Heat Study Exercise - I

HEAT Exercise - I Page # 20 UNSOLVED PROBLEMS Q.1 A ‘thermacole’ icebox is a cheap and efficient method for storing small quantities of cooked food in summer in particular. A cubical icebox of side 30 cm has a thickness of 5.0 cm. If 4.0 kg of ice is put in the box, estimate the amount of ice remaining after 6 h. The outside temperature is 45 oC, and co-efficient of thermal conductivity of thermacole is -1 -1 o -1 0.01 J s m C . -1 [Heat of fusion of water = 335 x 103 J kg ]. Q.2 A constant volume gas themometer using helium records a pressure of 20.0 kPa at the triple-point of water, and pressure of 14.3 kPa at the temperature 2 of ‘dry ice’ (solid CO ). What is the temperature of ‘dry ice’ ? 2 Q.2 A brass boiler has a base area of 0.15 m and thickness 1.0 cm. It boils water at the rate of 6.0 kg/ min when placed on a gas stove. Estimate the temperature of the part of the flame in contact with -1 the boiler. Thermal conductivity of brass = 109 J s -1 o -1 m C . 3 -1 [ Heat of vaporization of water = 2256 x 10 J kg ]. Q.4 The electrical resistance in ohms of acertain thermometer varies with temperature according to the -3 approximate law : R = R [1 + 5 x 10 (T – T )] o o The resistance is 101.6 W at the triple-point of water, and 165.5 W at the normalmelting point of lead (600.5 K ). What is the temperature when the resistance is 123.4 W ? Q.3 Estimate the temperature of the surface of the Sun from the following data : 8 average radius of the earth’s orbit = 1.5 x 10 km, 5 average radius of the Sun = 7.0 x 10 km, solar radiant power on the earth at noontime -2 = 1400 W m Assume the Sun to be a perfect black body. Will your estimate be greater of less than the actual surface temperature of the Sun ? Explain. Q.5 Answer the following : (a) The trople- point of water is a standard fixed point in modern thermometry. Why ? What is wrong intaking the melting point of ice and the boiling point of water as standard fixed points ( as was originally done inthe Celsius scale)? (b) There were two fixed point in the original Celsius scale as mentioned above which were assigned the o o number 0 C and 100 C respectively. On the absolute scale, one of the fixed points is the tr iple po int of water, which on the kelvin absolute scale is assigned the number 273.16 K. What is the other fixed point on this (Kelvin) scale ? (c) The absolute temperature (Kelvin scale) T is related to the temperature t on the Celsius scale by c t = T – 273.15 c Why do we have 273.15 in this relation, and not 273.16 (d) What is thetemperature of the triple-point of water on an absolute scale whose unit interval size is equal to that of Fahrenheit scale ? Q.4 Explain why : (a) a body with large reflectivity is a poor emitter (b) a brass tumbler feels much colder that a wooden tray on a chilly day (c) an optical pyrometer (for measuring high temperatures) calibrated for an ideal black body radiation gives too low a value for the temperature of a red hot iron piece in the open, but gives a correct value for the temperature when the same piece is in the furnace (d) the earth without its atmosphere would be inhospitably cold (e) heating systems based on circulation of steam are more efficient in warming a building than those based on circulation of hot water o o Q.5 A body cools from 80 C to 50 C in 5 minutes. o Calculate the time it takes to cool from 60 C to 30 o o C. The temperature of the surroundings is 20 C. Q.6 The triple points of neon and carbon dioxide are 24.57 K and 216.55 K respectively Express these temperatures on the Celsius and Fahrenheit scales. Q.3 Two absolute scale A and B have triple points of water defined ot be 200 A and 350 B. What is the relation betwwen T and T ? A B Q.6 Two ideal gas thermometers A and B use oxygen and hydrogen respectively. The follwing observations are made : (a) What is the absolute temperature of normal melting point of sulphur as read by thermometers A and B ? (b) What do you think is the reason for the slightly different answers from A and B ? (The thermometers are not faulty). What