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 of
water, 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,
temperature scales must be used?
, which of the following three
Celsius
Kelvin
Fahrenheit
B. What is the formula used to convert a temperature in degrees Celsius (
temperature in kelvins (
)?
Express
.
in terms of
) to the same
T_C + 273
or
=
T_C + 273.15
C. What is the formula used to convert a temperature in degrees Fahrenheit (
temperature in degrees Celsius (
Express
in terms of
) to the same
)?
.
= 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:
specific heat capacity
each. We will further assume a constant
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
for rod B are
and
and
respectively. Both rods are "normal" materials with
The temperature of the rods is now raised by
respectively; those
.
.
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 House
In this problem you will estimate the heat lost by a typical house, assuming that the temperature inside
is
and the temperature outside is
house are supported by
-inch wooden beams
. The walls and uppermost ceiling of a typical
with fiberglass insulation
in between. The true depth of the beams is actually
the thickness of the walls and ceiling to be
inches, but we will take
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
allowing for the fact that the
center to center.
Express
, the effective thermal conductivity of the wall (or ceiling),
beams are actually only
wide and are spaced 16 inches
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 the
northeastern 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 Person
In 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
does the body described in the introduction radiate?
.
, how much thermal power
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 Motion
Learning 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
constant.
is
, where
is Boltzmann's
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
about an axis and is rotating with angular velocity
energy
. If a molecule has moment of inertia
about that axis with associated rotational kinetic
, 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
dimensions. Find
, the rms speed if the gas is at an absolute temperature
Express your answer in terms of
=
—except in three
,
,
.
, 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
so you may assume that it has molecules of average atomic mass
.
molecules,
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
A. Find
, spaced a distance
apart.
, 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
)? Assume that
for a molecule like
for this molecule is
at room temperature (
. 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
as an atomic mass unit (
oxygen molecules.
and oxygen molecules have a mass of
, where
is defined
). Compare a gas of hydrogen molecules to a gas of
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,
that
, where
is the number of moles, and
),
and
are ideal gas constants such
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 gas
particles generate more pressure? This puzzle was explained by making a key assumption about the
connection between the microscopic world and the macroscopic temperature
called the Equipartition Theorem.
. This assumption is
The Equipartition Theorem states that the average energy associated with each degree of freedom in a
system at absolute temperature is
, where
is Boltzmann's
constant. 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
container with length
that is moving with speed
in a
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
= m*v_x^2/L_x
,
, and
.
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
d.
independent of
and
.
With just one particle in the container, the average force exerted on the particle by
the wall (at
) is independent of
and
) is
.
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
temperature
number
monatomic particles each of mass
all contained in a volume
. Your answers should be written in terms of the Boltzmann constant
rather than
.
at
and Avagadro's
A. Find
, the average x velocity squared for each particle.
Express the average x velocity squared in terms of the gas temperature
given quantities.
and any other
k_B*T/m
=
B. Find
, the average speed squared for each particle.
Express the average speed squared in terms of the gas temperature
given quantities.
and any other
3*k_B*T/m
=
C. Find
, the internal energy of the gas.
Express the internal energy in terms of the gas temperature
quantities.
and any other given
= 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
=
E.
and
.
(3/2)*N_A*k_B
Find the total heat capacity of the gas at constant volume.
Express the total heat capacity
introduction.
in terms of quantities given in the problem
= (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
?
to the old rms speed
?
3
B. What will be the ratio of the new 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 Away
Hot 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
fabric, and other equipment is
and the volume of the basket,
. 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
Express your answer in terms of ,
,
on the balloon?
, and
.
.
= g*rho_c*(V_1+V_2)
D. For the balloon to float, what is the minimum temperature
Express the minimum temperature in terms of
= T_c*(V_1/((V_1+V_2)-m_b/rho_c))
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,
,
of the hot air inside it?
,
, and
.