PHYSICS As LEVEL(FORM FIVE) NOTES – HEAT-4

A mono atomic gas initially at the temperature T = 25 and pressure of 2 atmospheres is expanded to a final pressure of 1.0 atmosphere.

1. Isothermally and reversibly
2. Isothermally against a constant pressure of 1.0 atmosphere. Calculate for each case:
1. The final temperature of the gas
2. The increase of internal energy

The figure below shows some details concerning the behavior of a fixed mass of a gas assumed to be an ideal one in a petrol engine. The gas starts at A with a volume 5 , temperature 300 K and a pressure of 1 . In the change from A to B it is compressed to volume of 1 , the pressure rises to 1.5 . And temperature 630K

1. Using the equation of state for an ideal gas, find the number of molecules in the fixed mass of a gas.
2. In the change from B to C the temperature of a gas rises from 630 K to 1500 K. The molar heat capacity at constant volume of the gas is 21. Calculate the internal energy of the gas. c) How much work is done by the gas in changing from B to C?

The figure below shows a sample of gas enclosed in a cylinder by a frictionless piston of area 100. The cylinder is now heated, so that 250J of energy is transferred to the gas, which then expands against atmospheric pressure of 1.00 x . And pushes the piston 15.0 cm along the cylinder as shown

1. The external work done by the gas
2. The increase in internal energy of the gas.

When 1.50kg of water is converted to steam (at 100) at standard atmospheric pressure of 1.01 , 3.39MJ of heat are required. During the transformation from liquid to vapor state, the increase in volume of the water is 2.50. Calculate the work done against the external pressure during the process of vaporization. Explain what happens to the rest of the energy.

Problem 102 A fixed mass of gas is cooled, so that its volume decreases from 4.0 liters to 2.5 liters at a constant pressure of 1.0 Pa.

The specific latent heat of vaporization of steam is 2.26 MJ . When 50 of water is boiled at standard atmospheric pressure of 1.01 Pa, 83 of steam are formed.

1. The mass of water boiled
2. The heat input needed
3. The external work done during vaporization
4. The increase in internal energy

Given that density of water 1000

56.0 kg of nitrogen is to be heated from 270 K to 310K. When this occurs in an insulated freely extensible container, 2.33 Kj of heat is required when contained in an insulated rigid container, 1.66KJ of heat is required. Calculate the principal molar heat capacities of nitrogen.

The specific heat capacity of a diatomic gas at constant volume is 0.410 KJ

1. The specific heat capacity of the gas at constant pressure.
2. The specific gas constant for the gas.

1. 
2. The value of
3. The heat required to raise the temperature of 4.00 mole from 300 K to 400 K at constant volume

Argon has a molar heat mass of 40 kg and a principal molar heat capacity, at constant volume, of 12.5 Calculate:

1. The valve of
2. The specific heat capacity at constant volume
3. The amount of heat required to raise the temperature of 1.00kg of argon by 80 K at constant volume.

2.00 mole of nitrogen, at 300K are in an insulated, freely extensible container, and the pressure outside the container is 1.00 . The principal molar heat capacity of nitrogen at constant pressure is 29.0

1. The heat required to raise its temperature to 340 K.
2. The increase in volume of the gas during this process.
3. The external work done
4. The internal energy change
5. The heat required to effect the temperature change at constant volume

Problem 109

The piston of a bicycle pump is slowly moved in until the volume of air enclosed is one-fifth of the total volume of the pump and is at room temperature (290K). The outlet is then sealed and the piston suddenly drawn out to full extension. No air passes the piston. Find the temperature of the air in the pump immediately after withdrawing the piston assuming that air is a perfect gas with = 1.

Problem 110

A fixed mass of gas, initially at 7 and a pressure of 1.00 , is compressed isothermally to one – third of i
ts original volume. Calculate the final temperature and pressure, assuming = 1.40

Problem 111

A fixed mass of gas is taken through the closed cycle A B C D A as shown in the figure below

1. Calculate the work done in the cycle.
2. How much heat transferred in the cycle?
3. Is the heat absorbed or emitted by the gas?

One mole of water, occupying a volume of 1.8 , is turned into steam in a boiler at a temperature of 373 K and a pressure of 1.0 Pa. The volume of steam generated i
s 0.031. The energy required is 41,000J.

1.00 kg of water at 100 is changing into steam at atmospheric pressure.

1. The external work done
2. The increase in internal energy
3. What happens to the internal energy absorbed during the vaporization process?

Given that:

Density of water at 100

Density of steam at 100 and at atmospheric pressure = 0.59 Atmospheric pressure, = 1.01 Pa.

Specific latent heat of vaporization of water = 2.26

At a temperature of 100 and a pressure of 1.01Pa, 1.00Kg of steam occupies 1.67 , but the same mass of water occupies only 1.04 . The specific latent heat of vaporization o water at 100 is 2.26

1. The heat supplied to the system
2. The work done by the system
3. The increase in internal energy of the system.

The ratio of the principal heat capacities of an ideal gas is , and the molar gas constant is R.

The specific heat capacity at constant volume of a certain ideal gas is 6 x and is independent of temperature.

Find the internal energy of 5.0 x kg of the gas at 27.

2. The temperature for this change is 300K. Wha
   t is the temperature of the gas at C?
3. What energy process takes place between B and C?
4. The change in internal energy of the sample in the process from B to C is 56KJ. Calculate the molar heat capacity at constant volume for helium.
5. Calculate the work done during the change from C to A. State and explain whether work is done on or by the gas during this part of the cycle. Justify your answer.
6. Determine the value of molar heat capacity at constant pressure for helium. Show Clearly how you arrive at your answer.
7. Use the figure to estimate the net work done during one complete cycle.

