Assignment 04

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course PHY 232

If your solution to stated problem does not match the given solution, you should self-critique per instructions athttp://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

Your solution, attempt at solution:

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

004. `query 4

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Question:

query intro set substance, water, both temperatures and masses known, final temperature known, find spec ht

Explain how the final temperature of the system is combined with the known initial temperatures and masses of both substances to obtain the unknownspecific heat

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Your Solution:

The change in thermal energy is equal to, dQ=m1c1dT1 + m2c2dT2. If the system is adiabatic or in a closed system, dQ=0, therefore there is no change in heat. The specific heats can then be solved for algebraically as long as the final temperature, initial temperatures, masses, and 1 of the specific heats are known. 0=m1c1dT1 + m2c2dT2c2= -m1c1dT1/m2dT2

confidence rating #$&*:

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confidence rating #$&*: (Type in a number from 0 to 3, indicating your level of confidence in your solution.

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3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Given Solution:

** The change in the thermal energy of an isolated system is 0. So assuming that the systems are isolated the thermal energy change of one object is equaland opposite to that of the other.

For an ideal substance the change in the thermal energy of an object is directly proportional to the change in its temperature and to its mass (i.e., more mass and more change in temperature implies more change in thermal energy). The specific heat is the proportionality constant for the substance. Using `dQ for thechange in the thermal energy of an object we can express this as

• `dQ = mass * specific heat * `dT.

(General College and University Physics students note that most substances do not quite behave in this ideal fashion; for most substances the specific heat is not in fact strictly constant and for most substances changes with temperature.)

For two objects combined in a closed system we have `dQ1 + `dQ2 = 0, which gives us the equation

• m1 c1 `dT1 + m2 c2 `dT2 = 0

or equivalently

• m1 c1 `dT1 = - m2 c2 `dT2.

That is, whatever energy one substance loses, the other gains.

In this situation we know the specific heat of water, the two temperature changes and the two masses. We can therefore solve this equation for specific heat c2 of the unknown substance. **

Your Self-Critique:

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Self-critique (if necessary): (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

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Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

• 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

• 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

• 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

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Question: prin and gen problem 14.3: 2500 Cal per day is how many Joules per day? At a dime per kilowatt hour, how much would this cost?

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Your Solution:

One Cal is about 4200 J, so 2500 Cal is about 4200* 2500 J = 13,000,000 J. W=J/sec, a kilowatt is 1000 J/sec and a kiliowatt-hour is 1000 J/sec*3600 sec=3,600,000 J. 13,000,000 J/(3,600,000 J / kwh) = 4 kwh. This is about 40 cents worth of electricity.

One Cal (with capital C) is about 4200 Joules, so 2500 Cal is about 4200 * 2500 Joules = 13,000,000 Joules. A watt is a Joule per second, a kilowatt is 1000 Joules / second and a kiliowatt-hour is 1000 Joules / second * 3600 seconds = 3,600,000 Joules.13,000,000 Joules / (3,600,000 Joules / kwh) = 4 kwh, rounded to the nearest whole kwh. This is about 40 cents worth of electricity. It's worth noting that you use 85% of this energy just keeping yourself warm, so the total amount of physical work you can produce in a day is worth less than a dime.

confidence rating #$&*:

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confidence rating #$&*: (Type in a number from 0 to 3, indicating your level of confidence in your solution.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Given Solution:

One Cal (with capital C) is about 4200 Joules, so 2500 Cal is about 4200 * 2500 Joules = 10,500,000 Joules.

A watt is a Joule per second, a kilowatt is 1000 Joules / second and a kiliowatt-hour is 1000 Joules / second * 3600 seconds = 3,600,000 Joules.

10,500,000 Joules / (3,600,000 Joules / kwh) = 3 kwh, rounded to the nearest whole kwh.

This is about 30 cents worth of electricity, and a dime per kilowatt-hour.

Relating this to your physiology:

• You require daily food energy equivalent to 30 cents’ worth of electricity.

• It's worth noting that you use 85% of the energy your metabolism produces just keeping yourself warm.

• It follows that the total amount of physical work you can produce in a day is worth less than a dime.

Your Self-Critique:

Self-critique (if necessary): (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

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Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

Your Self-Critique Rating:

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Question: prin phy and gen phy problem 14.07 how many Kcal of thermal energy would be generated in the process of stopping a 1200 kg car from a speed of 100 km/hr?

