Heat and Its Mechanical...

BCLN PHYSICS - Rev. Nov/2012

Thermodynamics ~ Learning Guide Name: _______________

Instructions: Using a pencil, answer the following questions. The guide is marked based on effort, completeness, thoughtfulness, and neatness (not accuracy). Do your best!

Heat and Its Mechanical Equivalent:

1. A birthday sparkler burns at temperatures around 2000 ºC. Why are they considered to be safe for a child’s birthday party ?

2. We have seen that a fire can be started by utilizing the friction generated when stroking two sticks together. A rough calculation for determining the temperature rise with each “stroke” might look like the one below:

½ mv2 = mcΔT a. The equation above is only works for ideal conditions. State the assumptions that are being

made with this equation. Comment on the variables used in the equation and their validity.

b. Modify the equation to accommodate for at least one assumption stated above.


3. The 1.20 kg head of a hammer is swung from a string attached to a rod like a pendulum as shown below

a. Describe the energy transformations that take place when the hammer is released from its

horizontal position until it strikes the nail.

b. If the distance from the rod to the head of the hammer is 2m, what will the final temperature of the 14 g nail be if it is struck in this fashion 10 times consecutively? The nail is initially at room temperature (20 ºC) and is made of iron (c = 450 J/(kg Cº). You can assume that it absorbs all of the energy. (ans: 57.3 ºC)

c. The actual final temperature of the nail would be higher/lower? (circle) Discuss why you think this is so by addressing the effects of any assumptions made above.


4. Redesign Joules Device (shown below) to maximize the temperature rise of the liquid as induced by the falling masses. State the change you would make and its effect in terms of energy.

Change to apparatus Effect (in terms of energy)

5. The specific heat capacity, c, is often called the mass’s “thermal inertia”. Explain why this nickname makes sense based on our current definition of inertia.


6. A 1200W kettle is 80% efficient. How long would this kettle take to boil 500g of water initially at room temperature? The specific heat of water is 4186 J/(kg Cº) (ans: 174 s or 2 min 54 s)

7. Explain why your car is better off using water as a coolant rather than mercury. Be sure to look up the two specific heat values and compare.

Heat Transfer and Thermal Expansion:

1. The video demonstration on conduction shows heating a rod with a candle at one end. Describe how the heat moves through the conductor by discuss the motion on the particles in the rod.


2. Having watched the videos found in the tutorial, design an experiment to test how the rate of heat flow, Q/t, is related to the difference in the temperature, ΔT. Sketch the apparatus and clearly label the parts. Discuss your variables, your constants and outline a basic procedure.

Diagram (fully labeled) Variables and Constants (brief explanation as to the nature of your set-up in terms of controls) Procedure

3. Watch the video that compares to rods of different materials. What is meant by thermal conductivity, k? Which metal has the higher thermal conductivity? How do you know?


4. Create an experiment to test the relationship between the area, A, of a poor conductor (or an insulator), and the rate of heat flow, Q/t.

Diagram (fully labeled) Variables and Constants (brief explanation as to the nature of your set-up in terms of controls) Procedure

5. A good absorber is said to be a good emitter. A Crookes radiometer is set into motion with a flashlight. After it gets up to speed, the flashlight is removed. The radiometer slows down and then begins to spin backwards. Why?


6. What is the main reason why convection does not occur in solids?

7. Discuss how the thermostat picture below works to allow us to read the temperature? Be sure to explain the choices of the metals used.

8. A square steel plate with sides of length 1.00 m has a hole in its center 0.100 m in diameter. If the entire plate is heated to such a temperature that its sides become 1.01 m long, the diameter of the hole will become what? Hint: if you understand the equation you need not calculate your final answer.

Ideal Gases and the Kinetic Molecular Theory:

1. a. Calculate the rms speed of an oxygen molecule at 0 ºC. (ans: 461 m/s)

b. Determine how many times per second it would move back and forth across a 7.0 m long room, on average, assuming that it made very few collisions with other molecules. (ans: 66 times/sec)


2.

a. If MW is the atomic mass of each gas, which gas contains more molecules and why?

b. What can we assume about the volume occupied by each gas mixed together in the beaker?

c. We assume the gas has reached equilibrium. What does this mean?

d. Based on your answer to (c) above, comment on the kinetic energy possessed by each gas (which is larger, or are they the same and why). Be sure to include the appropriate formula to back up your answer.

e. On average, discuss the velocity possessed by the molecules of each gas. Be sure to include the appropriate formula to back up your answer.

f. The Ideal Gas Law states that PV = nRT. Based on your answers above, which gas will exert more pressure or are they the same?


Ideal Gas Law:

1. Explain in words how Charles’s Law follows from the kinetic theory of gases and the relation between average kinetic energy and the absolute temperature. Let’s answer this question by first working through the questions below: a. What happens to the average kinetic energy of the molecules as the temperature increases?

