interconversion of temperature scales

7/18/2019 Interconversion of Temperature Scales

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Interconversion of Temperature Scales

Celsius

The world's most common temperature scale is Celsius. Abbreviated C, it is virtually the same as the

old centigrade scale and therefore has 100 degrees between the melting point and boiling point ofwater, taken to occur at 0 and 100 degrees, respectively.

Kelvin

Temperature is a measure of the thermal energy of a system. Thus cooling can proceed only to the

point at which all of the thermal energy is removed from the system, and this process defines the

temperature of absolute zero. The Kelvin scale, also called the absolute temerature scale, takes its zero

to be absolute zero. It uses units of kelvins (abbreviated K), which are the same size as the degrees on

the Celsius scale.

Fahrenheit

This anachronistic temperature scale, used primarily in the United States, has zero defined as the

lowest temperature that can be reached with ice and salt, and 100 degrees as the hottest daytime

temperature observed in Italy by Torricelli.

A. In the equation of state for the perfect gas, , which of the following three

temperature scales must be used?

Celsius

Kelvin

Fahrenheit

B. What is the formula used to convert a temperature in degrees Celsius ( ) to the same

temperature in kelvins ( )?

Express in terms of .

=

T_C + 273

or

T_C + 273.15

C. What is the formula used to convert a temperature in degrees Fahrenheit ( ) to the same

temperature in degrees Celsius ( )?

Express in terms of .

= Answer not displayed

D. It is possible to get a good "feel" for the Celsius scale because multiples of 10 have special

significance:

o : very cold weather;

o : water freezes;

o : a cool day, so wear a jacket outside;

o : room temperature;



o : a hot day, so drink extra water;

o : a high fever.

Convert these six temperatures into Fahrenheit.

Enter the temperatures to the nearest Fahrenheit degree, ordering them from smallest to

largest, separated with commas.

Temperatures = 14 , 32 , 50 , 68 , 86 , 104 degrees Fahrenheit

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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 C, are cooled down to skin temperature, 34 C, when

they come in contact with your skin. We will simply ask how much heat is transferred to the skin from

equal amounts (by weight) of steam and hot water: each. We will further assume a constant

specific heat capacity for both liquid water and steam.

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

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.

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

heat of vaporization for steam is .

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

= 63.3 (+/- 0.1%) kJ

C. How much heat is transferred by 25.0 g of water onto the skin?

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

= 6.91 (+/- 0.1%) kJ

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Hot Rods



Two circular rods, both of length and having the same diameter, are placed end to end between rigid

supports with no initial stress in the rods.

The coefficient of linear expansion and Young's modulus for rod A are and respectively; those

for rod B are and respectively. Both rods are "normal" materials with .

The temperature of the rods is now raised by .

A. After the rods have been heated, which of the following statements is true?

Choose the best answer.

The length of each rod is still .

The length of each rod changes but the combined length of the rods is still .

B. After the rods have been heated, which of the following statements is true?

Choose the best answer.

The stress in each rod remains zero.

A compressive stress arises that is the same for both rods.

A compressive stress arises that is different for the two rods.

A tensile stress arises that is the same for both rods.

A tensile stress arises that is different for the two rods.

A compressive stress arises in one rod and a tensile stress arises in the other rod.

C. What is the stress in the rods after heating?

Express the stress in terms of , , , , and .

=-(alpha_A+alpha_B)*DeltaT/(1/Y_A+1/Y_B)

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Conductive Heat Loss from a HouseIn this problem you will estimate the heat lost by a typical house, assuming that the temperature inside

is and the temperature outside is . The walls and uppermost ceiling of a typical

house are supported by -inch wooden beams with fiberglass insulation

in between. The true depth of the beams is actually inches, but we will take

the thickness of the walls and ceiling to be to allow for the interior and exterior

covering. Assume that the house is a cube of length on a side. Assume that the roof has

very high conductivity, so that the air in the attic is at the same temperature as the outside air. Ignore

heat loss through the ground.



A. The first step is to calculate , the effective thermal conductivity of the wall (or ceiling),

allowing for the fact that the beams are actually only wide and are spaced 16 inches

center to center.

Express numerically to two significant figures, in watts per kelvin per meter squared.

= 0.048 (+/- 2%)

B. What is , the total rate of energy loss due to heat conduction for this house?

Round your answer to the nearest 10 W.

= 2160 (+/- 0.4%) W

C. Let us assume that the winter consists of 150 days in which the outside temperature is 0 C.

This will give the typical number of "heating degree days" observed in a winter along thenortheastern US seaboard. (The cumulative number of heating degree days is given daily by

the National Weather Service and is used by oil companies to determine when they should fill

the tanks of their customers.) Given that a gallon (3.4 kg) of oil liberates

when burned, how much oil will be needed to supply the heat lost by

conduction from this house over a winter? Assume that the heating system is 75% efficient.

Give your answer numerically in gallons to two significant figures.