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Your Solution:

initial KE = final KE+ heat

100km/hr *3600*1/1000 = 360 m/s

100 km / hr is 100 * 1000 meters / (3600 sec) = 28 m/s. 100km/hr *3600 sec / hr *1/ (1000 m / km) = 360 km^2 sec / ( m hr^2).

KE = .5(1000 kg)(28 m/s)^2 = 400,000 J or 100 Kcal

confidence rating #$&*:

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confidence rating #$&*: (Type in a number from 0 to 3, indicating your level of confidence in your solution.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Given Solution:

NOTE: The given solution is based on student solutions for a previous edition of the text, in which the mass of the car was 1000 kg and its initial velocity 100 km/hr. The adjustments for the current mass and velocity (1200 kg and 95 km/hr as of the current edition) are easily made (see the student question below for a solution using these quantities).

**STUDENT SOLUTION for 1000 kg car at 100 km/hr WITH ERROR IN CONVERSION OF km/hr TO m/s:

The book tells that according to energy conservation

• initial KE = final KE + heat or (Q)

• 100km/hr *3600*1/1000 = 360 m/s

INSTRUCTOR COMMENT:

100km/hr *3600 sec / hr *1/ (1000 m / km) = 360 km^2 sec / ( m hr^2), not 360 m/s.

The correct conversion would be 100 km / hr is 100 * 1000 meters / (3600 sec) = 28 m/s or so.

STUDENT SOLUTION WITH DIFFERENT ERROR IN UNITS

Ke=0.5(1000Kg)(100Km)^2 = 5MJ

1Kcal=4186J

5MJ/4186J==1194Kcal

INSTRUCTOR COMMENT:

Right idea but 100 km is not a velocity and kg * km^2 does not give you Joules.

100 km / hr = 100,000 m / (3600 sec) = 28 m/s, approx.

KE = .5(1000 kg)(28 m/s)^2 = 400,000 J (approx.). or 100 Kcal (approx). **

STUDENT QUESTION:

The book and this problem originally states *1200kg* as the mass. The solution uses 1000kg, giving the answer as about 100kcal. The book uses 95km/hr, 1200kg, and gets an answer of 100kcal. This problem shows 100km/hr, with a mass of 1000kg in the solution and still an answer of 100kcal. I just want to make sure I am doing it correctly.

Where 26.39m/s should be used, I am using the conversion for this specific problem with 100km/hr = 1000m/1hr =

1hr/3600s = 27.78 ~28m/s.

KE = 1/2mv^2

= ½(1200kg)(28 m/s)^2 = 470,400kg*m/s^2 = 470,400 J

470,400J = 1 cal/4.186J = 1 kcal/1000cal = 112.37 kcal = 112kcal

INSTRUCTOR RESPONSE:

I apparently missed the change in the latest edition of the text; or perhaps I chose not to edit the student solutions posted here.

In any case your solution is good.

Your Self-Critique:

Self-critique (if necessary): (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

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Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

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Question: Openstax: The same heat transfer into identical masses of different substances produces different temperature changes. Calculate the final temperature when 1.00 kcal of heat transfers into 1.00 kg of the following, originally at 20.0ºC :

(a) water;

(b) concrete;

(d) mercury.

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Your solution:

dQ = m c `dT,

solving for `dT to get

`dT = `dQ / (m c).

In each case m would be 1.00 kg, `dQ would be 1.00 kCal, and c would depend on the substance.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

confidence rating #$&*: (Type in a number from 0 to 3, indicating your level of confidence in your solution.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Given Solution:

The given solution will use 1 kcal / (kg C) as the specific heat of water, and will assume that the specific heat of concrete is about 1/3 this value, while that of steel is about 1/10 of this value. Your solution should used more accurate accepted values for the specific heats.

1 kcal of heat transferred into 1 kg of water will therefore raise its temperature by 1 Celsius. This is exactly what it means to say that the specific heat of water is 1 kcal / (kg C). The water will therefore end up at temperature 21 C.

The specific heat of concrete being 1/3 that of water, the same heat transferred into 1 kg of concrete will raise its temperature 3 times as much, an increase of 3 Celsius, resulting in a final temperature of 23 C.

Similar reasoning would conclude that the transfer of the same heat to 1 kg of steel would raise its temperature by 10 C, to 30 C.

We could of course use the formula

`dQ = m c `dT,

solving for `dT to get

`dT = `dQ / (m c).

