# 1.6 Mechanisms of heat transfer  (Page 18/27)

 Page 18 / 27

At temperatures of a few hundred kelvins the specific heat capacity of copper approximately follows the empirical formula $c=\alpha +\beta T+\delta {T}^{-2},$ where $\alpha =349\phantom{\rule{0.2em}{0ex}}\text{J/kg}·\text{K,}$ $\beta =0.107\phantom{\rule{0.2em}{0ex}}\text{J/kg}·{\text{K}}^{2},$ and $\delta =4.58\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\text{J}·\text{kg}·\text{K}\text{.}$ How much heat is needed to raise the temperature of a 2.00-kg piece of copper from $20\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ to $250\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ ?

In a calorimeter of negligible heat capacity, 200 g of steam at $150\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ and 100 g of ice at $-40\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ are mixed. The pressure is maintained at 1 atm. What is the final temperature, and how much steam, ice, and water are present?

The amount of heat to melt the ice and raise it to $100\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ is not enough to condense the steam, but it is more than enough to lower the steam’s temperature by $50\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ , so the final state will consist of steam and liquid water in equilibrium, and the final temperature is $100\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ ; 9.5 g of steam condenses, so the final state contains 49.5 g of steam and 40.5 g of liquid water.

An astronaut performing an extra-vehicular activity (space walk) shaded from the Sun is wearing a spacesuit that can be approximated as perfectly white $\left(e=0\right)$ except for a $5\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}8\phantom{\rule{0.2em}{0ex}}\text{cm}$ patch in the form of the astronaut’s national flag. The patch has emissivity 0.300. The spacesuit under the patch is 0.500 cm thick, with a thermal conductivity $k=0.0600\phantom{\rule{0.2em}{0ex}}\text{W/m}\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ , and its inner surface is at a temperature of $20.0\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ . What is the temperature of the patch, and what is the rate of heat loss through it? Assume the patch is so thin that its outer surface is at the same temperature as the outer surface of the spacesuit under it. Also assume the temperature of outer space is 0 K. You will get an equation that is very hard to solve in closed form, so you can solve it numerically with a graphing calculator, with software, or even by trial and error with a calculator.

The goal in this problem is to find the growth of an ice layer as a function of time. Call the thickness of the ice layer L . (a) Derive an equation for dL / dt in terms of L , the temperature T above the ice, and the properties of ice (which you can leave in symbolic form instead of substituting the numbers). (b) Solve this differential equation assuming that at $t=0$ , you have $L=0.$ If you have studied differential equations, you will know a technique for solving equations of this type: manipulate the equation to get dL / dt multiplied by a (very simple) function of L on one side, and integrate both sides with respect to time. Alternatively, you may be able to use your knowledge of the derivatives of various functions to guess the solution, which has a simple dependence on t . (c) Will the water eventually freeze to the bottom of the flask?

a. $dL\text{/}dT=kT\text{/}\rho L$ ; b. $L=\sqrt{2kTt\text{/}\rho {L}_{\text{f}}}$ ; c. yes

As the very first rudiment of climatology, estimate the temperature of Earth. Assume it is a perfect sphere and its temperature is uniform. Ignore the greenhouse effect. Thermal radiation from the Sun has an intensity (the “solar constant” S ) of about $1370\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ at the radius of Earth’s orbit. (a) Assuming the Sun’s rays are parallel, what area must S be multiplied by to get the total radiation intercepted by Earth? It will be easiest to answer in terms of Earth’s radius, R . (b) Assume that Earth reflects about 30% of the solar energy it intercepts. In other words, Earth has an albedo with a value of $A=0.3$ . In terms of S , A , and R , what is the rate at which Earth absorbs energy from the Sun? (c) Find the temperature at which Earth radiates energy at the same rate. Assume that at the infrared wavelengths where it radiates, the emissivity e is 1. Does your result show that the greenhouse effect is important? (d) How does your answer depend on the the area of Earth?

Let’s stop ignoring the greenhouse effect and incorporate it into the previous problem in a very rough way. Assume the atmosphere is a single layer, a spherical shell around Earth, with an emissivity $e=0.77$ (chosen simply to give the right answer) at infrared wavelengths emitted by Earth and by the atmosphere. However, the atmosphere is transparent to the Sun’s radiation (that is, assume the radiation is at visible wavelengths with no infrared), so the Sun’s radiation reaches the surface. The greenhouse effect comes from the difference between the atmosphere’s transmission of visible light and its rather strong absorption of infrared. Note that the atmosphere’s radius is not significantly different from Earth’s, but since the atmosphere is a layer above Earth, it emits radiation both upward and downward, so it has twice Earth’s area. There are three radiative energy transfers in this problem: solar radiation absorbed by Earth’s surface; infrared radiation from the surface, which is absorbed by the atmosphere according to its emissivity; and infrared radiation from the atmosphere, half of which is absorbed by Earth and half of which goes out into space. Apply the method of the previous problem to get an equation for Earth’s surface and one for the atmosphere, and solve them for the two unknown temperatures, surface and atmosphere.

