1.
An absolute temperature of 300 K is
- how many degrees Fahrenheit?
- How many degrees Celcius?
Exercises 10.8 Exercise
Temperature & Expansion.
2.
Find the temeprature at which Fahrenheit and Celcius scales have the same reading.
3.
An aluminum wire 0.05 mm in diameter and 2 m long is heated from \(15^{o}C\) to \(50^{o}C\text{.}\) How much does it contract?
4.
A steel gas tank \((\alpha = 12\times 10^{-6} /^{o}C)\) is filled with 80 L (about 20 gallons) of gasoline \((\beta = 9.5\times 10^{-4} /^{o}C)\) at a temperature of \(0^{o}C\text{.}\)
- How much gasoline flows out of the tank when the temperature rises to \(30^{o}C\text{?}\)
- If the hose of the tank is 4 cm in diameter, what is the minimum length of the hose in order to keep the gasoline from spilling?
5.
A thin steel wire \((\alpha = 12\times 10^{-6} /^{o}C,\) \(\rho = 7.8 g/cm^{3}, \) \(Y = 20\times 10^{10} Pa)\) of length 60 cm and diameter 0.5 mm has its ends attached to a large brass rod \((\alpha = 19\times 10^{-6} /^{o}C,\) \(\rho = 8.6 g/cm^{3}, \) \(Y = 9\times 10^{10} Pa)\) of the same length. The system is heated from \(25^{o}C\) to \(100^{o}C\text{.}\)
- How much does the brass rod expand?
- How much would the steel wire have expanded if it had not been attached to the brass rod?
- What is the thermal strain in the steel wire attached to the brass rod?
- What is the thermal stress in the steel wire?
- What is the tension in the wire?
6.
The coefficient of linear expansion of lead is \(29\times 10^{-6} K^{-1}\text{.}\) What change in temperature will cause a 2 m long lead bar to change in length by 3.0 mm?
7.
An aluminum can is filled to the brim with a liquid. The can and the liquid are heated so their temperatures change by the same amount. The can’s initial volume at \(10^{o}C\) is \(4.5\times 10^{-4} \,m^{3}\text{.}\) The coefficient of volume expansion for aluminum is \(69\times 10^{-6} /^{o}C.\) When the can and the liquid are heated to \(70^{o}C,\) \(2.9\times 10^{-6} m^{3}\) of liquid spills over. What is the coefficient of volume expansion for the liquid.
Calorimetry.
8.
If 100 g of steam condense into 1 kg of water initially at \(25^{o}C\text{,}\) what is the final temperature of the combination? \((L_{v} = 540 \,cal/g)\)
9.
How much heat is required to take 10 moles of ice at \(0^{o}C\) to water at \(100^{o}C\text{?}\)
10.
A 400 g piece of iron is taken from an oven at \(300^{o}C\) and placed in a 100 g aluminum calorimeter containing 200 g of ice at \(0^{o}C\text{.}\)
- How much ice melts?
- What is the final temperature of the system?
11.
A water fall is 500 m high.
- What is the difference in temperature between the water at the top and bottom if no water evaporates while falling?
- What fraction of the water must evaporate to keep the temperatures the same?
12.
A man warms his hands by rubbing them together. Assume that his hands are made mostly of water, that they are 15 cm long and 5 cm wide, that he presses his hands together with 50 N of force, that the coefficient of friction between his hands is 0.5, that the hands move 10 cm with each stroke, that 30 strokes are made, and that the temperature of the surface of his hands increases \(20^{o}C\text{.}\) How thick is the surface of his hands which is being heated?
13.
A 1.55 g of methanol is burnt in a calorimeter. If the molar heat of combustion of methanol is -725 \,kJ/mole, and assuming that the 2.0 L of water absorbed all of the heat of combustion, what temperature change did the water experience?
14.
A 20g sample of iron at a temperature of \(120^{o}C\) is placed into a container of water. There are 300 ml of water in the container at a temperature of \(30^{o}C\text{.}\) What is the final temperature of the water? \(c_{iron}=0.444 \,J/g.^{o}C,\) \(c_{water}=4.184 \,J/g.^{o}C,\) and \(\rho_{water}=1 \,g/ml.\)
Heat Transfer.
15.
A copper rod \((k = 395 \,W/m.^{o}C)\) of length 40 cm and diameter 2 cm is placed end to end with an iron rod \((k = 79.5 \,W/m.^{o}C)\) of length 60 cm and diameter 2 cm. The combined rod is then insulated on its sides and the two ends subject to different temperatures. If the copper end is at \(100^{o}C\) and the iron end is at \(20^{o}C\text{,}\)
- what is the temperature at the copper-iron interface?
- How many joules of heat energy flow through the system in 1 hour?
16.
