Assignment 8: Temperature, Heat and Thermal PropertiesDue: 2:00am on Saturday, November 6, 2010

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A Sliding Crate of Fruit

A crate of fruit with a mass of 34.0 and a specific heat capacity of 3650 slides 9.00

down a ramp inclined at an angle of 36.6 below the horizontal.

Part A

If the crate was at rest at the top of the incline and has a speed of 2.05 at the bottom, how much

work was done on the crate by friction?

Hint A.1 How to approach the problem

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Hint A.2 Find the initial and final kinetic energies

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Hint A.3 Find the difference between initial and final potential energy

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Use 9.81 for the acceleration due to gravity and express your answer in joules.

ANSWER: = -1720

Correct

The frictional force opposes the motion of the crate, so the work done on the crate by frictionmust be a negative quantity.

Part B

If an amount of heat equal to the magnitude of the work done by friction is absorbed by the crate offruit and the fruit reaches a uniform final temperature, what is its temperature change ?

Hint B.1 Equation for temperature change

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ANSWER: = 1.38×10−2

Correct

Of course, the assumptions of "total heat absorption" and "uniform temperature change" are notvery realistic; still, this simplified model provides a useful reminder about the transformation ofmechanical energy into thermal energy when nonconservative forces are present.

mechanical energy into thermal energy when nonconservative forces are present.

Steam vs. Hot-Water Burns

Just about everyone at one time or another has been burned by hot water or steam. This problemcompares the heat input to your skin from steam as opposed to hot water at the same temperature.Assume that water and steam, initially at 100 , are cooled down to skin temperature, 34 , when they

come in contact with your skin. Assume that the steam condenses extremely fast. We will furtherassume a constant specific heat capacity for both liquid water and steam.

Part A

Under these conditions, which of the following statements is true?

ANSWER:Steam burns the skin worse than hot water because the thermalconductivity of steam is much higher than that of liquid water.

Steam burns the skin worse than hot water because the latent heat ofvaporization is released as well.

Hot water burns the skin worse than steam because the thermalconductivity of hot water is much higher than that of steam.

Hot water and steam both burn skin about equally badly.

Correct

The key point is that the latent heat of vaporization has to be taken into account for the steam.

Part B

How much heat is transferred to the skin by 25.0 of steam onto the skin? The latent heat of

vaporization for steam is .

Hint B.1 Determine the heat transferred from steam to skin

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Express the heat transferred, in kilojoules, to three significant figures.

ANSWER: = 63.3

Correct

Here we assumed that the skin continues to remain at 34 . Actually the local temperature in

the area where the steam condenses can be raised quite significantly.

Part C

How much heat is transferred by 25.0 of water onto the skin? To compare this to the result in

the previous part, continue to assume that the skin temperature does not change.

Hint C.1 Determine the heat transferred from water to skin

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Express the heat transferred, in kilojoules, to three significant figures.

ANSWER: = 6.91

Correct

The amount of heat transferred to your skin is almost 10 times greater when you are burned bysteam versus hot water. The temperature of steam can also potentially be much greater than 100

. For these reasons, steam burns are often far more severe than hot-water burns.

Dust Equipartitions

Small dust particles suspended in air seem to dance randomly about, a phenomenon called Brownianmotion.For this problem you will need to know Boltzmann's constant: .

Part A

What would you expect the mean translational kinetic energy of such particles to be

if they are in air at a temperature of 290 K?

Hint A.1 Kinetic energy in terms of average velocity

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Hint A.2 Equipartition Theorem

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Hint A.3 Use the Equipartition Theorem

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Express the mean translational kinetic energy numerically, in joules, to two significantfigures. Note that has been factored out already to make your answer simpler.

ANSWER: = 6.0

Correct

Part B

Find an expression for the rms (root-mean-square) speed of these particles, assuming them to be

spheres of diameter and density

Hint B.1 Find the rms speed in terms of temperature

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Hint B.2 Find the mass of the particles

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Express the rms speed in terms of , , , , and .

Express the rms speed in terms of , , , , and .

ANSWER:

=

Correct

Part C

Now calculate the rms (root-mean-square) speed of these particles, assuming them to be spheres

of diameter and density . The mass of such a dust particle is

.

