Tuesday, March 19, 2013

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Final exam (Thermodynamics 2014)

1. A 0.300 kg piece of metal is heated to 88°C and placed in a copper calorimeter with mass 0.150 kg which contains 0.500 L of water that are both initially at 12.6°C. Over a short period of time the mixture comes to an equilibrium temperature of 22.5°C, what is the specific heat of the metal?
A) c. 1500 J/(kg ∙ °C)
B) d. 3400 J/(kg ∙ °C)
C) b. 1075 J/(kg ∙ °C)
D) a. 1250 J/(kg ∙ °C)

The heat given by the metal and calorimeter = The heat taken by water
$c(Metal)*m(metal)*(Ti1-T_f) + c(copper)*m(calorimeter)*(Ti1-T_f) =$
$= m(water)*C(water)(T_f - Ti2)$
$C(copper) =385 J/(kg*K)$
$C_{water} = 4811 J/(kg*K)$
$C(Metal)*0.3*(88-22.5) = 385*0.15*(22.5-12.6) +0.5*4811*(22.5-12.6)$
$19.65*c(metal) =571.725 +23814.45$
$c(metal) =1241.02 J/(kg*K)$
Correct answer is D) 1250 J/kg/K

2. A gas is taken through the cycle illustrated here. During one cycle, how much work is done by an engine operating on this cycle?
A) 4PV
B) 3PV
C) 2PV

Infinitesimal work is
$d W= P*dV4$ where $d V$ is the infinitesimal change of volume
At constant volume (vertical transformations in the figure) there is no work done.
At the same pressure (horizontal transformations in figure) the work is
$W =2P*(4V-V) - P*(4V-V) = P*3V = 3PV$
Correct answer is B) 3PV

Show My Homework - PV 

3. An object has a weight of 400 N when it is dry. When it is completely submerged in water is has a weight of 150 N. What is the density of the material is the density of water is 1000 kg/m3?
A) c. 1200 kg/m3
B) b. 4500 kg/m3
C) d. 1600 kg/m3
D) a. 2300 kg/m3

$G = m*g = \rho*V*g$ for the dry object
$G(submerged) = [\rho-\rho(water)]*V*g$
$\rho/(\rho-\rho(water)) =G/G(submerged)$
$rho/(\rho-1000) =400/150$
$\rho = (\rho-1000)*2.67$
$\rho-2.67*\rho = -2670$
$1.67*\rho =2670$
$\rho =1598.8 kg/m^3$
Correct answer is C) 1600 kg/m^3

4. Using the standard form of a wave $x = A \cos(\omega t)$, for this wave: $2.5 \cos(6.0 t)$ please determine the following with the correct units:
a. The amplitude.
b. The frequency.
c. The maximum and minimum speed
d. The maximum acceleration

a. A is the amplitude so it is just 2.5 m.
b. the value next to the t is $\omega$, so $f = \omega/2p = 6.0 /(2p) = 0.955 (Hz)$
c. The maximum speed is $\omega*A$ so this is $(6.0 rad/s)(2.5 m) = 15.0 m/s$. The minimum speed is when it stops to “turn around” so it is 0.
d. The maximum acceleration is $-\omega*2A = -(6.0 rad/sec)2(2.5 m) = -90 m/s2$.

5. 4.50 moles of an ideal gas is at 560K. Is the gas undergoes an adiabatic compression and 3750 J of work is done on the gas, what is the final temperature of the gas?
A) b. 572 K
B) a. 627 K
C) c. 2590 K
D) d. 977 K

Adiabatic compression it means there is no heat exchanged.
variation of internal energy = work done
$\Delta(U) = Work$
$N*C_v*(T_f-T_i) = 3750$
where $N$ is the number of gas moles, $C_v = (3/2)*R =12.5 J/(mol*K)$
$4.5*12.5*(T_f-560) =3750$
$T_f-560 =66.67$
$T_f =626.67 K$
Correct answer is B) 627 K

6. A CD with a diameter of 12.5 cm goes from rest to a tangential velocity of 5.6 m/s. If it does this in 2.5 seconds, what is its angular acceleration?
A) d. 16 rad/s2
B) a. 13 rad/s2
C) c. 24 rad/s2
D) b. 18 rad/s2

