Mastering Physics HW 3 Ch 17, First Law of Thermodynamics

HW 3 Ch 17, First Law of Thermodynamics Due: 11:59pm on Wednesday, September 23, 2015 To understand how points are awarded, read the Grading Policy for Policy for this assignment.

Fast vs. Slow Tire Pumping Imagine the following desi design gn for a sim simple ple tire pump. pump. The pump is filled with with a volume of air air at atmos atmospheric pheric pressure and ambient temperature . When When you push push the pump handle, the air is compressed compress ed to a new (small (smaller) er) volume , raising its pressure. A valve is then opened, allowing air to flow from the pump into the tire until the remaining air in the pump reaches the pressure of the air in the tire, . In this problem, you will consider whether you can get more air into the tire per pump cycle by pushing the pump handle quickly or slowly. We will make the following simplifying assumptions: The pressure pressure of the air in the tire, , does does not change change significantly as air flows flows into the tire from from the pump. The temperature of the air in the pump does not change significantly while air is flowing into the tire (i.e., this is an isothermal process). The air is mainly composed of diatomic molecules with .

Part A First imagine that you push the pump handle quickly, so that the compression of air in the pump occurs adiabatically adiabatic ally.. Find the absolute temperature temperature of the air insi inside de the pump after a rapid compress ion from volume to volume , ass assuming uming an ambient temperature of . Express your answe answerr in terms of

,

,

, and

Hint 1. Properties of an adiabatic process For an adiab adiabatic atic pro process cess the pro product duct

is constant. Using the ideal gas law, you can derive the

equivalentt condition that the prod equivalen product uct

is constant throu throughou ghoutt the adiab adiabatic atic pro process cess..

ANSWER:

=

Correct

Part B Once the tire and pump pre pressures ssures have have equilibr equilibrated ated at will have ended up in the tire?

, what what fraction

Express the frac fraction tion in terms of your answer.

. The tempera temperatures tures

,

,

,

, and

of the gas particles initially in the pump and

should not appear in

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Hint 1. How to approach the problem Find an an express expression ion for the number of partic particles les in the pump pump initi initially ally ( pump after the pump and tire have have come into press pressure ure equilibrium ( fraction of the particles in the pump that are transferred into the tire:

) and another for the number in the ). Use these to determine the

.

Hint 2. Find an expression for the number of particles in a gas Using the ideal gas law, find an express expression ion for the number of particl particles es and temperature . Use

in a gas at pressure , volume

for Boltzm Boltzmann's ann's constant.

ANSWER: =

Hint 3. Find What is the ratio of the number of particles in the pump after it comes into pressure equilibrium with the tire to the number of gas particles initially in the pump (before pushing the handle)? Express the ratio in terms of in your answer.

,

,

,

, and

. The temperatures

and

should not appear

Hint 1. Find the initial number of particles in the pump How many gas particl particles es

were in the pump init initially ially? ?

Express your answer in terms of quantities given in the problem introduction and Boltzmann's Boltzm ann's constant . ANSWER: =

Hint 2. Find the final number of particles in the pump How many gas particl particles es

are in the pump after the pump and tire have equilibrated?

Express your answer in terms of quantities given in the problem introduction and Boltzmann's const constant ant . should not appear in your answe answer. r.

,















=

ANSWER:

=

ANSWER:

=

Correct

Part C Now imagine that you push the pump handle slowly, so that the compression of air in the pump occurs isothermally. isot hermally. Find the absolute temperature of the air insi inside de the pump ass assuming uming an ambient temperature of

Hint 1. Properties of an isothermal process Isothermal means "at constant temperature."

ANSWER: =

Correct

Part D Once the tire and pump pre pressures ssures have have equilibr equilibrated ated at will have ended up in the tire?

, what what fraction

Express the frac fraction tion in terms of

. The tempera temperatures tures

,

,

,

, and

of the gas particles initially in the pump and

should not appear in

.















.