In each case m would be 1.00 kg, `dQ would be 1.00 kCal, and c would depend on the substance.

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Self-critique (if necessary):

Self-critique (if necessary): (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

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Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

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Question: Openstax: A 0.250-kg block of a pure material is heated from 20.0ºC to 65.0ºC by the addition of 4.35 kJ of energy. Calculate its specific heat and identify the substance of which it is most likely composed.

The specific heat of .250 kg of material requiring 4.35 kJ to change its temperature by 45 Celsius is

4.35 kJ / (.250 kg * 45 C) = .4 kJ / (kg C), very approximately.

Copper and zinc are two elements with a specific heat in this range.

The given information is sufficient to calculate the specific heat to 3 significant figures, and this should be sufficient to determine the substance from a table of specific heats.

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Your solution:

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

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Self-critique Rating:

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Question: (optional for Principles of Physics)

Suppose identical amounts of heat transfer into different masses of copper and water, causing identical changes in temperature. What is the ratio of the mass of copper to water?

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Your solution:

I’m in university physics.

confidence rating #$&*:

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Given Solution: The specific heat of copper is about 1/11 that of water, so 11 times the mass of copper would be required.

This could also be formulated by an equation:

m_1 c_1 `dT_1 = m_2 c_2 `dT_2.

Since the temperature changes are identical, `dT_1 = `dT_2 so

m_1 c_1 = m_2 c_2.

Thus

m_2 / m_1 = c_1 / c_2

and the ratio of copper mass to water mass is equal to the ratio of water's specific heat to that of copper, which is

m_2 / m_1 = (4.18 J / (kg C) / (.386 J / (kg C) ), close to 1/11.

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Self-critique (if necessary):

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Self-critique Rating:

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Question: query gen phy problem 14.13 .4 kg horseshoe into 1.35 L of water in .3 kg iron pot at 20 C, final temp 25 C, final temp.

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Your Solution:

I’m in university physics.

confidence rating #$&*:

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Given Solution:

** STUDENT SOLUTION BY EQUATION (using previous version in which the amount of water was 1.6 liters; adjust for the present 1.35 liters):

M1*C1*dT1 + M2*C2*dT2 = 0 is the equation that should be used.

0.4kg * 450J/Kg*Celcius *(25celcius -T1) + 0.3kg * 450J/kg*celcius * (25 celcius - 20 celcius) + 1.6kg *4186J/kg*celcius * (25 celcius - 20 celcius) = 0

Solve for T1, T1 = 214.8 Celsius

Solution below is 189.8 C.

GOOD STUDENT SOLUTION:

This problem is very similar to the first video experiment. You can determine in the same way the thermal energy gained by the water. The pot gains a little thermal energy too,and you have to know how many Joules are gained by a kg of the iron per degree per kg, which is the specific heat of iron (give in text). You can therefore find the energy lost by the horseshoe, from which you can find the energy lost per degree per kg.

For this problem I think we are assuming that none of the water is boiled off ass the hot horse shoe is dropped into the water. First I will find the initial temperature of the shoe.

Since water's density is 1g/ml, 1 milliliter of water will weigh 1 gram. So 1.35 Liters of water will have a mass of 1.35 kg.

1.35kg of water is heated by 5 degrees

• The specific heat of water is 4186 J/kg/degrees celsius so 4186 J / kg / C *1.35 kg * 5 C = 28255 J of energy is necessary to heat the water but since it is in equilibrium with the bucket it must be heated too.

mass of bucket = 0.30 kg

• specific heat of iron = 450 J/kg/degrees

• 450 J / kg / C * 0.30 kg * 5 C = 675 J to heat bucket

So it takes

• 675 J to heat bucket to 25 degrees celsius

• 28255 J to heat water to 25 degrees celsius

so the horse shoe transferred 675+28255 = 28930 J of energy.

Mass of horse shoe = 0.40 kg

• horse shoe is also iron

• specific heat of iron = 450 J/kg/degree

• energy transferred / mass = 28930 J / 0.40kg =72,326 J / kg

• 72 330 J / kg, at 450 (J / kg) / C, implies `dT = 72,330 J/kg / (450 J / kg / C) = 160.7 C, the initial temperature of the horseshoe.

A symbolic solution:

m1 c1 `dT1 + m2 c2 `dT2 + m3 c3 `dT3 = 0.

