Unformatted text preview: 9/11/2015 Assignment 3 Chapter 19 Assignment 3 Chapter 19
Due: 1:00pm on Wednesday, September 16, 2015
You will receive no credit for items you complete after the assignment is due. Grading Policy Item 1
The diagram shows the pressure and volume of an ideal gas during one cycle of an engine. As the gas proceeds from state 1 to state 2, it is
heated at constant pressure. It is then cooled at constant volume, until it reaches state 3. The gas is then cooled at constant pressure to state 4.
Finally, the gas is heated at constant volume until it returns to state 1. Part A
Find W12 , the work done by the gas as it expands from state 1 to state 2.
Express the work done in terms of p0 and V 0 . Hint 1. Relating work, pressure, and volume
If the pressure of a gas is p, and its infinitesimal change in volume is dV , what is dW , the infinitesimal work done by the gas?
ANSWER:
dW = pdV 1/21 9/11/2015 Assignment 3 Chapter 19 Hint 2. Doing the integration
To find the work done by the gas as it expands from state 1 to state 2, multiply the gas pressure by the change in volume between states 1 and 2. When you do this, what value should you use
for the pressure p?
ANSWER:
p = 3p0 ANSWER:
W12 = 9p0 V 0 Correct Part B
Find W23 , the work done by the gas as it cools from state 2 to state 3.
Express your answer in terms of p0 and V 0 . Hint 1. Volume of the gas
Note that the volume of the gas remains constant during this part of the cycle.
ANSWER:
W23 = 0 Correct Part C
W
2/21 9/11/2015 Assignment 3 Chapter 19 Find W34 , the work done by the gas as it is compressed from state 3 to state 4.
Express your answer in terms of p0 and V 0 .
ANSWER:
W34 = −3p0 V 0 Correct Part D
Find W41 , the work done by the gas as it is heated from state 4 to state 1.
Express your answer in terms of p0 and V 0 .
ANSWER:
W41 = 0 Correct Part E
What is Wnet , the total work done by the gas during one cycle?
Express your answer in terms of p0 and V 0 .
ANSWER:
Wnet = 6p V 0
0 Correct
Notice that the net work done by the gas during this cycle is equal to the area of the rectangle that appears in the pV (pressurevolume) diagram. (The width of this rectangle is 3V 0 , and its height
is 2p0 .) In other words, the net work done by this gas is equal to the net area under its pV curve. For a cycle, this is equivalent to the (signed) area enclosed by the cycle.
3/21 9/11/2015 Assignment 3 Chapter 19 Part F
When the gas is in state 1, its temperature is T 1 . Find the temperature T 3 of the gas when it is in state 3. (Remember, this is an ideal gas.)
Express T 3 in terms of T 1 . Hint 1. Equation of state in terms of p0 and V 0
If an ideal gas is held at a fixed number of moles, then its equation of state is pV = cT , where c is some constant. For the gas given, find an expression for c in terms of given quantities. Express c in terms of p0 , V 0 , and T 1 .
ANSWER: c 3p 0 V 0 = T1 ANSWER:
T3 = 4
3 T1 Correct Item 2
A gas in a cylinder is held at a constant pressure of 2.29×105 Pa and is cooled and compressed from a volume of 1.70 m 3 to 1.28 m 3 . The internal energy of the gas decreases by 1.30×105 J . Part A
Find the work done by the gas.
Express your answer using three significant figures.
ANSWER:
4/21 9/11/2015 Assignment 3 Chapter 19
W = −9.62×104 J Correct Part B
Find the absolute value Q of the heat flow into or out of the gas.
Express your answer using three significant figures.
ANSWER:
Q = 2.26×105 J Correct Part C
State the direction of the heat flow.
ANSWER:
into the gas
out of the gas Correct Part D
Does it matter whether the gas is ideal?
ANSWER:
5/21 9/11/2015 Assignment 3 Chapter 19 yes
no Correct Item 3
We start with 5.00 moles of an ideal monatomic gas with an initial temperature of 129 ∘ C . The gas expands and, in the process, absorbs an amount of heat equal to 1240 J and does an amount of work
equal to 2120 J . Part A
What is the final temperature T f inal of the gas?