Hint 2. Find an expression for the number of particles in a gas Using the ideal gas law, find an express expression ion for the number of particl particles es and temperature . Use

in a gas at pressure , volume

for Boltzm Boltzmann's ann's constant.

ANSWER: =

Hint 3. Find What is the ratio of the number of particles in the pump after it comes into pressure equilibrium with the tire to the number of gas particles initially in the pump (before pushing the handle)? Express the ratio in terms of in your answer.

,

,

,

, and

. The temperatures

and

should not appear

Hint 1. Find the initial number of particles in the pump How many gas particl particles es

were in the pump init initially ially? ?

Express your answer in terms of quantities given in the problem introduction and Boltzmann's Boltzm ann's constant . ANSWER: =

Hint 2. Find the final number of particles in the pump How many gas particl particles es

are in the pump after the pump and tire have equilibrated?

Express your answer in terms of quantities given in the problem introduction and Boltzmann's const constant ant . should not appear in your answe answer. r. ANSWER: =

,















=

Correct

Part E Assume that and . Which method, f ast pumpin pumping g (ad (adiabatic iabatic pro process) cess) or slow s low pumping (isothermal process) will put a larger amount of air into the tire per pump cycle?

Hint 1. Adiabatic 1. Adiabatic process Whatt is the numer Wha numerical ical value of

for the fast (adiaba (adiabatic) tic) process? process?

Express your answer numerically, to two significant figures. ANSWER: =

Hint 2. Isothermal process What is the numerical value of

for the slow (isothermal) process process? ?

Express your answer numerically, to two significant figures. ANSWER: =

ANSWER: fast pumping slow pumping

Correct So when you pump quickly, not only do you get more pump cycles per unit time, but you also put more air in the tire per cycle (at least according to this simplified model). Of course you have to pump not only faster but















To practice Problem-Solving Strategy 17.1: Work in Ideal-gas Processes. A cyli cylinder nder with initial volume contains a sample of a gas at pre pressure ssure . On one end of the cyl cylinder inder,, a piston is let free to move so that the gas slowly expands in such a way that its pressure is directly proportional to its volume. After the gas rea reaches ches the volume and pre pressure ssure , the piston is pushed in so that the gas is compressed isobarically to its original volume . The gas gas is then cooled cooled isochorically until it returns returns to the origin original al volume and pre pressure. ssure. Find the work done on the gas during the entire process process.. MODEL Ass Assume ume that t hat

the gas is ideal and the pro process cess is quasi-static.

Show the process on a pV a pV diagram. Note whether it happens to be one of the basic gas processes: isochoric, isobaric, or isothermal. VISUALIZE

Calculate the work as the area under the pV the pV curve either geometrically or by carrying out the integration:

SOLVE

ASSESS

Check your signs.

when the gas is compressed. Energy is transferred from the environment to the gas. when the gas expands. Energy is transferred from the gas to the environment. No work is done if the volume doesn't change, .

Model Assume Ass ume the t he gas is ideal. Since the pro processes cesses take place slowly, you can also assume they are quasi-static.

Visualize

Part A Which of the following pV following pV diagrams diagrams correctly represents the entire process described in this problem? ANSWER:































Since the entire process consis consists ts of three three separ separate ate steps, the total work work done on the gas is given by the sum of the amounts of work done on the gas during each step of the process. For each step, you can calculate the work as the negative of the area under the corresponding branch of the pV curve curve either geometrically or by carrying out the appropriate integration. Be careful with signs. You may want to review how to distinguish between positive and negative areas.

Hint 2. Positive and negative areas When a gas expands from an initi initial al volume positive,, positive

to a larger volume

, the area under the curve is

When a gas is compressed compressed from from an initial volume to a smaller volume , the calculation of the are area a under the pV the pV curve curve is a little trickier because it requires to integrate “backward” along the V axis. axis. You learned in calculus that integrating from a larger limit to a smaller limit gives a negative result, so in this case the area under the pV the pV curve is a negative negative area. area. We express this mathematically as

Hint 3. Find the work done during expansion from Find

to

, the work done on the gas as it expands from volume

Express your answer in terms of

and

to volume

.