Let object 1 be the water, object 2 the pot and object 3 the horseshoe. Then `dT1 = `dT2 = + 5 C, and `dT3 = 25 C - T_03, where T_03 is the initial temperature of the horseshoe.

We easily solve for `dT3:

`dT3 = - (m1 c1 `dT1 + m2 c2 `dT2) / (m3 c3) so

`dT3 = - (m1 c1 `dT1 + m2 c2 `dT2) / (m3 c3) = - (1.35 kg * 4200 J / (kg C) * 5 C + .3 kg * 450 J / (kg C) ) / ((.4 kg * 450 J / (kg C) ) = -160 C, approx. so

25 C - T_03 = -160 C and

• T_03 = 160 C + 25 C = 185 C, approx..

STUDENT ERROR: MISSING COMMON KNOWLEDGE: Estimate 1.60L of water = 1KG. Could not find a conversion for this anywhere.

INSTRUCTOR RESPONSE: Each of the following should be common knowledge:

• 1 liter = 1000 mL or 1000 cm^3.

• Density of water is 1 gram/cm^3 so 1 liter contains 1000 g = 1 kg.

• Alternatively, density of water is 1000 kg / m^3 and 1 liter = .001 m^3, leading to same conclusion. **

Your Self-Critique:

Your Self-Critique Rating:

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Question: `q001. Which requires more energy, a 100 kg person climbing a hill 200 meters high or a cup of water heated from room temperature to the boiling point?

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Your Solution:

For the person hiking : W= F*d = m*a*d= 100*9.81m/s^2*200m= 196,200 joules

For the boiling of water: Q=m*c*dT = 236g*4.186J/g*C* (100-20)C = 79,031.68 Joules

I used 236 g being about the mass of the cup.

Therefore, it requires the person more energy to climb the hill.

confidence rating #$&*:

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(Type in a number from 0 to 3, indicating your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Self-Critique Rating:

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

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Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

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Question: `q002. A container holds 4 kilograms of water, 8 kilograms of concrete and 500 grams of ice, all at 0 Celsius. The system is heated to 30 Celsius.

Place in order the heat required to raise the temperature of the these three components of the system, given common knowledge and the fact that the specific heat of concrete is about 1/3 that of water.

At about what temperature would two of these components have gained the same amount of heat?

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Your Solution:

Qice= 500g*2.09J/g*C * 30C = 31,350 J

Qconcrete=8000g* 1.39J/g*C * 30C = 333,600 J

Qwater=4000g*4.18 J/g*C * 30C= 501,600 J

At none of the temperature would any two of these components gain the same amount of heat.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

(Type in a number from 0 to 3, indicating your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Self-Critique Rating:

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

------------------------------------------------

Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

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Question: query univ problem 18.62/18.61 / 18.60 (16.48 10th edition) 1.5 L flask, stopcock, ethane C2H6 at 300 K, atm pressure. Warm to 380 K, open system, then close and cool.

What is the final pressure of the ethane and how many grams remain? Explain the process you used to solve this problem.

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Your Solution:

PV= nRT

(1.03 *10^5 Pa)*(1.5*10^(-3))m^(3) =n* 8.314 J/mol*K * 380 K

n= 0.049 moles

0.049moles * 30g/m = 1.47 g

Now we set proportionality between temperature and volume.

V2/V1=T2/T1 : V2/1.5L= 380K/300K

V2= 1.9 L

The ratio between the initial volume and the increased volume time the mass of the substance will give us the remaining grams of ethane: 1.5L/1.9L * 1.47g = 1.2 g

Since all the values are constant except for Pressure and temperature we can set P and T proportionality and solve for final pressure : P1/T1 =P2/T2 1atm/300K= P2/380K P2= 1.27 atm

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

(Type in a number from 0 to 3, indicating your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Given Solution:

** use pV = nRT and solve for n.

• n = p V / (R T) = (1.03 *10^5 Pa )(1.5 * 10^-3 m^3 ) / [ (8.31 J / (mol K) )(380 K) ] = .048 mol, approx..

If the given quantities are accurate to 2 significant figures, then calculations may be done to 2 significant figures and more accurate values of the constants are not required.

The atomic masses of 2 C and 6 H add up to 30.1, meaning 30.1 grams / mol.