further procedure is needed in the experiment to reduce the discrepancy between the two readings? HEAT Q.7 A steel tape 1 m long is correctly calibrated for a temperature of 27.0 °C. The length of a steel rod measured by this tape is found to be 63.0 cm on a hot day when the temperature is 45.0 °C. What is the actual length of the steel rod on that day ? What is the length of the same steel rod on a day when the temperature is 27.0 °C ? Coefficient of linear expansion of steel = 1.20 × 10–5 °C–1. Q.8 A hole is drilled in a copper sheet. The diameter o of the hole is 4.24 cm at 27.0 C. What is the change in the diameter of the hole when the sheet is heated o to 227 C? Coefficient of linear expansion of copper -5 o -1 =1.77 x 10 C . o Q.9 A brass wire 1.8 m long at 27 C is held taut with little tension between two rigid supports. If the o wire is cooled to a temperature of -39 C, what is the tension developed in the wire, if its diameter is 2.0 mm ? Co - efficient of linear expansion of brass -5 o -1 = 2.0 x 10 C ; Young’s modulus of brass 11 = 0.91 x 10 Pa. Q.10 A brass or d of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined o o rod at 250 C, if the original lengths are at 40.0 C? Is there a ‘thermal stress’ developed at the junction ? The ends of the rod are free to expand ( Co-efficient o -1 of linear expansion of brass = 2.0 x 10-5 C , steel -5 o -1 = 1.2 x 10 C ). Q.11 The coefficient of volume expansion of glycerin -5 o -1 is 49 x 10 C . What is the fractional change in o its density for a 30 C rise in temperature ? Q.12 A 10 kW drilling machine is used to drill a bore in a small aluminium block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minutes, assuming 50% of power is used up in heating the machine itself or lost to the surroundings. Specific -1 o -1 heat of aluminium = 0.91 J g C . Q.13 A copper block of mass 2.5 kg is heated in a o furnace to a temperature of 500 C and then placed on a large ice block. What is the maximum amount of ice the can melt ? -1o -1 (Specific heat of copper = 0.39 J g C ; heat of fusion of water = 335 J g-1). Q.14 In an experiment on the specific heat of a metal, o a 0.20 kg block of the metal at 150 C is dropped in a copper calorimeter (of water equivalent 0.025 kg) 3 o containing 150 cm of water at 27 C. The final o temperature is 40 C. Compute the specific heat of the metal. If heat losses to the surroundings are not negligible, is your answer greater of smaller than the actual value for specific heat of the metal ? Page # 21 Q.15 A geyser heats water flowing at the rate of 3.0 o o liters per minute from 27 C to 77 C. If the geyser operates on a gas burner, what is the rate of consumption of the fuel is its heat of combustion is 4.0 x 104 J/g ? Q.16 In an experiment, the specific heats of some inert gases (at ordinary temperatures are measured to be as follows : Try to discover a regularity in the data and explain it on the basis of kinetic theory. Q.17 Given below are observations on molar specific heats at room temperature of some common gases. The measured molar specific heats of these gases are markedly different from those for monatomic gases. [Typically, molar specific heat of a monatomic gas is 2.92 cal/mol K, as you must have worked out in 12.16.] Explain this difference. What can you infer from the somewhat larger (then the rest) value for chlorine ? Q.18 The molar specific heat of hydrogen (H ) is about 2 (5/2) R in the temperature range of about 250 K to 750 K. At lower temperatures, molar specific heat of hydrogen decreases to the value typical of monatomic gases: (3/2 ) R. At higher temperatures, it tends to the value (7/2) R. What do you think is happening ? Q.19 What amount of heat must be supplied to 2.0 x -2 10 kg of nitrogen (at room temperature ) to raise o its temperature by 45 C at constant pressure -1 -1 ?(Molecular mass of N = 28; R = 8.3 J mol K .) 2 Q.20 Answer the following questions bases on the P– T phase diagram of carbon dioxide [Fig. 12.11 (b)] : (a) At what temperature and pressure can the solid, liquid and vapour phases of CO co-exist in 2 equilib rium ? (b) What is the effect of decrease of pressure on the fusion and boiling point of CO ? 