Back this up with an equation.

b. Describe how (a) affects the “strength” of the gas molecule collisions.

c. Explain how your answer above affects the average distance between the molecules.

d. Tie it all together – How does temperature affect volume in terms of the kinetic theory?

2. Explain in words how Gay-Lussac’s Law follows from kinetic theory (see above).


3. How might your answers above change if the collisions between the walls and the molecules were not perfectly elastic?

4.

This is a question about the basic mechanics of setting up the Ideal gas Law equation. Explain why you choose your answer briefly below.

5. Two containers hold equal masses of nitrogen gas at equal temperatures. You supply 10 J of heat to container A while not allowing its volume to change. You supply 10 J of heat to container B while not allowing its pressure to change. Afterward, is the temperature of container A greater than, less than, or equal to the temperature of container B? Justify your answer.


6. Rank the temperatures of the ideal gases below that contain a different number of molecules (N) at various pressures (P) and volumes (V).

What concepts in Physics are appropriate for this question (include equations)

Based on your answer above, what criteria did you use to rank the gases?

Greatest 1______ 2______ 3______ 4______ 5______ 6______ 7______ 8______ Least


7. Rank the temperatures of the ideal gases below that contain various amounts of internal energy (U) and various numbers of molecules (N) in various volumes.

What concepts in Physics are appropriate for this question (include equations)

Based on your answer above, what criteria did you use to rank the gases?

Greatest 1______ 2______ 3______ 4______ 5______ 6______ 7______ 8______ Least


8. Repeat the above question except, this time, rank them according to pressure. Explain the criteria that you used to rank the gases.

Greatest 1______ 2______ 3______ 4______ 5______ 6______ 7______ 8______ Least

9.

The inside of the cylindrical can shown above has cross-sectional area 0.005 m2 and length 0.15 m. The can is filled with an ideal gas and covered with a loose cap. The gas is heated to 363 K and some is allowed to escape from the can so that the remaining gas reaches atmospheric pressure (1.0 × 105 Pa). The cap is now tightened, and the gas is cooled to 298 K.

a. What is the pressure of the cooled gas? (ans: 8.2 × 104 Pa)

b. Determine the upward force exerted on the cap by the cooled gas inside the can. (ans: 410N)

c. If the cap develops a leak, how many moles of air would enter the can as it reaches a final equilibrium at 298 K and atmospheric pressure? (Assume that air is an ideal gas.) (ans: 5.45 × 10–3 mol)


The Laws of Thermodynamics:

1. Write the first law of thermodynamics, identifying each variable.

2. Which variable in the first law of thermodynamics is related to the temperature of the gas?

3. I put a container of gas, initially at room temperature, into an ice bath. a. Which variable in the first law of thermodynamics have I affected?

b. What is the sign of that variable?

4. The volume of a gas contracts. a. Was work done on, or by, the gas? Choose one of the following.

Work was done on the gas Work was done by the gas No work was done on or by the gas

b. What is the sign of the variable W in the first law of thermodynamics?

Positive Negative Zero

5. How do you determine the sign of ∆U from a pV diagram?


6.

Pressure

P0

P02

V0 2V0

Volume

I

IV II

III

T0

Four samples of ideal gas are each initially at a pressure P0, a volume V0, and a temperature T0 as shown on the diagram above. The samples are taken in separate experiments from this initial state to the final states I, II, III, and IV along the processes shown on the diagram.

a. Use the Ideal Gas law to determine which process ends with the highest final temperature.

b. What “normally” happens to the temperature of a gas when it expands quickly? Examine process II and III. Of these two processes, which one reaches the lowest final temperature?

c. One of the processes is isothermal. Identify this one and explain. In order to remain at a constant

temperature. Does heat needed to be added or removed during this process?

d. One of the processes is adiabatic. Identify this one and explain.

e. In which processes or processes is work done by the gas? Explain.

f. In which processes, II or III, was more work done by the gas? Use the first Law of Thermodynamics to explain your answer.

g. In which processes or processes does the internal energy of the gas increase? Explain.


7. Five thermodynamic processes are illustrated below. All five processes are for the same ideal gas starting at the same initial pressure and temperature (Po and To). Each of the five distinctly different processes results in different final equilibrium states. The labels on each of the diagrams are to be interpreted as follows:

A = adiabatic

L = linear IT = isothermal

IV = isovolumetric IB = isobaric

Rank these processes from greatest to least on the basis of the amount of work that is done by the gas. Positive work should be ranked higher than negative work.