So total mass of the gas is initially

• m(tot) = (.048 mol)(30.1 g/mol)

• m(tot) = 1.4 g

Now if the system is heated to 380 K while open to the atmosphere, pressure will remain constant so volume will be proportional to temperature. Therefore the volume of the gas will increase to

• V2 = 1.5 liters * 380 K / (300 K) = 1.9 liters.

Only 1.5 liters, with mass 1.5 liters / (1.9 liters) * 1.4 grams = 1.1 grams, will stay in the flask.

• The pressure of the 1.1 grams of ethane is 1 atmosphere when the system is closed, and is at 380 K.

As the temperature returns to 300 K volume and quantity of gas will remain constant so pressure will be proportional to temperature.

• Thus the pressure will drop to P3 = 1 atm * 300 K / (380 K) = .79 atm, approx.. **

Your Self-Critique:

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

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Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

Your Self-Critique Rating:

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Question: univ phy query problem (publisher has omitted this problem from the 12th edition) 18.62 (16.48 10th edition)

A uniform cylinder is .9 meters high, and contains air at atmospheric pressure. It is fitted at the top with a tightly sealed piston.

A little bit of mercury (density 13600 kg / m^3) is poured on top of the piston, which increases the force exerted by the piston. The piston therefore descends, compressing the confined air until the pressures equalize. Mercury continues to be added, further lowering the piston and compressing the air.

If this continues long enough, mercury will spill over the top of the cylinder. How high is the piston above the bottom of the cylinder when this occurs?

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Your Solution:

PV = nRT, and the temp is constant, so the product PV for the air is always equal to a constant C.

At the beginning, this product is P0*A*h, where P0 is atmospheric pressure, A is the cylinder's cross sectional area, and h is the cylinder height

Then when the air is pressed down to a height x, the product is P*A*x, where P is the new pressure, therefore P0*A*h = P*A*x and P = P0*h/x.

Now let's examine the pressure that the mercury exerts down on the air.

The force downward on the air is F = F_g + F_a, where F_g is the weight of the mercury and F_a is the force of the surrounding atmosphere (atmospheric pressure).

Now F = P*A

And F_g = m*g = p*V*g = p*A*(h-x)*g

Where m, p, and V are the mass, density, and volume of the mercury

And F_a = P0*A

So by dividing everything by A, we have P = p*(h-x)*g + P0

We can set this pressure equal to the pressure we found earlier

P = P0*h/x = p*(h-x)*g + P0

or P0*h/x - P0 = p*(h-x)*g

or P0*(h/x - 1) = p*(h-x)*g

P0*(h-x) = x*p*(h-x)*g

P0 = x*p*g

Therefore x = P0/(p*g)

Now solving for the value: x= 101300 pa/(13600 kg / m^3*9.81m/s^(2))= 0.759 m

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

(Type in a number from 0 to 3, indicating your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Given Solution:

** Let y be the height of the mercury column.

Since

• T and n for the gas in the cylinder remain constant we have P V = constant, and

• cross-sectional area remains constant V = A * h, where h is the height of the air column,

we have P * h = constant. Thus

• P1 h1 = P2 h2, with

P1 = atmospheric pressure = Patm and

h1 = .9 m, P2 = Patm + rho g y.

Mercury spills over when the depth of the mercury plus that of the air column is .9 m, at which point h2 = h1 - y. So the equation becomes

• Patm * h1 = (Patm + rho g y) * (h1 - y).

We can solve this equation for y (the equation is quadratic).

We obtain two solutions:

• one solution is y = 0; this tells us what when there is no mercury (y = 0) there is no deflection below the .9 m level.

• The other solution is

y = (g•h1•rho - Pa)/(g•rho) = .140 m,

which tells us that .140 m of mercury will again bring us to .9 m level.

We might assume that this level corresponds to the level at which mercury begins spilling over. To completely validate this assumption we need to show that the level of the top of the column will be increasing at this point (if the height is not increasing the mercury will reach this level but won’t spill over).

• The level of the top of the mercury column above the bottom of the cylinder can be regarded as a function f (y) of the depth of the mercury.

• If mercury depth is y then the pressure in the cylinder is Patm + rho g y and the height of the gas in the cylinder is Patm / (Patm + rho g y ) * h1. The level of the mercury is therefore

f(y) = Patm / (Patm + rho g y) * h1 + y

The derivative of this function is f ' ( y ) = 1 - Patm•g•h1•rho/(g•rho•y + Patm)^2, which is a quadratic function of y.