A Styrofoam ice chest \((k = 0.05 \,W/m.^{o}C)\) is cubic in shape with each edge 30 cm long. Each face is a square sheet of Styrofoam 2 cm thick. If a 4 kg bag of ice is placed in the chest and the outside temperature is a constant \(30^{o}C\text{,}\)
- how long does it take the ice to melt?
- How long after the ice melts does it take for the interior of the ice chest to heat up to \(15^{o}C\text{?}\)
17.
A solar home has 4 solar collectors each 1 m by 2 m with an average efficiency of \(20\%\text{.}\) If the sun radiates \(1.36 kW/m^{2}\) of solar power upon the earth,
- how many Btu’s of heat are collected on an average day?
- If the house is well insulated on the sides and bottom and if the roof/ceiling is 20 ft by 40 ft consisting of lapped shingles (R = 0.87), dead air space (R = 1.01), 6 inch fiberglass insulation (R = 18.80), and sheetrock (R = 0.45), what will be the average interior temperature of the house if the outside temperature is a freezing \(32^{o}F\text{?}\)
18.
Two rods, one of aluminum and the other of copper are joined end to end. The cross-sectional area of each is \(4.0\times10^{-4} \,m^{2},\) and the length of each is 0.04 m. The free end of the aluminum rod is kept at \(302^{o}C,\) while the free end of the copper rod is kept at \(25^{o}C\text{.}\) The loss of heat through the sides of the rods may be ignored.
- What is the temperature at the aluminum-copper interface?
- How much heat is conducted through the unit in 2.0 s? \((\kappa_{al} = 238 \,J/(sm.^{o}C), \quad \kappa_{cu} = 397 \,J/(s m.^{o}C)).\)
19.
The surface of the Sun has a temperature of \(5500^{o}C.\)
- Treating the Sun as a perfect blackbody, with an emissivity of 1.0, find the power that it radiates into space. The radius of the sun is \(7.0\times 10^{8} \,m\) and the temperature of space can be taken to be 3.0 K.
- The solar constant is the number of watts of sunlight power falling on a square meter of the Earth’s upper atmosphere. Use your result from part (a) to calculate the solar constant, given that the distance from the Sun to the Earth is \(1.5\times10^{11} \,m.\)
20.
A long copper rod of diameter 2.0cm is initially at a uniform temperature of \(100 \,^{o} C.\) It is now exposed to an air stream at \(20\,^{o} C\) with a heat transfer coefficient of \(200 \,W/m^{2}K.\) How long would it take for the copper rod to cool at an average temperature of \(25\,^{o} C.\)
KTG.
21.
How many molecules are in a cube of aluminum 2 cm on a side if the density of aluminum is \(2.7 \,g/cm^{3}\) and the atomic mass of aluminum is 26.98 u?
22.
What is the rms speed of a diatomic oxygen molecule at a room temperature of 300 K? Assume \(k_{B} = 1.380\times 10^{-23} \,J/K\text{,}\) \(1 \,u = 1.66\times10^{-27} \,kg, \) \(Z(oxygen) = 8, \quad M(oxygen) = 15.999 \,u,\) \(pa = 1 \,atm = 1.013\times10^{5} \,Pa\text{.}\)
23.
On first day a tank of oxygen \((M = 32.0 \,g/mol)\) having a volume of 100 L is filled to a gauge pressure of \(4.00\times 10^{5} \,Pa\) at a temperature of \(43.0 \,^{o}C\text{.}\) On the second day the temperature is \(22.0 \,^{o}C\) and the gauge pressure of the tank is \(3.00\times 10^{5} \,Pa\text{.}\)
- How many moles of oxygen were initially in the container?
- How many grams of oxygen were used before the second day measurements were made?
24.
At high enough temperatures a diatomic molecule like hydrogen has three degrees of translational freedom, two degrees of rotational freedom, and two degrees of vibrational freedom. How much heat Q must be added to n moles of hydrogen to raise its temperature from such a high temperature to one even higher by an amount \(\Delta T\) if the volume of the gas remains constant?
25.
At what temperature does the oxygen molecules have the same rms speed as those of hydrogen at \(-72 \,^{o}C.\) Molar mass of oxygen and hydrogen are 32 and 2, respectively.
I Law of Thermodynamics.
26.
An ideal monatomic gas containing 10 moles of Helium initially at \(0\,^{o}C\) and one atmosphere is allowed to expand to twice its initial volume. If the gas is first cooled at a constant volume to half its initial pressure and then allowed to expand at constant pressure to its final volume, find each of the following:
- the initial volume of the gas,
- the final pressure of the gas,
- the final volume of the gas,
- the final temperature of the gas,
- the work done by the gas,
- the heat added to the gas,
- the change in internal energy of the gas.
27.