Use R = 8.3145 J/(mol ⋅ K) for the ideal gas constant. Hint 1. First law of thermodynamics
The first law of thermodynamics for an ideal gas system is given by the equation
U 2 − U 1 = ΔU = Q − W,
where U 2 is the final total internal energy, U 1 is the initial total internal energy, 1240 J is the heat added to the system, and 2120 J is the work the gas system does on its surroundings. Thus,
when we add heat without the gas doing any work, the internal energy goes up. If the gas does work on the surroundings by expanding without any addition of heat, the internal energy goes
down. Hint 2. Find the change in internal energy
What is the total change in internal energy ΔU of the gas?
Express your answer in joules.
ANSWER:
ΔU = 880 J Hint 3. Calculate the change in temperature
ΔT
6/21 9/11/2015 Assignment 3 Chapter 19 Calculate the change in temperature ΔT of the gas.
Express your answer in degrees Celsius. Hint 1. Equipartition Theorem
For an ideal gas with three degrees of freedom (one for each of the dimensionsx, y, and zthat the atoms can move in) the Equipartition Theorem states that the internal energy of the
gas is given by
E int = 3
2 nRT , where 5.00 is the amount of gas in moles, 8.3145 J/(mol ⋅ K) is the ideal gas constant, and 129 ∘ C is the temperature.
ANSWER:
ΔT = 14.1 ∘ C ANSWER:
T f inal = 115 ∘ C Correct Item 4
On a warm summer day, a large mass of air (atmospheric pressure 1.01 × 10 5 Pa ) is heated by the ground to a temperature of 26.0 ∘ C and then begins to rise through the cooler surrounding air. Part A
Calculate the temperature of the air mass when it has risen to a level at which atmospheric pressure is only 8.10×104 Pa . Assume that air is an ideal gas, with γ
rising air, corresponding to roughly 1 ∘ C per 100 m of altitude, is called the dry adiabatic lapse rate.) . (This rate of cooling for dry, = 1.40 ANSWER: 7/21 9/11/2015 Assignment 3 Chapter 19
T = 7.72 ∘ C Correct Item 5
A large research balloon containing 2.00 × 10 3 m 3 of helium gas at 1.00 atm and a temperature of 15.0 ∘ C rises rapidly from ground level to an altitude at which the atmospheric pressure is only 0.900 atm (the figure ). Assume the helium behaves like an ideal gas and the balloon's ascent is too rapid to permit much heat exchange with the
surrounding air. Part A
Calculate the volume of the gas at the higher altitude.
ANSWER:
V = 2130 m 3 Correct
8/21 9/11/2015 Assignment 3 Chapter 19 Part B
Calculate the temperature of the gas at the higher altitude.
ANSWER:
T = 3.07 ∘ C Correct Part C
What is the change in internal energy of the helium as the balloon rises to the higher altitude?
ANSWER:
ΔU = −1.25×107 J Correct Item 6
Learning Goal:
To understand the meaning and the basic applications of pV diagrams for an ideal gas.
As you know, the parameters of an ideal gas are described by the equation
pV = nRT , where p is the pressure of the gas, V is the volume of the gas, n is the number of moles, R is the universal gas constant, and T is the absolute temperature of the gas. It follows that, for a portion of an
ideal gas,
pV
T = constant . One can see that, if the amount of gas remains constant, it is impossible to change just one parameter of the gas: At least one more parameter would also change. For instance, if the pressure of the gas
9/21 9/11/2015 Assignment 3 Chapter 19 is changed, we can be sure that either the volume or the temperature of the gas (or, maybe, both!) would also change.
To explore these changes, it is often convenient to draw a graph showing one parameter as a function of the other. Although there are many choices of axes, the most common one is a plot of pressure as
a function of volume: a pV diagram.
In this problem, you will be asked a series of questions related to different processes shown on a pV diagram . They will help you become familiar
with such diagrams and to understand what information may be obtained from them. One important use for pV diagrams is in calculating work. The product pV has the units of Pa × m 3 = (N/m 2 ) ⋅ m 3 = N ⋅ m = J; in fact, the absolute value of the work done by the gas (or on the gas)
during any process equals the area under the graph corresponding to that process on the pV diagram. If the gas increases in volume, it does positive work; if the volume decreases, the gas does negative
work (or, in other words, work is being done on the gas). If the volume does not change, the work done is zero.