.

Hint 1. How to compute the work Identify the branch of the pV curve that corresponds to this process. Compute the area under that portion of the curve either geometrically or by carrying out the appropriate integration. Be careful to distinguish between positive and negative areas. Finally, the work done on the gas during this process will be equal to the negative of that area. Note that in this particular problem, where the curve of interest is a straight line, it is easier to use geometry, rather than calculus, to compute areas.

ANSWER:















ANSWER: =

Hint 5. Find the work done during isochoric process Find

, the work done on the gas during isoc isochoric horic process process..

Express your answer in terms of

and

, or appropri appropriate ate constants.

Hint 1. Work in isochoric processes Recall that wor work k is the nega negative tive integral integral of . Since the volume is not not changing dur during ing an isochoric pro process cess,, wha whatt value will the integra integrand nd have at any instant in this process? Wha Whatt would then the integral of be equal to?

ANSWER: =

ANSWER: =

Correct

Assess















Correct In both the initial expansion and the isobaric process, the gas interacts with the environment both mechanically (the gas expands when the piston is allowed to move and is compressed when the piston is pushed in) and thermally (the gas gains or loses thermal energy). Since we can tell what is happening to the overall energy of the gas by paying attention to the temperature we can tell which part (work or heating) was the dominant process in each part.

Isobaric, Isochoric, Isothermal, and Adiabatic Processes Learning Goal: To recognize various ty types pes of process processes es on

diagrams and to understand the relationshi relationship p between

-diagram















can be added to the gas in the form of heat by applying a flame to the outside of the container. Conversely, energy can also be removed from the gas in the form of heat by immersing the container in ice water. Energy can be added to the system in the form of work by pushing the piston in, thereby compressing the gas. Conversely, if the gas pushes the piston out, thereby pushing some atmosphere aside, the internal energy of the gas is reduced by the amount of work done. The internal energy of an ideal gas gas is directly proportional to its absolute absolut e temperature . An ideal gas also obeys the ideal gas law , so the absolute temperature is directl directly y propor proportional tional to the product of the absolute pressure and the volume . Here denotes denotes the amoun amountt of gas gas in moles, moles, which is a constant because because the gas is confined, confined, and is the universal universal gas gas constant. A diagram diagr am is a convenient way to track the pre pressure ssure and volume of a system. Energy transfers by heat and/or work are associated with processes, which are lines or curves on the diagram diagr am taking the sys system tem from one state (i.e., one point on the diagram) to another. Work corresponds geometrically geometrical ly to the area under the curve on a diagram. If the volume increases (i.e., the system expands) the work will be classified as an energy output from the system.

Part A What is the sign of What proportional propor tional to .

as the sys system tem of ideal gas goes from point A to point B on the gra graph? ph? Recall that

Hint 1. How to approach the problem

is

















Correct The value of depends depends only on the state of the system. sys tem. Thus depends depends only on the endpoint endpoint states, not on the process followed that determines the path between the endpoint states.

One possible way for the system to get from state A to state B is to follow a hyperbolic curve through point C, along which the prod product uct of is a constant. Temper Temperature ature is proportion pro portional al to the pro product duct , so this is a constanttemperature path, also known as an isothermal process.

Part B















Both

and

equal zero.

Both

and

provide energy input.

Both

and

provide energy output.

provides energy energy output, while while provides energy energy input, while while

provides energy input. They are equal in magnitude. provides energy energy output. They are equal in magnitude.

Correct You can tell that the system is losing internal energy due to work because its volume is increasing. The internal energy change during any isothermal process involving an ideal gas is zero, so here the system must gain as much energy in the form of heat as it loses by doing work during this process.

Another way to get from state A to state B is to go vertically v ertically from A to point D, holding volume constant, and then go horizontally to point B, holding pressure constant. A constantvolume path is called an isochoric process. A constantpressure path is called an isobaric process.