Express your answer in millimeters per second to one decimal place only.

ANSWER: = 0.3

Correct

This speed is several orders of magnitude smaller than the typical velocities of gas molecules atthis temperature (which are of the order of hundreds of meters per second). This is simplybecause the mass of these particles is much larger than the mass of typical gas molecules. Forparticles larger than the ones described here, the weight can no longer be ignored. Such particlestend to settle quite quickly on account of their weight. Then such a calculation is no longer valid.

Heating a Room

Imagine you've been walking outside on a cold winter's day. When you arrive home at your studioapartment, you realize that you left a window open and your room is only slightly

warmer than the outside. You turn on your 1.0- space heater right away and wait impatiently for the

room to warm up.In this problem, make the following assumptions:The entire output of the space heater goes into warming the air in the room.

The air in the room is an ideal gas with five degrees of freedom per particle (three translational degreesof freedom and two rotational degrees of freedom—about right for nitrogen and oxygen).The air in the room is at a constant pressure of 1.00 .

At room temperature and atmospheric pressure, 1 of air fills a volume of 23 . This is slightly

larger than the volume of air at standard temperature and pressure, because room temperature is hotterthan 0 .

Part A

How long will it be before the heater warms the air in the room by 10. ?

Hint A.1 How to approach the problem

Since you know the power output of the heater, you know how much energy per unit time is beingadded to the air in the room. To determine how long it will take to warm up the room, then, youneed to determine the total energy needed to raise the temperature of the air in the room by 10 .

Once you have this value, simply divide by the power of the heater to determine the time:

Once you have this value, simply divide by the power of the heater to determine the time:

.

Hint A.2 Find the energy needed to raise the temperature

How much energy (in joules) is needed to raise the temperature of the room by 10. ?

Hint A.2.1 How to approach the problem

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Hint A.2.2 Find the heat capacity

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Hint A.2.3 Find the amount of air in the room

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Express your answer in joules to two significant figures.

ANSWER: energy = Answer not displayed

Express your answer in minutes to two significant figures.

ANSWER: time =

16Correct

In practice, it would probably take more than an hour to heat the room by 10. because the

walls and any items in the room are in thermal contact with the air and would have to be warmedup also.

Melting Point of Platinum

The ratio of the pressure of a gas at the melting point of platinum to its pressure at the triple point ofwater, when the gas is kept at constant volume, is found to be 7.476.

Part A

What is the Celsius temperature of the melting point of platinum?

Hint A.1 How to approach the problem.

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Hint A.2 Find an expression for the pressure ratio

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Hint A.3 Triple-point temperature of water

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Express your answer in degrees Celsius to four significant figures.

ANSWER: = Answer not displayed

Adding Ice to Water

An insulated beaker with negligible mass contains liquid water with a mass of 0.330 and a

temperature of 64.8 .

Part A

How much ice at a temperature of -14.0 must be dropped into the water so that the final

temperature of the system will be 29.0 ?

Hint A.1 How to approach the problem

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Hint A.2 Calculate the heat lost by the water

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Hint A.3 How to calculate the heat gained by the ice

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Hint A.4 Heat gained by the ice

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Take the specific heat of liquid water to be 4190 , the specific heat of ice to be 2100

, and the heat of fusion for water to be 334 .

ANSWER: = Answer not displayed

Thermal Energy from Friction on a Rope

A capstan is a rotating drum or cylinder over which a rope or cord slides to provide a great amplificationof the rope's tension while keeping both ends free . Since the added tension in the rope is due to friction,the capstan generates thermal energy.

Part A

If the difference in tension ( ) between the two ends of the rope is and the capstan has

a diameter of and turns once in , find the rate at which thermal energy is

being generated.

Hint A.1 Torque applied to the rope

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Hint A.2 Power due to the torque

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Hint A.3 Power from force

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Hint A.4 How both methods work equally well

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Give a numerical answer, in watts, rounded to the nearest 10 W.

ANSWER: = Answer not displayed W

Part B

If the capstan is made of iron (with a specific heat capacity ) and has a mass of

, at what rate does its temperature rise? Assume that the temperature in the capstan

is uniform and that all the thermal energy generated flows into it.Note that is a temperature.