Angular velocity $\omega = V/R = 5.6/0.125 = 44.8 rad/sec$
angular acceleration $\epsilon = \omega/t = 44.8/2.5 =17.92 rad/s^2$
Correct answer is D) 18 rad/s^2

7. A piece of gold with a mass of 5.50 kg and density of 19300 kg/m3 is suspended from a string and then totally immersed in a beaker of water. Using density of water is 1000 kg/m3
a) Determine the volume of the piece of gold.
b) Determine buoyant force on the gold when it is submersed.

a) \$rho = mass/volume$ so, $volume = mass/density = 5.50 kg/19300 kg/m^3 =$
$= 2.85*10^{-4} m^3 = 285 cm^3$.
b) BF = weight of water displaced, so this equals the volume of gold times the density of water:
$2.85*10{-4} m^3 x 1000 kg/m^3 = 0.285 N$

8. If a car horn honking that is 450 m is heard 1.3 seconds after it goes off, what is the temperature in degrees Celsius?
A) a. 25 °C
B) c. 56 °C
C) b. 34 °C
D) d. 12 °C

speed of sound $V = Distance/Time = 450/1.3 =346.15 (m/s)$
dependence of speed of sound with temperature
$V(T) = 331.3 + 0.606*T$ , T in Celsius degree
$346.15 =331.3 +0.606*T$
$T =24.5$ Celsius degrees
Correct answer is A) 25 Celsius degrees
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Volume of rotation

Show My Homework - Volume of Rotation

a) the 3D object obtained by revolving the shaded area about y axis is a cone, having as basins a circle od radius r and with height h
b) the general equation of the line is
$y = m*x+n$
$m = (y-y0)/(x-x0) = (h-0)/(0- r) = - h/r$ is the slope
$n = h$ is the free term
$y = -h/r*x +h$ is the eq of the line
the infinitesimal area under the line is
$dA = y*dx = [(-h/r)*x +h]*dx$ (see the attached figure)
By revolving this infinitesimal area aroud y one gets the infinitesimal volume
$dV = (2*pi*x)*dA =2*pi*x*[(-h/r)*x+h]*dx$
The volume of the generated cone is simply
$V = \int_0^r {2\pi x[(-h/r)*x +h]*dx} = [-2\pi (h/r)(x^3)/3 +2\pi h(x^2)/2]_0^r =$
$=-2*\pi h(r^3)/3 +\pi h(r^2) = \pi h(r^3)/3$

The volume of a cone having the basis radius r and height h is indeed
$V(cone)  =(1/3)*\pi h(r^2)$

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Monday, March 4, 2013

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Electricity (Homework 4)

38. Calculate the current in the 240 ohm  filament of a bulb connected to a 120-V line.
Current = Voltage/ Resistance
$I =U/R =120/240=0.5 A$

40. Calculate the power of a hair dryer that operates on 120 V and draws a current of 10 A.
$Power = Voltage*Current = U*I = 120*10 =1200 W$

70. Why is charge usually transferred by electrons rather than by protons?
 A transferred charge means that a  charge is taken away from an atom and moves to s second atom and so on. Since the forces that keep together the electrons in the atoms (electric forces) are much weaker than the forces that keep together the protons in a atom (nuclear forces) than the charge is transferred by electrons rather than protons.

72. Why do clothes often cling together after tumbling in a clothes dryer?
Because the became electrically charged by the friction between them. Different materials become electrically charged with different (opposite) charges. Opposite charges attract each other.

86. What happens to the brightness of light emitted by a light bulb when the current in the filament increases?
Since the electrical power delivered is proportional to the current, then the electrical power increase. Since this electrical power is transformed into light with in a certain proportion, then the brightness of the light emitted by the bulb will increase.

88. Why are compact fluorescent lamps (CFLs) more efficient than incandescent lamps?
CFL are more efficient than the incandescent lamps because the fluorescence mechanism of emitting light by atoms is in itself more efficient than incandescence mechanism of emitting light. This first mechanism implies atoms of the emitting material are excited by impact with moving IONS and then un-excite emitting photons, the second mechanism  implies the moving ELECTRONS collide with atoms, excite them and then these atoms un-excite emitting photons. Since mass of moving IONS is larger than mass of moving ELECTRONS and then the efficiency of excitation the atoms is higher in the first mechanism.