2 (c) What are the critical temperature and pressure for CO ? What is their significance ? 2 o (d) Is CO solid, liquid or gas at (a) -70 C under 1 2 o o atm, (b) -60 C under 10 atm, (c) 15 C under 56 atm ? Q.21 Answer the following questions based on the P–T phase diagram of CO [Fig. 12.11 (b)] : 2 o (a) CO at 1 atm pressure and temperature -60 C is 2 compressed isothermally. Does it go through a liquid phase ? 2 (b) What happens when CO at 4 atm pressure is cooled from room temperature at constant pres sure (c) Describe qualitatively the changes in a give mass of solid CO at 10 atm pressure and tempera ture -65 2 o C as it is heated up to room temperature at constant pressure. 2 o (d) CO is heated to a temperature 70 C and HEAT compressed isothermally. What changes in its proper ties do you expect to observe ? Q.22 A fat man is used to consuming about 3000 kcal worth of food everyday. His food contains 50 g of butter plus a plate of sweets everyday, besides items which proved him with other nutrients (proteins, vitamins, minerals, etc) in addition to fats and carbohydrates. The caloric value of 10 g of butter is 60 kcal and that of a plate of sweets is averagely 700 kcal. What dietary strategy should be adopt to cut down his calories to about 2100 kcal per day ? Assume the man cannot resist eating the full plate of sweets once it is offered to him ! o Q.23 A child running a temperature of 101 F is given an antipyrin (i.e. a medicine that lowers fever) which causes an increase in the rate of evaporation of sweat o from his body. If the fever is brought down to 98 F in 20 min, what is the average rate of extra evaporation causes, by the drug. Assume the evaporation mechanism to be the only way by which heat is lost. The mass of the child is 30 kg. The specific heat of human body is approximately the same as that of water, and latent heat of evaporation of water at -1 that temperature is about 580 cal g . Q.24 What is the relative humidity on a day in July in Mumbai when the partial pressure of water vapour is 5 o 0.013 × 10 Pa and the temperature is 15 C ? The vapour pressure of water at this temperature is 5 0.0169 x 10 Pa. Q.25 Answer the following questions: (a) A vessel with a movable piston maintained at a constant temperature by a thermostat contains a certain amount of liquid in equilibrium with its vapour. Does this vapour obey Boyle’s law ? In other words, what happens when the volume of vapour is decreases ? Does the vapour pressure increase ? (b) What is meat by ‘superheated water’ and ‘supercooled vapour’ ? Do these states of water lie on its P-V-T surface ? Give some practical applications of these state of water. Q.26 Explain why (a) Two bodies at different temperature T and T if 1 2 brought in thermal contact do not necessarily settle to the mean temperature (T + T )/2. 1 2 (b) The coolant in a chemical or a nuclear plant (i.e. the liquid used to prevent the different parts of a plant from getting too hot) should have high specific heat. (c) Air pressure in a car tyre increases during driving. (d) The climate of a harbour town is more temperate than that of town in a desert at the same latitude. Page # 22 Q.27 A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume? Q.28 In changing the state of a gas adiabatically from an equilibrium state A to another equilibrium state B, an amount of work equal to 22.3 J is done on the system. If the gas is take from state A to B via a process in which the net heat absorbed by the system is 9.35 cal, how much is the net work done by the system in the latter case ? ( Take 1 cal = 4.19 J) Q.29 Two cylinders A and B of equal capacity are connected to each other via a stopcock. A contains a gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally insulated. The stopcock is suddenly opened. Answer the following : (a) What is the final pressure of the gas in A and B (b) What is the change in internal energy of the gas ? (c) What the change