Gallons consumed = 270 (+/- 3%) gallons per winter

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Heat Radiated by a PersonIn this problem you will consider the balance of thermal energy radiated and absorbed by a person.

Assume that the person is wearing only a skimpy bathing suit of negligible area. As a rough

approximation, the area of a human body may be considered to be that of the sides of a cylinder of

length and circumference .

For the Stefan-Boltzmann constant use .

A. If the surface temperature of the skin is taken to be , how much thermal power

does the body described in the introduction radiate?

Take the emissivity to be .

Express the power radiated into the room by the body numerically, rounded to the

nearest 10 W.

= 460 W



B.

The basal metabolism of a human adult is the total rate of energy production when a person is

not performing significant physical activity. It has a value around 125 W, most of which is

lost by heat conduction to the surrounding air and especially to the exhaled air that was

warmed while inside the lungs. Given this energy production rate, it would seem impossible

for a human body to radiate 460 W as you calculated in the previous part.

Which of the following alternatives seems to best explain this conundrum?

The human body is quite reflective in the infrared part of the spectrum (where it

radiates) so is in fact less than 0.1.

The surrounding room is near the temperature of the body and radiates nearly the same

power into the body.

C. Now calculate , the thermal power absorbed by the person from the thermal radiation

field in the room, which is assumed to be at . If you do not understand the role

played by the emissivities of room and person, be sure to open the hint on that topic.

Express the thermal power numerically, giving your answer to the nearest 10 W.

= 400 (+/- 2%) W

D. Find , the net power radiated by the person when in a room with temperature

.

Express the net radiated power numerically, to the nearest 10 W.

= 60 W

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Equipartition Theorem and Microscopic MotionLearning Goal: To understand the Equipartition Theorem and its implications for the mechanical

motion of small objects.

In statistical mechanics, heat is the random motion of the microscopic world. The average kinetic or

potential energy of each degree of freedom of the microscopic world therefore depends on the

temperature. If heat is added, molecules increase their translational and rotational speeds, and the

atoms constituting the molecules vibrate with larger amplitude about their equilibrium positions. It is a

fact of nature that the energy of each degree of freedom is determined solely by the temperature. The

Equipartition Theorem states this quantitatively:

The average energy associated with each degree of freedom in a system at absolute

temperature is , where is Boltzmann's

constant.

The average energy of the ith degree of freedom is , where the angle brackets

represent "average" or "mean" values of the enclosed variable. A "degree of freedom" corresponds to



any dynamical variable that appears quadratically in the energy. For instance, is the kinetic

energy of a gas particle of mass with velocity component along the x axis.

The Equipartition Theorem follows from the fundamental postulate of statistical mechanics—that

every energetically accessible quantum state of a system has equal probability of being populated,

which in turn leads to the Boltzmann distribution for a system in thermal equilibrium. From the

standpoint of an introductory physics course, equipartition is best regarded as a principle that is

justified by observation.

In this problem we first investigate the particle model of an ideal gas. An ideal gas has no interactions

among its particles, and so its internal energy is entirely "random" kinetic energy. If we consider the

gas as a system, its internal energy is analogous to the energy stored in a spring. If one end of the gas

container is fitted with a sliding piston, the pressure of the gas on the piston can do useful work. In fact,

the empirically discovered perfect gas law, , enables us to calculate this pressure. This

rule of nature is remarkable in that the value of the mass does not affect the energy (or the pressure) of

the gas particles' motion, only the temperature. It provides strong evidence for the validity of the

Equipartition Theorem as applied to a particle gas:

or

for a particle constrained by a spring whose spring constant is . If a molecule has moment of inertia

about an axis and is rotating with angular velocity about that axis with associated rotational kinetic

energy , that angular velocity represents another degeree of freedom.

A. Consider a monatomic gas of particles each with mass . What is , the root

mean square (rms) of the x component of velocity if the gas is an at an absolute temperature

?

Express your answer in terms of , , , and other given quantities.

=sqrt(k_B*T/M)

B. Now consider the same system—a monatomic gas of particles of mass —except in three

dimensions. Find , the rms speed if the gas is at an absolute temperature .

Express your answer in terms of , , , and other given quantities.

=sqrt(3*k_B*T/M)

C. What is the rms speed of molecules in air at ? Air is composed mostly of molecules,so you may assume that it has molecules of average atomic mass

.

Express your answer in meters per second, to the nearest integer.

= 484 (+/- 0.2%)

Now consider a rigid dumbbell with two masses, each of mass , spaced a distance apart.

A. Find , the rms angular speed of the dumbbell about a single axis (taken to be the x

axis), assuming that it is in equilibrium at temperature .



Express the rms angular speed in terms of , , , , and other given quantities.

=sqrt(2*k_B*T/(m*d^2))

B. What is the typical angular frequency for a molecule like at room temperature (

)? Assume that for this molecule is . Take the atomic mass of to be

.

Express numerically in hertz, to three significant figures.