92. Are automobile headlights wired in parallel or in series? What is your evidence?
The automobile headlights are wires in parallel. The evidence of parallel wiring is that all the bulbs have the same working Voltage (12 V- battery voltage). If they were wired in series, the current through bulbs had been the same and the working voltage had been less than 12 V, the battery voltage.

108. Your tutor tells you that an ampere and a volt really measure the same thing, and the different terms only make a simple concept seem confusing. Why should you find another tutor?
Let’s compare the flow of the electric current to a river. Then the voltage would be equivalent with the height of a water fall and the current with the quantity of water that falls. There is no equivalence (in other words it is not the same thing) between the height of the fall and the quantity of water that falls.

110. Does more current “flow” out of a battery than into it? Does more current “flow” into a light bulb than out of it? Explain.
The answer is no. Since the electric current represents the movement of electric charges, and since there is no charge created outside the battery or inside the light bulb then the current is the same in and out.

112. Does a lamp with a thick filament draw more current or less current than a lamp with a thin filament? Defend your answer.
The current in the filament is inversely proportional to the resistance of the filament as ohm law states. $I = U/R$. Since Resistance is inversely proportional to the thickness ($S$- transverse area of filament)  $R = \rho*L/S$  then this implies $I$ proportional to $S$ (thickness). It means the thicker the filament, the greater current.

120. As more and more bulbs are connected in series to a flashlight battery, what happens to the brightness of each bulb? Assuming that the heating inside the battery is negligible, what happens to the brightness of each bulb when more and more bulbs are connected in parallel?
The brightness of a bulb is proportional to the current through it. If you connect in series, the total resistance increases ($R = R1+R2+R3+...$). The current $I = U/R$ ($U$ is the battery voltage) decreases, and therefore the brightness decreases.
When you connect in parallel bulbs to a battery, if the battery is IDEAL then the voltage and current through the bulbs stays the same and therefore the brightness stays is the same.

Chapter 9
31. Write a letter to a relative or friend saying that you have discovered the answer to what has been a mystery for centuries— the nature of light. State how light is related to electricity and magnetism.
Beginning with the 18th century laws of electrostatics were discovered (electrical charges create an electric field) .  At the beginning of the 19th century a new chapter of physics began to be studied, i.e. the moving electric charges (which create a magnetic field)  became to be studied and the chapter in physics that deals with this was named electrodynamics. At the end of 19th century electrodynamics discovered the laws of by which an electrical VARIABLE field transforms itself into a magnetic VARIABLE field,  and named these laws the Maxwell  laws.  This way the electric and magnetic fields  were reunited into a single field ELECTROMAGNETIC field. The Maxwell laws predict for the speed of an electromagnetic field the same value as the speed of light has. Light by its nature is electromagnetic and has the same laws as Maxwell laws.

48. All atoms have moving electric charges. Why, then, aren’t all materials magnetic?
The answer is that the electrons moving inside the atoms have also a property named SPIN. The electric currents generated by the movement of the electrons depends on the value of the electron spins. Of these spins couples themselves (are parallel in other words) then the material is magnetic otherwise (if the spins are opposite, as in the majority of materials) and the moving electrons does not couple each other the material is nonmagnetic.

52. What kind of force field surrounds a stationary electric charge? What additional field surrounds it when  it moves?
See answer to question 31.
 A stationary electric charge creates am ELECTRIC field around it. A moving electric charge is equivalent to a current and creates a MAGNETIC field around it. Both these fields are reunited into a single filed named ELECTROMAGNETIC field.

92. Discuss why a motor also tends to act like a generator.
A motor is constructed of rotor windings through which current flows creating a magnetic field. This field interacts with a second magnetic field (of the stator) created by permanent magnets and creates motion (i.e. applies an internal force to exterior). This a a motor.
A generator is constructed of the same rotor windings that  moves (because of an external force) into the same second magnetic field created by the permanent magnets of the stator. As Maxwell laws says a  variable magnetic field (because of the movement of the rotor) creates an electric field. Therefore in the windings there will be an electric current. This is a generator.

In other words a motor transforms the electrical force  in mechanical force, and the same device can perform in an opposite direction to transform mechanical force into electric force.
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Ionization energy

1. Atomic potassium emits light very weakly at 365 nm when an excited electron moves from a 4d-orbital to a 4s-orbital , and at 689 nm when an electron moves from a 4d-orbital to a 4p-orbital. What is the energy separation (in kJ/mol) between the 4s- and 4p-orbitals in a potassium atom?