The following questions may seem repetitive; however, they will provide practice. Also, the results of these calculations may be helpful in the final section of the problem. Part A
Calculate the work W done by the gas during process 1→2 .
Express your answer in terms of p0 and V 0 .
ANSWER:
W = 6p V 0
0 Correct
10/21 9/11/2015 Assignment 3 Chapter 19 Part B
Calculate the work W done by the gas during process 2→1 .
Express your answer in terms of p0 and V 0 .
ANSWER:
W = −6p V 0
0 Correct
Compare your result with that from part A. The work WAB done during a process A→B is equal to −WBA , the work done during the reverse process B→A . Part C
Calculate the work W done by the gas during process 5→6 .
Express your answer in terms of p0 and V 0 .
ANSWER:
W = 2p V 0
0 Correct Part D
Calculate the work W done by the gas during process 1→3→6 .
Express your answer in terms of p0 and V 0 .
ANSWER:
W = 4p V 0
0 11/21 9/11/2015 Assignment 3 Chapter 19 Correct Part E
Calculate the work W done by the gas during process 2→6 .
Express your answer in terms of p0 and V 0 .
ANSWER:
W = 0 Correct
No work is done during a process, if the gas does not experience a change in volume. The absolute value of the work done by the gas during a cycle (a process in which the gas returns to its original state) equals the area of the loop corresponding to the cycle. One must be careful, though, in
judging whether the work done by the gas is positive or negative. One way to determine the total work is to calculate directly the work done by the gas during each step for the cycle and then add the
results with their respective signs. Part F
Calculate the work W done by the gas during process 1→2→6→5→1 .
Express your answer in terms of p0 and V 0 .
ANSWER:
W = 4p V 0
0 Correct
This result can be obtained either by calculating the area of the region 1265 or by adding the amounts of work done by the gas during each process of the cycle. The latter method helps verify that
the net work done by the gas is, indeed, positive.
As discovered earlier, The work W15621 done during a process 1→5→6→2→1 is equal to −W12651 , the work done during the reverse process 1→2→6→5→1 .
12/21 9/11/2015 Assignment 3 Chapter 19 Part G
Calculate the work W done by the gas during process 1→2→6→3→1 .
Express your answer in terms of p0 and V 0 .
ANSWER:
W = 2p0 V 0 Correct Item 7
Five moles of an ideal monatomic gas with an initial temperature of 121 ∘ C expand and, in the process, absorb an amount of heat equal to 1280 J and do an amount of work equal to 2080 J . Part A
What is the final temperature of the gas?
ANSWER:
T = 108 ∘ C Correct Item 8 Part A
An ideal gas expands through an adiabatic process. Which of the following statements is/are true?
Check all that apply.
13/21 9/11/2015 Assignment 3 Chapter 19 Hint 1. How to approach the problem
To determine the correct statement(s) you need to apply the first law of thermodynamics. Note that when a gas expands it does work on its surroundings. Hint 2. First law of thermodynamics
When heat Q is added to a system, some of this added energy goes to increase the internal energy of the system by an amount ΔU . The remaining energy leaves the system as the system
does work W on its surroundings. Thus, we have
ΔU = Q − W .
Since W and Q may be positive, negative, or zero, we can also expect ΔU to be positive, negative or zero, depending on the process. Hint 3. Adiabatic process
An adiabatic process is a thermodynamic process in which no heat exchange occurs.
ANSWER:
The work done by the gas is negative, and heat must be added to the system.
The work done by the gas is positive, and no heat exchange occurs.
The internal energy of the system has increased.
The internal energy of the system has decreased. Correct Part B
After the adiabatic expansion described in the previous part, the system undergoes a compression that brings it back to its original state. Which of the following statements is/are true?
Check all that apply. Hint 1. Internal energy in cyclic processes
A process, or a sequence of processes, that brings the system back to its original state is called a cyclic process. In a cyclic process the total internal energy change is zero. 14/21 9/11/2015 Assignment 3 Chapter 19 ANSWER:
Thetotal change in internal energy of the system after the entire process of expansion and compression must be zero.