Multiplying both sides by (rho g y + Patm)^2 we solve for y to find that y = sqrt(Patm)•(sqrt(g* h1 * rho) - sqrt(Patm) )/ (g•rho) = .067 m approx., is a critical point of f(y).

The second derivative f '' (y) is 2 Patm•g^2•h1•rho^2/(g•rho•y + Patm)^3, which is positive for y > 0.

This tells us that any critical point of f(y) for which y > 0 will be a relative minimum.

So for y = .0635 m we have the minimum possible total altitude of the air and mercury columns, and for any y > .0635 m the total altitude is increasing with increasing y.

This proves that at y = .140 m the total height of the column is increasing and additional mercury will spill over.

To check that y = .140 m results in a total level of .9 m:

• We note that the air column would then be .9 m - .140 m = .760 m, resulting in air pressure .9 / .760 * 101300 Pa = 120,000 Pa.

• The pressure due to the .140 m mercury column is 19,000 Pa, which when added to the 101,300 Pa of atmospheric pressure gives us 120,000 Pa, accurate to 3 significant figures.

The gauge pressure will be 19,000 Pa.

A more direct but less rigorous solution:

The cylinder is originally at STP. The volume of the air in the tube is inversely proportional to the pressure and the altitude of the air column is proportional to the volume, so the altitude of the air column is inversely proportional to the pressure.

If you pour mercury to depth y then the mercury will exert pressure rho g y = 13,600 Kg/m^3 * 9.8 m/s^2 * y = 133,000 N / m^3 * y.

Thus the pressure in the tube will thus be atmospheric pressure + mercury pressure = 101,000 N/m^2 + 133,000 N/m^3 * y. As a result the altitude of the air column will be the altitude of the air column when y cm of mercury are supported:

• altitude of air column = atmospheric pressure / (atmospheric pressure + mercury pressure) * .9 m =101,000 N/m^2 / (101,000 N/m^2 + 133,000 N/m^3 * y) * .9 m.

At the point where mercury spills over the altitude of the air column will be .9 m - y. Thus at this point

• 101,000 N/m^2 / (101,000 N/m^2 + 133,000 N/m^3 * y) * .9 m = .9 m - y.

This equation can be solved for y. The result is y = .14 m, approx.

The pressure will be 101,000 N/m^2 + 133,000 N/m^3 * .14 m = 120,000 N/m^2.

The gauge pressure will therefore be 120,000 N/m^2 - 101,000 N/m^2 = 19,000 N/m^2. **

Your Self-Critique:

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

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Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

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Question: query univ phy 18.79 was 18.75 (16.61 10th edition) univ phy problem 16.61 for what mass is rms vel .1 m/s; if ice how many molecules; if ice sphere what is diameter; is it visible?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your Solution:

KE = 3/2*k*T

½ *m*v^(2) = 3/2*k*T

m = 3/2*k*T/(v^(2)*1/2)

We can change the equation to allow us to solve for volume by dividing both side by the density of water: m/D = 3/2*k*T/(0.5*v^(2)*D)

Now we have V = 3*k*T/(v^(2)*D), in which we can set it equal to the volume of the sphere : 4/3* pi*r^(3) = 3/2*k*T/(0.5*v^(2)*D).

Since we are solving for diameter we must derive r = [ 9 k T / ( 4 v^2 rho) ] ^(1/3).

Now solving for kinetic energy, KE = 3/2 *k*T= 3/2*(1.38*10^(-23))J/K *273 K= 5.65*10^-21 J.

To find how many molecules there are we must solve for mass using the derived equation: m = 3/2*k*T/ (v^(2)*1/2)= (5.65*10^-21 J)/ (0.001m/s^2 *(1/2)) = 1.13 *10^-14 kg= 1.13*10^-11 g

Finding the amount the number of molecules: 1.13*10^(-11) g * (1mole /18 g)= 6.28*10^-13 moles. 6.28*10^-13 moles* (6.02*10^23) = 3.78*10^11 molecules.

Now about the diameter: density of ice at 273 K = 1000 kg/m^3, 1000 kg/m^3= (1.13 *10^-14 kg)/ V, V= 1.13 *10^-17 kg/m^2

V=4/3* pi*r^(3)

1.13 *10^-17 kg/m^2 = 4/3* pi*r^(3)

r = 1.39*10^-6 m

Diameter = 2r = 2.78*10^-6 m

This is not visible to the naked eye.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

(Type in a number from 0 to 3, indicating your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Given Solution:

** We can solve this problem knowing that ave KE per particle is 3/2 k T so the .5 m v^2 = 3/2 k T, where v is RMS velocity. Thus

• m = 3 k T / v^2.