Both

and

equal zero.

provides energy input, while provides energy output, while provides energy input, while provides energy output, while

equals zero. equals zero. provides energy output. provides energy input.

Correct You can tell that that the system sys tem is losing internal internal ener energy gy since its temperature temperature goes down down (since goes down). down). No work is done during any isochoric process, since no area accumulates under a vertical curve. Hence energy transfer in the form of heat must account for the entire internal energy change.

Part D How are

and

related during the isobaric part of the overall path from st state ate D to st state ate B?

ANSWER: Both

and

provide energy input.

Both

and

provide energy output.

provides energy energy output, while while

provides energy input. They are equal in magnitude.

provides energy output, while

provides energy input;

is larger.















Part E Which of the following statements are true about the isochoric part of the overall path, from state E to state B? Check all that apply.

Hint 1. How to approach the problem Recall that the total internal energy change from state A to state B is zero. This means that the isochoric process must undo any changes to internal energy made during the adiabatic process.















Part F Which of the following statements are true about the first half of this process, just going from state A to state F? Check all that apply. ANSWER: Both

and

increase.

provides energy input.

















Isobaric : . The pressure does not change. (A barometer measures the pressure.) Isochoric : . The volume does not change change.. (This process is infrequently used.)

The key idea in determining which of these processes is occurring from a pV a pV plot plot is to recall that an ideal gas must obey the ideal gas equation of st state: ate: (where (wher e the constant is the Boltzmann constant , which has the value in SI units). Gene Generally rally , the number of gas particles, is held constant, so you can determine what happens to at various points along the curve on the pV the pV diagram. diagram. Note that in this problem, as is usually assumed, the processes happen slowly enough that the gas remains in equilibrium without hot spots, without propagating pressure waves from a rapid change in volume, or without involving















Hint 2. Use the ideal gas law Recall the ideal gas equa equation tion of state for a fixed amoun amountt of gas: , whe where re is some constant. Solving this equa equation tion for yields . If is to be pro proportion portional al to , the temper temperature ature must remain constant throughout the process. What is the name of a process in which temperature remains constant?

ANSWER: adiabatic isobaric isochoric isothermal

Correct















greater than zero less than zero equal to zero

Correct

Part E The temperature of the system in state B is __________ the temperature of the system in state O.

Hint 1. Compare curve OB to an isothermal process Does curve OB lie above or below a curve representing an isothermal (constant-temperature) process? Use















greater than less than equal to

Correct

Piston in Water Bath Conceptual Work-Energy Problem Imagine a piston containing a sample of ideal gas in thermal equilibrium with a large water bath. Assume that the piston head is perfectly free to move unless locked in place, and the walls of the piston readily allow the transfer of energy via heat unless wrapped in insulation. The piston head is unlocked and the gas is in an equilibrium state.















Heat energy can either flow into or out of a gas sample. When the energy flows into the gas, it is considered positive heat; when the energy flows out of the gas it is considered negative heat.

ANSWER:

Hint 3. Find the sign of the change in internal energy of the gas Based on the signs of work Give the sign of

and heat

, what must be the sign of change in internal energy

?

. Answer with +, ‐, or 0.

Hint 1. First law of thermodynamics The first law of thermo thermodynamics dynamics states that the change

in interna internall ener energy gy of a gas is equal equal to

















considered positive, and when the work is done on the gas this work is considered negative.

ANSWER:















Part C Action: Lock the piston head in place. Plunge the piston into very cold water water.. Enter the signs of is negative negative,, and

, , and . Use , , or 0 separ separated ated by commas. For example, if is zero, you would type +,‐,0.

Hint 1. Find the sign of the work done by the gas

is positive,















Hint 1. Find the sign of the work done by the gas If the piston head is pulled up, will the work done by the gas be positive, negative, or zero? Give the sign of ANSWER:

. Answer with +, ‐, or 0.





























Hint 2. Find the change in volume





























































































































Mastering Physics HW 3 Ch 17, First Law of Thermodynamics

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