State the physics concepts and/or equations appropriate for the above question. Based on the concepts above, what criteria was used to determine the rankings of the gas processes? Greatest 1______ 2______ 3______ 4______ 5______ Least


8. An ideal gas initially has pressure po, volume Vo, and absolute temperature To. It then undergoes the following series of processes:

I. It is heated, at constant volume, until it reaches a pressure 2po.

II. It is heated, at constant pressure, until it reaches a volume 3 Vo.

III. It is cooled, at constant volume, until it reaches a pressure po.

IV. It is cooled, at constant pressure, until it reaches a volume Vo.

a. On the axes below i. draw the p-V diagram representing the series of processes; ii. label each end point with the appropriate value of absolute temperature in terms of To.

b. For this series of processes, determine the following in terms of po and Vo. i. The net work done on the gas (ans: –2P0V0)

ii. The net change in internal energy (ans: 0)

iii. The net heat absorbed (ans: +2P0V0)

c. Determine the heat transferred during process 2 in terms of po and Vo. (ans: +10P0V0)


9. A cylinder with a movable piston contains 0.1 mole of a monatomic ideal gas. The gas, initially at state a, can be taken through either of two cycles, abca or abcda, as shown on the PV diagram below.

The following information is known about this system:

Qc→a = 685 J along the curved path Wc→a = –120 J along the curved path Ua – Ub = 450 J Wa→b→c = 75 J

a. Determine the change in internal energy, Ua – Uc, between states a and c. (ans: 565 J) i. Is heat added to or removed from the gas when the gas is taken along the path abc ?

_____added to the gas _____removed from the gas

Explain your reasoning ii. Calculate the amount added or removed. (ans: –640 J)

b. How much work is done on the gas in the process cda? (ans: –150 J)

d. Is heat added to or removed from the gas when the gas is taken along the path cda? _____added to the gas _____removed from the gas Explain your reasoning.


10. A 0.03 mol sample of helium is taken through the cycle shown in the diagram below. The temperature of state A is 400 K.

a. For each process in this cycle, indicate in the table below whether the quantities W, Q,

and ΔU are positive (+), negative (–), or zero (0). W is the work done on the helium sample.

b. Explain your response for the signs of the quantities for process AB.

c. Calculate Vc.


11.

You are given a cylinder of cross-sectional area A containing n moles of an ideal gas. A piston fitting closely in the cylinder is lightweight and frictionless, and objects of different mass m can be placed on top of it, as shown in the figure above. In order to determine n, you perform an experiment that consists of adding 1 kg masses one at a time on top of the piston, compressing the gas, and allowing the gas to return to room temperature, T, before measuring the new volume V. The data collected are given in the table below.

m (kg) V (m3) 1/V (m–3) P (Pa)

0 6.0 × 10–5 1.7 × 104

1 4.5 × 10–5 2.2 × 104

2 3.6 × 10–5 2.8 × 104

3 3.0 × 10–5 3.3 × 104

4 2.6 × 10–5 3.8 × 104

a. Write a relationship between total pressure P and volume V in terms of the given quantities and fundamental constants that will allow you to determine n.

You also determine that A = 3.0 × 10–4 m2 and T = 300 K.

b. Calculate the value of P for each value of m and record your values in the data table above. (remember to add-on the additional pressure from the room itself, atmospheric pressure, Patm = 1.013x105 Pa) note: P = F/A


Plot the data on the graph below, labeling the axes with appropriate numbers to indicate the scale.

c. Using your graph in part (c), calculate the experimental value of n. (ans: n = .0025 moles)

Heat Engines and Efficiency:

1. An automobile engine has an efficiency of 22.0% and produces 2510 J of work. a. Define what efficiency means (in general). he input heat

b. What is the input heat for this engine? (ans: 11 400 J)

c. How much heat is rejected (exhausted) by the engine? (ans: 8900 J)


2. A PV diagram is shown below depicting a typical Carnot Cycle for an ideal heat engine.

a. Label each process for each line, ABCD, above. You may label directly onto the graph. b. Label (neatly and clearly), the sign of each term in our first law of thermodynamics. For each

cell in the table use either an equation or a simple explanation (one or two words) for your choice of sign

ΔU (explain) W (explain) Q (explain) Process A

Process B

Process C

Process D

c. Is the overall work done by this engine positive or negative? How can you tell?

d. Which processes occur at lower temperatures? AB? or CD?

e. Based on your answers for c and d above, why is it important for a heat engine to have a two different operating temperatures? What would be the result if TH = TC? What would the Carnot Efficiency be if this was the case? Explain the significance in terms of work.


3. A heat engine operating between temperatures of 500 K and 300 K is used to lift a 10-kilogram mass vertically at a constant speed of 4 meters per second. a. Determine the power that the engine must supply to lift the mass. (ans: 400 W) b. Determine the maximum possible efficiency at which the engine can operate. (ans; 40%) c. If the engine were to operate at the maximum possible efficiency, determine the following.

i. The rate at which the hot reservoir supplies heat to the engine (ans: 1000 W) ii. The rate at which heat is exhausted to the cold reservoir (ans: 600 W)

4. An engine operates on the cycle shown in the PV diagram below. The working substance of the engine is an ideal monatomic gas. The processes A → B and C → D are isobaric, while processes B → C and D → A are isochoric.

a. What is the efficiency of a Carnot engine operating between the same maximum and

minimum temperatures as this engine? (ans: e = 5/6)

b. What is the actual efficiency of this engine? (ans: e = 4/21)

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