 2. Look up the ionization energy for hydrogen, in kJ/mol (cite your source; this value is widely available, but can also be found in chapter 8 of your textbook). Compare this quantity with the value determined by calculating the ΔE for an electron transition from n = 1 to n = ∞ using the Bohr formula for the energy levels (express the result in kJ/mol).
Ionization Energy for H: ________ (Source:______________) Calculated value:_____ 

3. A student mixed 50Ml of water containing 0.010moles of HCl with 60Ml of water containing 0.012 moles of NaOH in a coffee cup calorimeter. Both the solutions were initially 25C but after reaction upon mxing the temperature rose to 26.2C. The specific heat of the solution is 4.18J/g C, and the calorimeter has a het capacity of 35J/C. The reaction is shown below. How much heat is produced in the reaction? How much heat is produced if one mole of water is formed in this chemical reaction?

1. the separation energy between levels 4d and 4s is
$E(d-s) = h*F_1 = (h*c)/\lambda_1 = (6.626*10^{-34})*(3*10^8)/(365*10^{-9}) =5.44*10^{-19} J$
the separation energy between levels 4d and 4p is
$E(d-p) = h*c/\lambda_2 = (6.626*10^{-34})*(3*10^8)/(689*10^{-9}) =2.885*10^{-19} J$
The separation energy between 4p and 4s is
$E(p-s) =5.44*10^{-19} -2.885*10^{-19} =2.55*10^{-19} J =1.597 eV$

2. http://en.wikipedia.org/wiki/Molar_ionization_energies_of_the_elements
For Hydrogen $E=1312 KJ/mol$
in one mol there are Avogadro number atoms. $N_A = 6.023*10^{23} atoms$
Energy per atom $E =1312*10^3/6.023*10^{23} =2.178*10^{-18} (J/atom)$

From Bohr model of atom transition energy between states n and m is
$E(n-m) = E_1 *(1/n^2  -  1/m^2)$ where $E_1=-13.6 eV $ is the energy of first level
Therefore the ionization energy for Hydrogen is ($n= \infty$ , $m=1$) from Bohr model is
$E = 13.6 eV  = 13.6*1.6*10^{-19} =2.176*10^{-18} J$
For one mole the energy is
$E(one mole) = E*N_A =2.176*10^{-18}*6.023*10^{23} =1310.6 KJ/mol$
Ionization Energy for H: 1312 KJ/mol
Calculated value: 1310.6 KJ/mol

$m(solution)*c(solution)(Tf-Ti)  + C(calorimeter)*(Tf-Ti) = Q$
We suppose the solution has density of 1 g/ml
$Q = (50+60)*4.18*(26.2-25) +35*(26.2-25) =593.76 J$

The eq. of reaction between HCl and NaOH is
$HCl +NaOH = H2O + NaC$
from here one can see that to form one mole of water it is necessary one mole of HCl and one mole of NaOH . Therefore in the reaction need to enter not 0.010 mol HCl and 0.012 mol NaOH but 1 mol HCl and 1 mol NaOH.

Heat of reaction is
$H = H(products) - H(reactants)$

$H(water) = 285.8 KJ/mol$
$H(NaCl) =385.92 KJ/mol$
$H(HCl) = 92.30 kJ/mol$
$H(NaOH) =469.15 KJ/mol$
$H = (285.8+385.92)-(469.15+92.30) =110.27 KJ/mol$
Since it is formed one mole of water the total heat produced is approximately equal to
$Q =H = 110.27 KJ$
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X Ray (Maximum Wavelength)

What is the cut off wavelength in an X-ray production tube if the accelerating voltage is 5 kV?

The cutoff frequency is when all the energy of the incoming electrons (e*U) is transferred to the photons (h*F). e is the electron charge, U is the accelerating Voltage, h is the plank constant, F is the frequency.
$e*U = h*F$
$F = C/\lambda$
C is the speed of light, lambda is the wavelength
$e*U = h*C/\lambda$
$\lambda = h*C/(e*U) = (6.626*10^-34)*(3*10^8)/[(1.6*10^-19)*(5*10^3)] =$
$=2.485*10^{-10} m =24.85 nm$
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