1. In terms of Earth’s radius, the constant $\sigma$ , and the unknown temperature ${T}_{s}$ of the surface, what is the power of the infrared radiation from the surface?
2. What is the power of Earth’s radiation absorbed by the atmosphere?
3. In terms of the unknown temperature ${T}_{e}$ of the atmosphere, what is the power radiated from the atmosphere?
4. Write an equation that says the power of the radiation the atmosphere absorbs from Earth equals the power of the radiation it emits.
5. Half of the power radiated by the atmosphere hits Earth. Write an equation that says that the power Earth absorbs from the atmosphere and the Sun equals the power that it emits.
6. Solve your two equations for the unknown temperature of Earth.
For steps that make this model less crude, see for example the lectures by Paul O’Gorman.

a. $\sigma \left(\pi {R}^{2}\right){T}_{\text{s}}{}^{4}$ ; b. $e\sigma \pi {R}^{2}{T}_{\text{s}}{}^{4}$ ; c. $2e\sigma \pi {R}^{2}{T}_{\text{e}}{}^{4}$ ; d. ${T}_{s}^{4}=2{T}_{e}^{4}$ ; e. $e\sigma {T}_{\text{s}}^{4}+\frac{1}{4}\left(1-A\right)S=\sigma {T}_{\text{s}}^{4}$ ; f. 288 K

#### Questions & Answers

What is differential form of Gauss's law?
Rohit Reply
help me out on this question the permittivity of diamond is 1.46*10^-10.( a)what is the dielectric of diamond (b) what its susceptibility
OLUWA Reply
a body is projected vertically upward of 30kmp/h how long will it take to reach a point 0.5km bellow e point of projection
Abu Reply
i have to say. who cares. lol. why know that t all
Jeff
is this just a chat app about the openstax book?
Lord Reply
kya ye b.sc ka hai agar haa to konsa part
MPL Reply
what is charge quantization
Mayowa Reply
it means that the total charge of a body will always be the integral multiples of basic unit charge ( e ) q = ne n : no of electrons or protons e : basic unit charge 1e = 1.602×10^-19
Riya
is the time quantized ? how ?
Mehmet
What do you meanby the statement,"Is the time quantized"
Mayowa
Can you give an explanation.
Mayowa
there are some comment on the time -quantized..
Mehmet
time is integer of the planck time, discrete..
Mehmet
planck time is travel in planck lenght of light..
Mehmet
it's says that charges does not occur in continuous form rather they are integral multiple of the elementary charge of an electron.
Tamoghna
it is just like bohr's theory. Which was angular momentum of electron is intral multiple of h/2π
Aditya
determine absolute zero
OFERE Reply
The properties of a system during a reversible constant pressure non-flow process at P= 1.6bar, changes from constant volume of 0.3m³/kg at 20°C to a volume of 0.55m³/kg at 260°C. its constant pressure process is 3.205KJ/kg°C Determine: 1. Heat added, Work done, Change in Internal Energy and Change in Enthalpy
Opeyemi Reply
U can easily calculate work done by 2.303log(v2/v1)
Abhishek
Amount of heat added through q=ncv^delta t
Abhishek
Change in internal energy through q=Q-w
Abhishek
please how do dey get 5/9 in the conversion of Celsius and Fahrenheit
Gwam Reply
what is copper loss
timileyin Reply
this is the energy dissipated(usually in the form of heat energy) in conductors such as wires and coils due to the flow of current against the resistance of the material used in winding the coil.
Henry
it is the work done in moving a charge to a point from infinity against electric field
Ashok Reply
what is the weight of the earth in space
peterpaul Reply
As w=mg where m is mass and g is gravitational force... Now if we consider the earth is in gravitational pull of sun we have to use the value of "g" of sun, so we can find the weight of eaeth in sun with reference to sun...
Prince
g is not gravitacional forcé, is acceleration of gravity of earth and is assumed constante. the "sun g" can not be constant and you should use Newton gravity forcé. by the way its not the "weight" the physical quantity that matters, is the mass
Jorge
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Prince
Thats why it can't have a constant value of g ....
Prince
not true. you must know Newton gravity Law . even a cloud of gas it has mass thats al matters. and the distsnce from the center of mass of the cloud and the center of the mass of the earth
Jorge
please why is the first law of thermodynamics greater than the second
Ifeoma Reply
every law is important, but first law is conservation of energy, this state is the basic in physics, in this case first law is more important than other laws..
Mehmet
First Law describes o energy is changed from one form to another but not destroyed, but that second Law talk about entropy of a system increasing gradually
Mayowa
first law describes not destroyer energy to changed the form, but second law describes the fluid drection that is entropy. in this case first law is more basic accorging to me...
Mehmet
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
Mateshwar Reply
explain the lack of symmetry in the field of the parallel capacitor
Phoebe Reply
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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