PC1431 MasteringPhysics Assignment 7

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Assignment 8: Temperature, Heat and Thermal Properties

Due: 2:00am on Saturday, November 6, 2010

Note: To understand how points are awarded, read your instructor's Grading Policy. [Switch to Standard Assignment View]

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

Hint not displayed Hint A.2 Find the initial and final kinetic energies

Hint not displayed

Hint A.3 Find the difference between initial and final potential energy Hint not displayed

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 friction must 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 of fruit and the fruit reaches a uniform final temperature, what is its temperature change ?

Hint B.1 Equation for temperature change

Hint not displayed ANSWER:

= 1.38×10−2

Correct

Of course, the assumptions of "total heat absorption" and "uniform temperature change" are not very realistic; still, this simplified model provides a useful reminder about the transformation of mechanical energy into thermal energy when nonconservative forces are present.

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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 problem compares 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 further

assume 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 thermal conductivity of steam is much higher than that of liquid water.

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

Hot water burns the skin worse than steam because the thermal conductivity 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 Hint not displayed

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 Hint not displayed

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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 by steam 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 Brownian motion.

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 Hint not displayed Hint A.2 Equipartition Theorem

Hint not displayed Hint A.3 Use the Equipartition Theorem

Express the mean translational kinetic energy numerically, in joules, to two significant figures. 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 Hint not displayed Hint B.2 Find the mass of the particles

Hint not displayed Express the rms speed in terms of , , , , and .

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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 at this temperature (which are of the order of hundreds of meters per second). This is simply because the mass of these particles is much larger than the mass of typical gas molecules. For particles larger than the ones described here, the weight can no longer be ignored. Such particles tend 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 studio

apartment, 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 degrees of 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 hotter than 0 .

Part A

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

Since you know the power output of the heater, you know how much energy per unit time is being added to the air in the room. To determine how long it will take to warm up the room, then, you need 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:

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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

Hint not displayed Hint A.2.2 Find the heat capacity

Hint not displayed Hint A.2.3 Find the amount of air in the room

Hint not displayed 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 = 16_Correct

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 warmed up 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 of water, 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.

Hint not displayed Hint A.2 Find an expression for the pressure ratio

Hint not displayed 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 not displayed Hint A.2 Calculate the heat lost by the water

Hint not displayed Hint A.3 How to calculate the heat gained by the ice

Hint not displayed Hint A.4 Heat gained by the ice

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: ₌

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 amplification of 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.

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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

Hint not displayed Hint A.2 Power due to the torque

Hint not displayed Hint A.3 Power from force

Hint not displayed Hint A.4 How both methods work equally well

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

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

= _{Answer not displayed}

Particle Gas Review

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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

Hint not displayed Hint A.2 Find for each particle

Hint not displayed Hint A.3 Relating the , , and velocities

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

ANSWER: ₌

Part B

Find , the internal energy of the gas.

Hint B.1 How to approach the problem

Hint not displayed Hint B.2 Kinetic energy of a single gas particle

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

Part C

Part not displayed Part D

Part not displayed

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

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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:

Part F

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

Hint F.1 Definition of rms speed

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

pV Diagram for a Piston

A container holds a sample of ideal gas in thermal equilibrium, as shown in the figure. One end of the container is sealed with a piston whose head is

perfectly free to move, unless it is locked in place. The walls of the container readily allow the transfer of energy via heat, unless the piston is wrapped in insulation.

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Refer to the pV diagram presented to answer the questions below. In each case, the piston head is initially unlocked and the gas is in equilibrium

at the pressure and volume indicated by point 0 on the diagram.

Part A

Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"Lock the piston head in place, and hold the container above a very hot flame."

Hint A.1 Understand the graph

Hint not displayed Hint A.2 Find the change in volume

Hint not displayed ANSWER: _{point 1} point 2 point 3 point 4 point 5 point 6 point 7 point 8

Part B

"Immerse the container into a large water bath at the same temperature, and very slowly push the piston head further into the container."

Hint B.1 Find the change in volume

Hint not displayed Hint B.2 Find the change in temperature

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

Part C

"Lock the piston head in place and plunge the piston into water that is colder than the gas."

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

Part D

"Wrap the piston in insulation. Pull the piston head further out of the container."

Hint D.1 Find the change in volume

Hint not displayed Hint D.2 Find the change in temperature

Hint not displayed ANSWER: point 1 point 2 point 3 point 4 point 5 point 6 point 7 point 8

Average Spacing of Gas Molecules

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

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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 not displayed Hint A.2 Calculate the volume per mole

Hint not displayed Hint A.3 Calculate the volume per molecule

Hint not displayed Hint A.4 The edge length of a cube

Hint not displayed Express your answer numerically in meters.

ANSWER: ₌

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 Hint not displayed

Hint A.2 Total translational kinetic energy

Hint not displayed Express the temperature numerically in kelvins.

ANSWER: ₌

Answer not displayed K

Part B

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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

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 the

temperature of the gas. A relationship between the microscopic properties of the gas molecules and the macroscopic 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 several different ways to describe statistically the average velocity of a molecule in a gas. The most obvious measure 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 average to 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 distribution of 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

Hint not displayed Hint A.2 More on computing the average

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Express your answer numerically to three significant digits.

ANSWER: ₌

Part B

Because the kinetic energy of a single molecule is related to its velocity squared, the best measure of the 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 not displayed Hint B.2 Find the mean square velocity

Hint not displayed Express your answer numerically to three significant digits.

ANSWER: ₌

Part C

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

Hint C.1 How to approach the problem

Hint not displayed Hint C.2 Find the molar mass of N2

Express your answer in degrees Celsius to three significant figues. ANSWER: ₌

Part D

Part not displayed

Pressure Cooker

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

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Part A

If an otherwise empty pressure cooker is filled with air of room temperature and then placed on a hot stove, 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

Hint not displayed Hint A.2 Relating pressure and force

Hint not displayed Hint A.3 Determine the role of the outside pressure

Hint not displayed Express the force in terms of given variables.

ANSWER: ₌

Part B

The pressure relief valve on the lid is now opened, allowing hot air to escape until the pressure inside the 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 not displayed Hint B.2 What stays constant when the cooker is opened?

Hint not displayed Hint B.3 Calculate the pressure inside

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

ANSWER: ₌

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Your score on this assignment is 99.6%.