Physics 102 --- Midterm 1 --- 2015 --- Solutions
Starting with 3000 cm3 of isolated ideal gas at atmospheric pressure (take that to be 105 Pa) and room
temperature (22° C), we heat it at constant volume to twice atmospheric pressure, then expand it
isothermally until it attains atmospheric pressure again, and then compress it isobarically back to its
initial condition. The cycle shown is shown in the figure.
a) How much work is done on the gas in the constant volume heating (A ® B)?
b) How much work is done on the gas in the isothermal expansion (B ® C)?
c) How much work in done on the gas in the isobaric compression (C ® A)?
d) How my work is done by the gas in one whole cycle?
e) How much heat is put into the gas in one whole cycle?
VA = 0.003 ;
PA = 100 000. ;
TA = 22 + 273. ;
TB = 2 TA ;
R = 8.314 ;
n = PA * VA HR * TAL ;
Print@"aL Zero work."D
WBC = - n * R * TB * Log@2D ;
Print@"bL WorkHB to CL = ", WBC, " J"D
VC = 2 VA ;
WCA = - PA * HVA - VCL ; Print@"cL WorkHC to AL = ", WCA, " J"D
Wcycle = - HWBC + WCAL; Print@"dL WorkHby gas, one cycleL = ", Wcycle, " J"D
Print@"eL Heat in = ", Wcycle, " J"D
aL Zero work.
bL WorkHB to CL = -415.888 J
cL WorkHC to AL = 300. J
dL WorkHby gas, one cycleL = 115.888 J
eL Heat in = 115.888 J
2. A concrete roadway is laid down in 12 meter sections at 2° C.
a) Anticipating that the temperature will sometime be 40° C, how much of a gap should be left between
sections to prevent buckling?
b) Suppose no gap is left, and the roadway buckles in a simple way: the original 12 meter sections break
in half, and the pieces rise up in the middle so that they can fit into the original length allowance. See the
diagram. How high, h, will the middle rise?
L0 = 12 ;
Α = 12. * 10 ^ - 6 ;
Tf = 40 ;
Ti = 2 ;
DL = Α L0 HTf - TiL ; Print@"aL gap = DL = ", 100 DL, " cm"D
H* h^2 + L0^24 = HL0+DLL^24 LL^24 + L0*DL2 + small *L
H* h^2 = L0*DL2 *L
h = Sqrt@L0 * DL 2D; Print@"bL bump height = h = ", 100 h, " cm"D
aL gap = DL = 0.5472 cm
bL bump height = h = 18.1196 cm
3. We have a cup of coffee at 80° C. The amount of coffee is 120 gm (about 4 oz.). The day is hot and we
want iced coffee at 15° C.
a) How much ice do we have to add to the coffee to cool it as required? Assume that the ice starts at 0° C,
that the heat transfer involves only the ice and the liquid in the cup, and that coffee has the same heat
2
Midterm1_P102_2015a.nb
3. We have a cup of coffee at 80° C. The amount of coffee is 120 gm (about 4 oz.). The day is hot and we
want iced coffee at 15° C.
a) How much ice do we have to add to the coffee to cool it as required? Assume that the ice starts at 0° C,
that the heat transfer involves only the ice and the liquid in the cup, and that coffee has the same heat
capacity as water.
b) How many ice cubes (nearest whole number) would this require? Take each ice cube to be 8 grams, or
roughly 2 cm on a side.
Th = 80;
Tc = 0;
Tf = 15;
c = 1;
Lfusion = 80. ;
mcoffee = 120;
Qcoffee = c * mcoffee * HTf - ThL;
H* Qice=Lfusion mice + c*mice*HTf-TcL *L
mice = - Qcoffee HLfusion + c * HTf - TcLL;
Print@"aL mice = ", mice, " gm"D
Print@"bL No. 8 gm ice cubes = ", IntegerPart@mice 8D,
"
Hfor whole number obtained by rounding down; one higher will be acceptableL"D
aL mice = 82.1053 gm
bL No. 8 gm ice cubes = 10 Hfor whole number obtained by rounding down; one higher will be acceptableL
Midterm1_P102_2015a.nb
{4.} The average outside temperature during a given period is 0° C. A heat pump, which we shall view as
an ideal Carnot engine running backwards, is installed to keep the inside temperature of a house a 20° C.
Measurements show that heat is needed at a rate of 3,000 Watts to maintain this temperature difference.
The heat pump is run by a motor that can work at the rate of 320 Watts.
a) What is the efficiency of an ideal Carnot engine operating between these two temperatures?
b) What is the maximum heat the heat pump can deliver to the house?
c) Will the heat pump suffice to keep the house warm?
Suppose there is a cold snap, and the average temperture becomes -8° C over certain period of time. For
the same inside temperature of the house and the same heat pump, answer the same questions again:
d) What is the efficiency of an ideal Carnot engine operating between the two relevant temperatures?
e) What is the maximum heat the heat pump can deliver to the house?
f) Will the heat pump suffice to keep the house warm?
Th = 20. + 273 ;
Tc = 0 + 273 ;
WorkRate = 320 ;
eff = 1 - Tc Th ; Print@"aL Efficiency = ", effD
H* CoP=1eff *L
Qh = WorkRate eff; Print@"bL Heat pump heat delivery rate = ", Qh, " Watts"D
Print@"cL Yes."D
Th2 = 20. + 273 ;
Tc2 = - 8 + 273 ;
eff2 = 1 - Tc2 Th2 ; Print@"dL Efficiency now = ", eff2D
Qh2 = WorkRate eff2; Print@"eL Heat pump heat delivery rate = ", Qh2, " Watts"D
Print@"fL No. The temperature gap is
40% larger and 4200 Watts are needed to heat the house."D
aL Efficiency = 0.0682594
bL Heat pump heat delivery rate = 4688. Watts
cL Yes.
dL Efficiency now = 0.0955631
eL Heat pump heat delivery rate = 3348.57 Watts
fL No. The temperature gap is 40% larger and 4200 Watts are needed to heat the house.
3