in the temperature of the gas (d) Do the intermediate states of the system (before settling to the final equilibrium state) lie on its P-V-T surface ? Q.1 Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the radius of an oxygen molecule to be roughly 3 Å. Q.2 Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP : 1 atmosphere pressure, 0 °C). show that it is 224. litres. Q.3 Figure shows plot of PV/T versus P for 1.00 × 10 –3 kg of oxygen gas at two different temperatures. (a) What does the dotted plot signify ?] (b) Which is true : T1 > T2 or T1 < T2 ? (c) What is the value of PV / T where the curves meet on the y-axis ? (d) If we obtained similar plots for 1.00 × 10–3 kg of hydrogen, would we get the same value of PV/T at the point where the curves meet on the y-axis ? If not, what mass of hydrogen yields the same value of PV/T (for low pressure high temperature region of the plot ) ? HEAT Page # 23 (Molecular mass of H2 = 2.02 u, of O2 = 32.0 u, R = 8.31 J mol–1 K–1). y PV –1 (JK ) T T1 P T2 x Q.4 An oxygen cylinder of volume 30 litres has an initial gauge pressure of 15 atm and a temperature of 27°C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17°C. Estimate the mass of oxygen taken out of the cylinder. (R = 8.31 J mol–1 K– 1, molecular mass of O = 32 u) 2 Q.5 An air bubble of volume 1.0 cm3 rises from the bottom of a lake 40 m deep at a temperature of 12°C. To what volume does it grow when it reaches the surface, which is at a temperature of 35°C ? Q.6 Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m3 at a temperature of 27°C and 1 atm pressure. Q.7 Estimate the average thermal energy of a helium atom at (i) room temperature (27°C), (ii) the temperature on the surface of the sun (6000 K), (iii) the temperature of 10 million kelvin (the typical core temperature in the case of a star). Q.8 Three vessel of equal capacity have gases at the same temperature and pressure. The first vessel contains noon (monoatomic), the second contains chlorine (diatomic), equal number of respective molecules ? Is the root mean square speed of molecules the same in the three cases ? If not, in which case is vrms the largest ? Q.9 At what temperature does the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at – 20°C ? (Atomic mass of Ar = 40 u, of He = 4.0 u). Q.10 Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17°C. Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of N2 = 28.0 u). Q.11 A metre long narrow bore held horizontally (and closed at one end) contains at 76 cm long mercury thread, which traps a 15 cm column of air. What happens if the tube is held vertically with the open end at the bottom ? Q.12 Estimate the root mean square speed of the suspended particles in Brownian motion if the particle mass is 10–6 kg and the temperature of the liquid is 27°C. Would you expect the answer the change if the liquid were replaced by another liquid of different density and viscosity keeping the temperature fixed ? Q.13 Explain qualitatively how the extent of Brownian motion is affected by the (a) Size of the Brownian particle (b) Density of the medium, (c) Temperature of the medium (d) Viscosity of the medium ? Q.14 In an experiment on Brownian motion using a torsion pendulum a small mirror of area 1 mm2 mounted on a thin torsion fibre with torsion constant = 1.8 × 10–17 J/rad2) the mean value of angular displacement  was found to by nearly zero, and the fluctuation in , that is the mean square of was found to be 2.0 × 10–4 rad2. Estimate the value of Boltzmann constant from this data and compare it with the correct value. The temperature is 300 K, [Hint : Use the relation (1/2)  < 2 > = (1/2) kB T where a is the torsion constant. The mean potential energy of the torsion pendulum is (1/2) < 2 >. In thermal equilibrium it equals 1/2 kBT according the law of equipartition of energy.] Q.15 The specific heat at constant volume of a certain metal is approximately 0.1 cal/g °C. Write the chemical formula of its chloride if it contains 0.345 fraction of the metal. [Hint : Use Dulong and Petit's law to get