, the cycle shown in figure to the right is run backwards so that the Q (^) in shown flowing into

the gas from the hot bath for leg 1 → 2 now becomes a refrig Qout flowing into the hot bath for a 2 → 1 leg, the Qout

shown flowing out of the gas into the cold bath for leg

3 → 4 becomes a Qin refrig flowing into the gas from the

cold bath, and the previous total work done by the Carnot engine, Wby total , now becomes the total work done on the gas

( Won total ) from an external power source. Qin refrig , Qout refrig ,

and Won total are positive quantities. Which of the following

refrigerator parameter connections is valid? Hint: It is suggested that you redraw arrows and re- label parameters on the diagram to help your reasoning.

TB4 For a Carnot refrigerator, the cycle shown in figure to the right is run backwards so that the Q (^) in shown flowing into

the gas from the hot bath for leg 1 → 2 now becomes a refrig Qout flowing into the hot bath for a 2 → 1 leg, the Qout

shown flowing out of the gas into the cold bath for leg

3 → 4 becomes a Qin refrig flowing into the gas from the

cold bath, and the previous total work done by the Carnot engine, Wby total , now becomes the total work done on the gas

( Won total ) from an external power source. Qin refrig , Qout refrig ,

and Won total are positive quantities. The “Coefficient of

Performance” or “COP” fir a refrigerator is defined as the ratio of the heat extracted from the cold bath divided by the total work input to the system. Which of the equations below express this result?