An ideal monatomic gas containing 10 moles of Helium initially at \(0\,^{o}C\) and one atmosphere is allowed to expand to twice its initial volume. If the gas is first allowed to expand at constant pressure to its final volume and then cooled at constant volume to half its initial pressure, find each of the following:
- the initial volume of the gas,
- the final pressure of the gas,
- the final volume of the gas,
- the final temperature of the gas,
- the work done by the gas,
- the heat added to the gas,
- the change in internal energy of the gas.
28.
An ideal monatomic gas containing 10 moles of Helium initially at \(0\,^{o}C\) and one atmosphere is allowed to expand to twice its initial volume. If the gas is allowed to expand adiabadically \((\gamma = 5/3)\) to twice its initial volume, find each of the following:
- the initial volume of the gas,
- the final pressure of the gas,
- the final volume of the gas,
- the final temperature of the gas,
- the work done by the gas,
- the heat added to the gas,
- the change in internal energy of the gas.
29.
- With the assumption that the air temperature is a uniform \(27\,^{o}C\text{,}\) what is the atmospheric pressure at an altitude of 5000 m?
- With the assumption that a bubble of air rises adiabatically from the surface of the earth to the pressure obtained in part (a) of this problem, what would be the temperature of the bubble at that altitude?
- Which of the previous two inconsistent assumptions is more realistic and why?
II Law of Thermodynamics.
30.
A heat engine absorbs 1000 J of heat at a temperature of 400 K and deposits 900 J of heat at 300 K. What is the efficiency of this engine?
31.
A Carnot engine absorb 1000 J of heat at temperature of 400 K and rejects heat at 300 K. How much heat does it reject?
32.
An ideal heat pump pumps heat from an outside temperature of \(0\,^{o}C\) into an inside temperature of \(30\,^{o}C\text{.}\) If 1000 J of heat are deposited into the house every second and if electricity costs 10 cents per kilowatt-hour, then what is the cost of operating this heat pump for 24 h?
33.
A refrigerator requires an amount of work W to extract heat Qc from a refrigerator and reject an amount of heat Qh into the surroundings. What is the coefficient of performance of the refrigerator?
34.
What is the coefficient of performance of a Carnot refrigerator whose inside temperature is \(0\,^oC\) an whose outside temperature is \(30\,^{o}C\) if it takes 500 W of electrical power to operate it?
35.
A gasoline engine operates approximately according to the Otto cycle which takes in a gas/air mixture at ambient temperature \(T_{A}\text{,}\) compresses that mixture adiabatically to a higher temperature \(T_{B}\text{,}\) ignites the mixture at constant volume to an even higher temperature \(T_{C}\text{,}\) expands the gas adiabatically to a cooler temperature \(T_{D}\text{,}\) reduces the pressure at constant volume back to \(T_{A}\text{,}\) and then expels the gas into the atmosphere. The efficiency of an Otto cycle is \(1 - \frac{T_{A}}{T_{B}}\text{.}\) Assume \(T_{A} = 27\,^{o}C,\quad T_{B} = 127\,^{o}C\text{,}\) and \(T_{C} = 227^{o}C\text{.}\)
- What is the efficiency of this gasoline engine?
- What would be the maximum possible efficiency of an ideal heat engine utilizing these same temperatures?
- What is the efficiency ratio for this gasoline engine (i.e. the ratio of its efficiency to the maximum possible efficiency)?
Entropy.
36.
The heat of fusion of a substance is \(L_{f}\text{.}\) What is the change in entropy of a mass m of the substance if it freezes at temperature \(T_{f}\text{.?}\)
37.
A mass m with specific heat c is heated from temperature \(T_{i}\) to \(T_{f}\text{.}\) What is its change in entropy?
38.
Water has a specific heat capacity of 4.18 J/g, a heat of fusion of 333 J/g, and a heat of vaporization of \(2.26\times10^{3} \,J/g\text{.}\) If 100 g of ice at \(0\,^{o}C\) is dropped into 100 g of water at \(27\,^{o}C\text{,}\) then what is the change in entropy of the universe?
39.
Which has the greatest entropy
- ice below freezing temperature,
- ice at freezing temperature,
- liquid water at freezing temperature,
- water above freezing temperature,
- No general statement can be made.
40.
A house has the least entropy when it is
- neat,
- cluttered,
- clean and neat,
- cluttered and dirty,
- No general statement applies in this situation.
41.
A gas expands adiabatically and reversibly. What is its change in entropy?
42.
A Carnot engine in one cycle takes in 2000 J of heat at 500 K and discards heat at 300 K.
- How much work does it do?
- How much heat does it discard?
- What is its change in entropy during its isothermal expansion?
- What is its change in entropy during its adiabatic expansion?
- What is its change in entropy during its isothermal compression?
- What is its change in entropy during its adiabatic compression?
- What is its total change in entropy?
- What is the change in entropy of the hot reservoir?
- What is the change in entropy of the cold reservoir?
- What is the change in entropy of the universe?
- Is this result consistent with the second law of thermodynamics?