Hint B.1 Use the chain rule

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Give a numerical answer, in degrees Celsius per second, rounded to two significant figures.

ANSWER: = Answer not displayed

Particle Gas Review

A particle gas consists of monatomic particles each of mass all contained in a volume at

A particle gas consists of monatomic particles each of mass all contained in a volume at

temperature . Your answers should be written in terms of the Boltzmann constant and Avagadro's

number rather than .

Part A

Find , the average speed squared for each particle.

Hint A.1 how to approach the problem

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Hint A.2 Find for each particle

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Hint A.3 Relating the , , and velocities

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Express the average speed squared in terms of the gas temperature and any other given

quantities.

ANSWER: = Answer not displayed

Part B

Find , the internal energy of the gas.

Hint B.1 How to approach the problem

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Hint B.2 Kinetic energy of a single gas particle

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Express the internal energy in terms of the gas temperature and any other given quantities.

ANSWER: = Answer not displayed

Part C

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Part D

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Now imagine that the mass of each gas particle is increased by a factor of 3. All other informationgiven in the problem introduction remains the same.

given in the problem introduction remains the same.

Part E

What will be the ratio of the new molar mass to the old molar mass ?

ANSWER: = Answer not displayed

Part F

What will be the ratio of the new rms speed to the old rms speed ?

Hint F.1 Definition of rms speed

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ANSWER: = Answer not displayed

Part G

What will be the ratio of the new molar heat capacity to the old molar heat capacity ?

Hint G.1 How to approach the problem

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ANSWER: = Answer not displayed

pV Diagram for a Piston

A container holds a sample of ideal gas in thermal equilibrium, as shown in the figure. One end of thecontainer is sealed with a piston whose head isperfectly free to move, unless it is locked inplace. The walls of the container readily allowthe transfer of energy via heat, unless the pistonis wrapped in insulation.

Refer to the pV diagram presented to answer the questions below. In each case, the piston head isinitially unlocked and the gas is in equilibriumat the pressure and volume indicated by point0 on the diagram.

Part A

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after thefollowing process?"Lock the piston head in place, and hold the container above a very hot flame."

Hint A.1 Understand the graph

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Hint A.2 Find the change in volume

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ANSWER:point 1point 2point 3point 4point 5point 6point 7point 8

Answer not displayed

Part B

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after thefollowing process?"Immerse the container into a large water bath at the same temperature, and very slowly push thepiston head further into the container."

Hint B.1 Find the change in volume

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Hint B.2 Find the change in temperature

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ANSWER:

ANSWER:point 1point 2point 3point 4point 5point 6point 7point 8

Answer not displayed

Part C

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after thefollowing process?"Lock the piston head in place and plunge the piston into water that is colder than the gas."

ANSWER:point 1point 2point 3point 4point 5point 6point 7point 8

Answer not displayed

Part D

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after thefollowing process?"Wrap the piston in insulation. Pull the piston head further out of the container."

Hint D.1 Find the change in volume

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Hint D.2 Find the change in temperature

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ANSWER:point 1point 2point 3point 4point 5point 6point 7point 8

Answer not displayed

Average Spacing of Gas Molecules

Consider an ideal gas at 27.0 degrees Celsius and 1.00 atmosphere pressure. Imagine the molecules tobe uniformly spaced, with each molecule at the center of a small cube.

be uniformly spaced, with each molecule at the center of a small cube.

Part A

What is the length of an edge of each small cube if adjacent cubes touch but don't overlap?

Hint A.1 How to approach the problem

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Hint A.2 Calculate the volume per mole

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Hint A.3 Calculate the volume per molecule

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Hint A.4 The edge length of a cube

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Express your answer numerically in meters.

ANSWER: = Answer not displayed

Velocity and Energy Scaling

Hydrogen molecules have a mass of and oxygen molecules have a mass of , where is defined

as an atomic mass unit ( ). Compare a gas of hydrogen molecules to a gas of

oxygen molecules.

Part A

At what gas temperature would the average translational kinetic energy of a hydrogen molecule be

equal to that of an oxygen molecule in a gas of temperature 300 K?