From the density of water and the mass of the particle we can determine its volume, which is equal to 4/3 pi r^3. From this we find r.

• We obtain volume m / rho = 3 k T / (v^2 rho), where rho is the density of water.

• Setting this equal to 4/3 pi r^3 we get the equation 4/3 pi r^3 = 3 k T / (v^2 rho). The solution is

r = [ 9 k T / ( 4 v^2 rho) ] ^(1/3).

From the mass, Avogadro's number and the mass of a mole of water we determine the number of molecules.

The following analysis shows the intermediate quantities we obtain in the process. Some of the calculations, which were done mentally, might be in error so you should redo them using precise values of the constants.

At 273 Kelvin we have ave KE = 3/2 k T = 5.5 * 10^-21 Joules.

mass is found by solving .5 m v^2 = 3/2 k T for m, obtaining m = 3/2 k T / (.5 v^2) = 5.5 * 10^-21 J / (.5 * (.001 m/s)^2 ) = 1.2 * 10^-14 kg.

The volume of the sphere is therefore 1.2 * 10^-14 kg / (1000 kg / m^2) = 1.2 * 10^-17 m^3.

Setting this equal to 4/3 pi r^3 we obtain radius r = ( 1.2 * 10^-17 m^3 / 4.2)^(1/3) = ( 2.8 * 10^-18 m^3)^(1/3) = 1.4 * 10^-6 m. Diameter is double this, about 2.8 * 10^-6 m. This is only 3 microns, and is not visible to the naked eye, though it could easily be viewed using a miscroscope.

A water molecule contains 2 hydrogen and 1 oxygen molecule with total molar mass 18 grams = .018 kg.

The 1.2 * 10^-14 kg mass of particle therefore consists of 1.2 * 10^-14 / (.018 kg / mole) = 6.7 * 10^-13 moles. With about 6 * 10^23 particles in a mole this consists of

6.7 * 10^-13 moles * 6 * 10^23 particles / mole = 4* 10^11 particles (about 400 billion water molecules). **

STUDENT COMMENT:

I'm still not sure about the 'visible' thing.

INSTRUCTOR COMMENT:

In any case, visible light has a wavelength between about .4 microns and .7 microns. Nothing smaller than this is visible even in principle, in the sense that its image can't be resolved by visible light.

If we mean 'visible to the naked eye', that limit occurs between 10 and 100 microns.

So this object is in principle visible (wouldn't be hard to resolve with a microscope), but not to the naked eye.

Your Self-Critique:

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

------------------------------------------------

Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

*********************************************

Question: `q003. A long U-tube, open at both ends, holds water to a level 40 cm above the bottom of the tube. Oil of density 900 kg / m^3 is added to one side. The oil floats on top of the water, forming an oil column on top of the water column on that side.

• What will be the height of the oil column when the difference in levels of the two sides is 3 centimeters?

• What will be the height of the water column when the water has been completely displaced from one side?

• What will happen if oil continues to be slowly added?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your Solution:

I have assumed that the water’s height on both side of the u-tube is 40 cm from the bottom.

P=D*g*h

A is what we are looking for.

P = P0+(Dw*g*h2)= P0+(DOil*g*A)

Dw*g*h2=DOil*g*A

So we are given the height of the water and the difference so: h2=A-3cm

Now solving for the derived equation:

Dw*g*h2=DOil*g*A

h2=Doil/Dw *A

Now substituting h2=A-3cm:

A-3cm = Doil/Dw *A

A-3cm= 900/1000 *A

0.1A =3 cm

A= 30cm

The height of the water will be 80 cm since from the beginning it was considered that the water level was 40 cm in both sides so therefore if it was moved to one side then another 40 cm would be added to the current 40 cm level giving 80 cm.

If the oil continues to be added then the water level would increase on one side with the bottom of the water level filled with the oil.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

(Type in a number from 0 to 3, indicating your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

Response: 3

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Self-Critique Rating:

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.)

------------------------------------------------

Self-critique Rating: (If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

Self-critique (if necessary):

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Self-critique rating:

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

• 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

Self-critique (if necessary):

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Self-critique rating:

#*&!

Assignment 04

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