The total change in internal energy of the system after the entire process of expansion and compression must be negative.
The total change in temperature of the system after the entire process of expansion and compression must be positive.
The total work done by the system must equal the amount of heat exchanged during the entire process of expansion and compression. Correct Item 9
In an experiment to simulate conditions within an automobile engine, 0.170 mol of air at a temperature of 710 K and a pressure of 3.10×106 Pa is contained in a cylinder of volume 320 cm 3 . Then 630 J
of heat is transferred to the cylinder. Part A
If the volume of the cylinder is constant while the heat is added, what is the final temperature of the air? Assume that the air is essentially nitrogen gas.
ANSWER:
T = 888 K Correct Part B
If instead the volume of the cylinder is allowed to increase while the pressure remains constant, find the final temperature of the air.
ANSWER:
T = 837 K 15/21 9/11/2015 Assignment 3 Chapter 19 Correct Item 10
A thermodynamic system is taken from state a to state c in the figure along either path abc or path adc. Along path abc the work W done by the
system is 450 J. Along path adc,W is 120 J. The internal energies of each of the four states shown in the figure are U a = 150 J, U b = 240 J, U c = 680 J, and U d = 330 J. Part A
Calculate the heat flow Q for the process ab.
ANSWER:
Q = 90 J Correct Part B
In the process ab, does the system absorb or liberate heat?
16/21 9/11/2015 Assignment 3 Chapter 19 ANSWER:
System absorbs heat
System liberates heat Correct Part C
Calculate the heat flow Q for the process bc.
ANSWER:
Q = 890 J Correct Part D
In the process bc, does the system absorb or liberate heat?
ANSWER:
System absorbs heat
System liberates heat Correct Part E
Calculate the heat flow Q for the process ad.
17/21 9/11/2015 Assignment 3 Chapter 19 ANSWER:
Q = 300 J Correct Part F
In the process ad, does the system absorb or liberate heat?
ANSWER:
System absorbs heat
System liberates heat Correct Part G
Calculate the heat flow Q for the process dc.
ANSWER:
Q = 350 J Correct Part H
In the process dc, does the system absorb or liberate heat?
ANSWER: 18/21 9/11/2015 Assignment 3 Chapter 19 System absorbs heat
System liberates heat Correct Item 11
Three moles of argon gas (assumed to be an ideal gas) originally at a pressure of 1.50 × 10 4 Pa and a volume of 3.00×10−2 m 3 are first heated and expanded at constant pressure to a volume of
4.30×10−2 m 3 , then heated at constant volume until the pressure reaches 3.50 × 10 4 Pa, then cooled and compressed at constant pressure until the volume is again 3.00×10−2 m 3 , and finally cooled at
constant volume until the pressure drops to its original value of 1.50 × 10 4 Pa. Part A
Calculate the total work done by the gas during the cycle.
ANSWER:
W = 260 J Correct Part B
Calculate the net heat exchanged with the surroundings.
ANSWER:
Q = 260 J Correct 19/21 9/11/2015 Assignment 3 Chapter 19 Part C
Does the gas gain or lose heat overall?
ANSWER:
gas gains heat
gas loses heat Correct Item 12
Six moles of an ideal gas are in a cylinder fitted at one end with a movable piston. The initial temperature of the gas is 27.3 ∘ C and the pressure is constant. Part A
As part of a machine design project, calculate the final temperature of the gas after it has done 1720 J .
Express your answer using three significant figures.
ANSWER:
T = 61.8 ∘ C Correct Item 13
An experimenter adds 940 J of heat to 1.75 mol of an ideal gas to heat it from 10.0 ∘ C to 25.0 ∘ C at constant pressure. The gas does 224 J of work during the expansion. Part A
Calculate the change in internal energy of the gas.
20/21 9/11/2015 Assignment 3 Chapter 19 ANSWER:
ΔU = 716 J Correct Part B
Calculate γ for the gas.
ANSWER:
γ = 1.31 Correct
Score Summary:
Your score on this assignment is 99.6%.
You received 99.61 out of a possible total of 100 points. 21/21 ...
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