Hint A.1 Find the energy associated with one degree of freedom

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Hint A.2 Total translational kinetic energy

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Express the temperature numerically in kelvins.

ANSWER: = Answer not displayed K

Part B

At what gas temperature would the root-mean-square (rms) speed of a hydrogen molecule be

At what gas temperature would the root-mean-square (rms) speed of a hydrogen molecule be

equal to that of an oxygen molecule in a gas at 300 K?

Hint B.1 Find the rms speed

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State your answer numerically, in kelvins, to the nearest integer.

ANSWER: = Answer not displayed K

The Speed of Nitrogen Molecules

The kinetic theory of gases states that the kinetic energy of a gas is directly proportional to thetemperature of the gas. A relationship between the microscopic properties of the gas molecules and themacroscopic properties of the gas can be derived using the following assumptions:The gas is composed of pointlike particles separated by comparatively large distances.The gas molecules are in continual random motion with collisions being perfectly elastic.The gas molecules exert no long-range forces on each other.One of the most important microscopic properties of gas molecules is velocity. There are severaldifferent ways to describe statistically the average velocity of a molecule in a gas. The most obviousmeasure is the average velocity . However, since the molecules in a gas are moving in random

directions, the average velocity is approximately zero. Another measure of velocity is , the

average squared velocity. Since the square of velocity is always positive, this measure does not averageto zero over the entire gas. A third measure is the root-mean-square (rms) speed, , equal to the

square root of . The rms speed is a good approximation of the the typical speed of the molecules

in a gas.This histogram shows a theoretical distributionof speeds of molecules in a sample of nitrogen (

) gas. In this problem, you'll use the

histogram to compute properties of the gas.

Part A

What is the average speed of the molecules in the gas?

Hint A.1 How to use the histogram

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Hint A.2 More on computing the average

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Express your answer numerically to three significant digits.

ANSWER: = Answer not displayed

Part B

Because the kinetic energy of a single molecule is related to its velocity squared, the best measure ofthe kinetic energy of the entire gas is obtained by computing the mean squared velocity, , or its

square root . The quantity is more common than because it has the dimensions of

velocity instead of the less-familiar velocity-squared.What is the rms speed of the molecules in the nitrogen gas?

Hint B.1 How to approach the problem

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Hint B.2 Find the mean square velocity

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Express your answer numerically to three significant digits.

ANSWER: = Answer not displayed

Part C

What is the temperature of the sample of gas described in the histogram?

Hint C.1 How to approach the problem

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Hint C.2 Find the molar mass of N2

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Express your answer in degrees Celsius to three significant figues.

ANSWER: = Answer not displayed

Part D

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Pressure Cooker

A pressure cooker is a pot whose lid can be tightly sealed to prevent gas from entering or escaping.

Part A

Part A

If an otherwise empty pressure cooker is filled with air of room temperature and then placed on a hotstove, what would be the magnitude of the net force on the lid when the air inside the cooker had

been heated to ? Assume that the temperature of the air outside the pressure cooker is

(room temperature) and that the area of the pressure cooker lid is . Take atmospheric pressure to be

.

Treat the air, both inside and outside the pressure cooker, as an ideal gas obeying .

Hint A.1 Calculate the pressure inside

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Hint A.2 Relating pressure and force

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Hint A.3 Determine the role of the outside pressure

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Express the force in terms of given variables.

ANSWER: = Answer not displayed

Part B

The pressure relief valve on the lid is now opened, allowing hot air to escape until the pressure insidethe cooker becomes equal to the outside pressure . The pot is then sealed again and removed from

the stove. Assume that when the cooker is removed from the stove, the air inside it is still at .

What is the magnitude of the net force on the lid when the air inside the cooker has cooled back

down to ?

Hint B.1 How to approach the problem

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Hint B.2 What stays constant when the cooker is opened?

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Hint B.3 Calculate the pressure inside

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Express the magnitude of the net force in terms of given variables.

ANSWER: = Answer not displayed

Score Summary:

Your score on this assignment is 99.6%.You received 39.85 out of a possible total of 40 points.