an approximate atomic mass of the metal. Then use the second piece of data to arrive at a simple solution for its chemical formula. You can feed this back to get a more correct value of its atomic mass. The atomic mass of chlorine is 35.5 u] Q.16 From a certain apparatus, the diffusion rate of hydrogen has an average value of 28.7 cm3 s–1. The diffusion of another gas under the same conditions is measured to have an average rate of 7.2 cm3 s–1. Identify the gas. [Hint : Use Graham's law of diffusion R1/R2 = M2 / M1 . where R1, R2 are diffusion rates of gases 1 and 2, and M1 and M2 their respective molecular masses. The law is a simple consequence of kinetic theory.] HEAT Page # 24 Q.17 For Brownian motionof particles of suspensions Q.20 Given below are data on a gum-resin suspension in liquids, answer the following : in water at 22°C. Average radius of a grain of (a) What should be the typical size o suspended suspension = 0.2 m. 1 m = 10–6 m, average mass of particles ? Whjy should not the size of the particles a grain = 6.2 × 10–17 kg, average concentration in a too small (say of atomic dimensions 10–10 m) or too layer = 43 particles per unit area at some reference large (say of the order of 1 m) ? level, average concentration in a layer = 100.3 particles (b) Bombardments of the suspended particle by per unit area at a level 11 m lower. Estimate the molecules of the liquid are suspended particle from all value of Avogadro's number from this data and directions. Why is not the net impact zero ? (c) Can the assembly of suspended particles be, compare you answer with the correct value. considered a gas' of 'heavy molecules' ? If so, what Q.21 Given below are densities of some solids and isthe temperature of this 'gas', if the temperature of liquids. Give rough estimates of the size of their atoms the liquid is T ? Density Substance Atomic mass(u) (10 kgm ) of the most compelling evidences (through necessarily Carbon (diamond 12.01 2.22 indirect since we can never hope to 'see' the atoms Gold 197 19.32 directly) of the existence of atoms. Do you appreciate Nitrogen (Liquid) 14.01 1 Lithium 6.94 0.53 Fluorine (liquid) 19 1.14 Q.18 (a) Brownian motion is generally regarded as one this claim ? Why is it a better evidence than that obtained from the laws of chemical combinations or from the specific heats of gases and solids at ordinary temperatures ? (b) "Brownian motion is observable since Avogadro's number is finite". Comment on this statement. 3 –3 [Hint : Assume the atoms to be 'tightly packed' in a solid or liquid phase, and use the known value Avogadro's number. You should, however, not take the actual numbers you obtain for various atomic sizes Q.19A gas in equilibrium has uniform density and too literally. Because of the crudeness of the tight pressure throughout its volume. This is strictly true packing approximation, the results only indicate that only if there are no external influences. A gas atomic sizes are in the range of a few Å] column under gravity, for example, does not have uniform density (and pressure). As you might expect, Q.22 (a) What is the simplest evidence in nature that its density decreases with height. The precise you can think of to suggest that atoms are not point dependence is given by the so-called 'law of particles, but have finite (non-zero) size ? (b) Does  mg  atmospheres' n2 = n1 exp  k T (h 2  h1 ) where n2,  B  n1 refer to number density at heights h2 and h1 atomic size increase monotonically with increase in respectively. Use this relation to derive the equation hydrogen atom ? for sedimentation equilibrium of a suspension in a (0.5) Å ? [Note, this question, among other things,   mgN A     liquid column : n2 = n1 exp  RT 1    (h 2  h1 )     where  is the density of the suspended particle, had puzzled Niels Bohr before he arrived at his famous and ' that of surrounding medium. [NA is Avogadro's number, and R the universal gas constant]. atomic mass. If not, can you guess why not ? (c) What determines fundamentally the size of, say, a quantum model of the hydrogen atom].