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Material Type: Exam; Professor: Echt; Class: General Physics II; Subject: Physics; University: University of New Hampshire-Main Campus; Term: Fall 2010;
Typology: Exams
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Your name: __________________SOLUTION_________________________________________
first last your recitation section
Please detach the cheat sheet (at the end of this exam), and then write your name on the back of the last page
Problem # max points your score
1 30
2 15
3 20
4 15
5 20
total 100
Multiple choice, 3 points each, only one correct answer for each problem.
1.1 At room temperature, C V of nitrogen (N 2 ) is larger than that of helium (He) because a) N 2 also has translation energy b) N 2 also has rotational energy c) N 2 also has vibrational energy d) N 2 has a larger mass than He
1.2 Consider containers of equal volume, all filled with one mole of an ideal gas at temperature T = 300 K. For which of the following gases would the average translational energy per molecule be smallest? a) neon, atomic mass 20 u, b) oxygen O 2 , molecular mass 32 u, c) water H 2 O, molecular mass 18 u, d) all equal.
1.3 The equipartition theorem states that a) for a given temperature, all gases have the same thermal energy per molecule b) all gases have the same number of degrees of freedom c) all gases have an energy of 0.5 k B T per degree of freedom d) all gases have the same molar specific heat at constant volume, C V.
1.4 For this cyclic process heat is added to the engine a) from 1 to 2 b) from 2 to 3 c) from 3 to 1 d) a and b e) a and c f) b and c
1.5 For this cyclic process heat 30 + 84 J are added to the engine while in the third process 38 J are discharged. The efficiency of this engine is approximately a) 10 % b) 33 % c) 50 % d) 67 % e) impossible to tell without knowing the amount of work done.
Solution
a) W cyc = |enclosed area| = ½ (300 – 100) kPa * (600 – 200) cm^3 = ½ 200E3 Pa * 400E-6 m^3 = 40 J
b) First law => W cyc = Q H – Q C => Q H = W cyc + Q C = 40 + 100 + 180 = 320 J
c) efficiency = benefit/cost = W (^) cyc / QH = 40/320 = 0.125 = 12.5 %
1 1
2 2 2 1 1
1 1 2
(temperature at point 2)
e) The efficiency of a Carnot engine is H
H C
η =. With the temperatures computed in (d) that would be
η=
b) Compute the root-mean-square speed of the argon atoms.
c) Another rigid container (volume 0.03 m^3 ) contains 84 g of nitrogen (N 2 ) with an initial thermal energy of 18 kJ. What is the temperature of the gas?
d) The two containers are brought into thermal contact with each other but they are still thermally insulated from their surroundings. What is the final temperature of the gas after thermal equilibrium has been established?
e) How much heat has been absorbed by the colder gas?
Solution
a) Argon has A = 40 => mass^80
g
2
th V
b)
3 1 2 3 31.38 23481 m 2 2 40*1.66 27 548 s
k TB (^) E ε (^) avg k TB mvrms vrms (^) m E − = = → = = (^) − =
c) Similar to part a, but n = 84g/ 28 g/mol = 3, C v = 5 R /2 for N 2 => (^5) 2
th V
d) Energy is conserved. Initial total energy is E tot = 12 kJ + 18 kJ = 30 kJ; this equals the total energy of the
two gases when they have come into equilibrium at temperature T f :
tot f Ar V Ar N V N
e) Q = ∆ E th because W = 0 (rigid containers, no change in volume). N 2 is the colder gas; we have
Solution
a) The point P is at the center of the square. Symmetry =>
the fields due to the pair of 2 q charges cancel, the fields due to the pair of - q charges cancel, the fields due to the pair of -3 q charges cancel. So, we are left with the fields due to charge q at the upper right corner and 3 q at the lower left corner. These two charges produce fields at P that are opposite in direction. The contribution from the 3q charge is larger => the field points at P points diagonally to the upper right.
b) The magnitude of E is the difference between the two contributions,
where I have used the Pythagorean theorem to write r^2 = a^2 + a^2 = 2a^2.
c) E computed in (b) forms an angle of 45° with the x-axis.
Cheat sheet
You don’t have to return this sheet.
Units: SI units: m = meters; s = seconds; kg = kilograms; J = joules; Pa = pascals; K = kelvins Non-SI units: 1 g (gram) = 10 -3^ kg; 1 L (liter) = (10 cm)^3 ; 1 atm (atmosphere) = 1.013× 105 Pa; 1 cal (calorie) = 4.186 J, 1 atomic mass unit u = 1.661× 10 -27^ kg.
Constants: Boltzmann constant k B = 1.38× 10 -23^ J/K; universal gas constant R = 8.31 J/(K mol); Avogadro’s number N A = 6.02× 1023 /mol, gravitational acceleration on earth g = 9.81 m/s^2 , speed of light c = 3× 10 8 m/s, speed of sound in air at 20 °C: v = 343 m/s, permittivity constant ε 0 = 8.85× 10 -12^ C^2 /(m^2 N), K = 1/(4πε 0 ) = 8.99× 109 N m^2 /C^2 , elementary charge e = 1.60× 10 -19^ C, electron mass m e = 9.11× 10 -31^ kg, proton mass m p = 1.67× 10 -27^ kg
Equations Isochoric process: (^) Q = nCV ∆ T with C V = 3/2 R for a monatomic ideal gas, C V = 5/2 R for a diatomic ideal gas,
Isobaric process: (^) Q = nC (^) p ∆ T for an ideal gas at constant pressure, (^) C (^) p = CV + R ,
pV^ γ^ = const for an adiabatic process of an ideal gas; the adiabatic coefficient^ γ^ is shorthand for^ Cp / CV = ( f +2)/ f
Pressure on one side of a wall due to molecular collisions 2 3 rms
p mv V
Thermal energy per particle:
1 2 1 2
for idealized systems (fails at low temperature for all real systems)
for real systems over a limited temperature range
th B
th B
f k T
∆ f k ∆T
Number of degrees of freedom: f = 3 for monatomic ideal gas, 5 for diatomic ideal gas, 6 for solid.
Efficiency of Carnot engine H^ C H
η = −
Some properties of matter mass density of water 1000 kg/m^3 mass density of air under standard conditions 1.28 kg/m^3 mass density of helium gas under standard conditions: 0.18 kg/m^3 mass density of aluminium is 2700 kg/m^3 mass density of silicon is 2400 kg/m^3 mass density of ethyl alcohol is 890 kg/m^3 specific heat of ice c ice = 2090 J/ (kg K) specific heat of water (soda, coffee, tea...) c w = 4190 J/ (kg K) specific heat of glass c g = 750 J/ (kg K) latent heat of melting ice L f = 3.33 × 10 5 J/kg (heat of fusion) latent heat of vaporization of water L v = 2.26 × 10 6 J/kg Approximate atomic masses of some elements: He (4 u), C (12 u), N (14 u), O (16 u), Ne (20 u), Ar (40 u).