= 6.58*10^11 (+/- 0.2%) Hz

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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.

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?

Express the temperature numerically in kelvins.

= 300 K

B. 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?

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

= 19 (+/- 1%) K

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The Ideal Gas Law Derived

The ideal gas law, discovered experimentally, is an equation of state that relates the observable state

variables of the gas—pressure, temperature, and density (or quantity per volume):

(or ),

where is the number of atoms, is the number of moles, and and are ideal gas constants such

that , where is Avogadro's number. In this problem, you should use Boltzmann's

constant instead of the gas constant .

Remarkably, the pressure does not depend on the mass of the gas particles. Why don't heavier gasparticles generate more pressure? This puzzle was explained by making a key assumption about the



connection between the microscopic world and the macroscopic temperature . This assumption is

called the Equipartition Theorem.

The Equipartition Theorem states that the average energy associated with each degree of freedom in a

system at absolute temperature is , where is Boltzmann'sconstant. A degree of freedom is a term that appears quadratically in the energy, for instance

for the kinetic energy of a gas particle of mass with velocity along the x axis. This

problem will show how the ideal gas law follows from the Equipartition Theorem.

To derive the ideal gas law, consider a single gas particle of mass that is moving with speed in a

container with length along the x direction.

A. Find the magnitude of the average force in the x direction that the particle exerts on the

right-hand wall of the container as it bounces back and forth. Assume that collisions between

the wall and particle are elastic and that the position of the container is fixed. Be careful of the

sign of your answer.

Express the magnitude of the average force in terms of , , and .

= m*v_x^2/L_x

B. Imagine that the container from the problem introduction is now filled with identical gas

particles of mass . The particles each have different x velocities, but their average x velocity

squared, denoted , is consistent with the Equipartition Theorem.

Find the pressure on the right-hand wall of the container.

Express the pressure in terms of the absolute temperature , the volume of the

container (where ), , and any other given quantities. The lengths of the

sides of the container should not appear in your answer.



= (N/V)*k_B*T

C. Which of the following statements about your derivation of the ideal gas law are true?

a. The Equipartition Theorem implies that .

b. owing to inelastic collisions between the gas molecules.

c. With just one particle in the container, the pressure on the wall (at ) is

independent of and .

d. With just one particle in the container, the average force exerted on the particle by

the wall (at ) is independent of and .

Enter t for true or f for false for each statment, separating your answers with commas.

t , f , f , t

D. If you heat a fixed quantity of gas, which of the following statements are true?

a. The volume will always increase.

b. If the pressure is held constant, the volume will increase.

c. The product of volume and pressure will increase.

d. The density of the gas will increase.

e. The quantity of gas will increase.

Enter t for true or f for false for each statment, separating your answers with commas.

f , t , t , f , f

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Particle Gas Review

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 .

A. Find , the average x velocity squared for each particle.

Express the average x velocity squared in terms of the gas temperature and any other

given quantities.

=k_B*T/m

B. Find , the average speed squared for each particle.

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

given quantities.



=3*k_B*T/m

C. Find , the internal energy of the gas.

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

= 3/2*N*k_B*T

D. Find , the molar heat capacity (heat capacity per mole) of the gas at constant volume.

Express the molar heat capacity in terms of and .

=(3/2)*N_A*k_B

E. Find the total heat capacity of the gas at constant volume.

Express the total heat capacity in terms of quantities given in the problem

introduction.

= (3/2)*N*k_B

F. Find , the pressure of the gas.

= N*k_B*T/V

G. Express the pressure of the gas in terms of its energy density .

= (2/3)*(U/V)

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.

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

=3

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

= sqrt(1/3)

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

=1

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Up, Up, and AwayHot air balloons float in the air because of the difference in density between cold and hot air. Consider

a balloon in which the mass of the pilot basket together with the mass of the balloon fabric and other

equipment is . The volume of the hot air inside the balloon is and the volume of the basket,



fabric, and other equipment is . The absolute temperature of the cold air outside the balloon is and

its density is . The absolute temperature of the hot air inside the balloon is (where ). The

balloon is open at the bottom, so that the pressure inside and outside of the balloon is the same.

Assume that we can treat air as an ideal gas. Use for the magnitude of the acceleration due to gravity.

A.What is the density of hot air inside the balloon?

Express the density in terms of , , and .

= rho_c*T_c/T_h

B. What is the total weight of the balloon plus the hot air inside it?

Express your answer in terms of quantities given in the problem introduction and/or .

=

(m_b+rho_h*V_1)*g

or

(m_b+(rho_c*T_c/T_h)*V_1)*g

C. What is the magnitude of the buoyant force on the balloon?

Express your answer in terms of , , , and .

= g*rho_c*(V_1+V_2)

D. For the balloon to float, what is the minimum temperature of the hot air inside it?

Express the minimum temperature in terms of , , , , and .

= T_c*(V_1/((V_1+V_2)-m_b/rho_c))

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interconversion of temperature scales

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