A fixed mass of an ideal gas has a volume at an initial temperature of 300 K and an initial pressure of

1.2 Pa.

1. Isothermal expansion from its initial volume to a volume 2
2. Expansion at constant pressure to a volume 4 C. Isothermal compression

2. The temperature at the end of process B
3. The volume at the end of process C

A fixed mass of an ideal gas at an initial temperature of 20 and at a pressure of 1.00 Pa is compressed until its volume is one- quoter of its original volume.

1. The compression is isothermal
2. The compression is adiabatic

Given that = = 1.40

Problem 120

In a diesel engine, fuel oil is injected into a cylinder in which air has been heated by adiabatic compression to above the ignition temperature of the oil. The ignition temperature of a certain fuel is 630 , and the air enters the cylinder, which has an initial volume of 5.0 at a pressure of 1.0 Pa and a temperature of 28

1. What minimum compression ratio (the ratio of the initial to the final volume of the cylinder) is required to heat the air to the fuel ignition temperature?
2. How much work is done in compressing the air
   ?

Given that for air = 1.40

(a) A cylinder fitted with a piston which can move without friction contains 0.05 mole of a mono atomic ideal gas at a temperature of 27 and a pressure of 1.0 Pa.

( b) The temperature of the gas in (a) above is raised to 77, the pressure remaining constant.

Given that molar gas constant = 8.3.

Problem 122

1. Give one practical example of each of the following: (i) A process in which heat is supplied to a system without causing an increase in temperature.

2. What happens to the energy added to an ideal gas when it is heated:

1. At constant volume?
2. At constant pressure?

3. Deduce an expression for the difference between the specific heat capacities of a gas at constant pressure and at constant volume.
4. If the ratio of the principal specific heat capacities of a certain gas is 1.40 and its density at S.T.P is 0.09, calculate the values of the specific heat capacity at constant pressure and at constant volume. Standard atmospheric pressure = 1.01 x

Problem 123

A steel pressure vessel of volume 2.2 contains 4.0 Pa and temperature 300 K. An explosion suddenly releases 6.48 J of energy, which raises the pressure instantaneously to 1.0 Pa. Assuming no loss of heat to the vessel, and ideal gas behaviour,

1. The maximum temperature attained
2. The two principal specific heat capacities of the gas.

1. Explain why an ideal gas can have infinity number of molar heat capacities and define the principal values.
2. A thermally – insulated tube through which a gas may be passed at constant pressure contains an electric heater and thermometers for measuring the temperature of the gas as it enters and as it leaves the tube. 3.0 of gas of density 1.8 flows into the tube in 90 seconds and, when electrical power is supplied to the heater at a rate of 0.16W, the temperature difference between the out let and inlet is 2.5 K. Calculate a value for the specific heat capacity of the gas at constant pressure.

1. The internal energy remains constant when the gas expands isothermally.
2. The heat capacity at constant pressure is greater than the heat capacity at constant volume.

(b) A vessel of volume 1.0 contain an ideal gas at a temperature of 300 K and pressure 1.5 Pa. Calculate the mass of gas, given that the density of the gas at temperature

Pa. Assuming ideal gas behavior, and no loss of heat to the containing vessel, Calculate the temperature rise, and hence the specific heat capacity at constant volume of the gas.

A vessel of volume 8.00 contains an ideal gas at a pressure of 1.14 Pa.

A stopcock in the vessel is opened and the gas expands adiabatically, expelling some of its original mass, until its pressure is equal to that outside the vessel 1.01 Pa. The stopcock is then closed and the vessel is allowed to stand until the temperature returns to its original value; in this equilibrium state, the pressure is 1.06 Pa.

3. Sketch a graph showing the way in which the pressure and volume of the mass of gas left in the vessel changed during the operations described above:
4. What is the value of , the ratio of the principal heat capacities of th
   e gas.
5. What can you deduce about the molecules of the gas? Give your reasons.

then heated a constant volume and returned to its original

state

( b) Calculate the heat given out by the gas in the process A B

( c) Calculate the heat absorbed in the process B C

Given that R = 8.3. and

The specific latent heat of vaporization of particular liquid at 130 and a pressure of 2.60 Pa is

1.84 .

vapor is 5.66 .

1. Explain what is meant by a reversible change.
2. A mass of 0.35 kg of ethanol is vaporized at its boiling point of 78 and a pressure of 10

Pa. At this temperature. The specific latent heat of vaporization of ethanol is 0.95 and the densities of the liquid and vapor are 790 and 1.6 respectively. Calculate:

1. The temperature of the gas changes
2. There is heat transfer to or from the gas
3. Work is done on or by the gas

An ideal gas at 17 has a pressure of 760mmHg, and is compressed.

1. Isothermally,
2. Adiabatically until its volume is halved, in each case reversibly. Calculate in each case the final pressure and temperature of the gas, assuming =

2100 = 1500.

Problem 132

1. Show that for an ideal gas the
   curves relating pressure and volume for an adiabatic change have a greater slope than those for an isothermal change, at the same pressure.
2. A gas in a cylinder initially at a temperature of 17 and a pressure of 1.01

, is to be compressed to one-eighth of its volume. What would be the

1. Isothermally
2. Adiabatically?

Given that = 1.40

Problem 133

Given that the volume of a gas at S.T.P is 2.24 and
that standard pressure is 1.01, calculate the molar gas constant R and use it to find the difference between the quantities of heat required to raise the temperature of 0.01kg of oxygen from 0 to 10 when.

1. The pressure is kept constant
2. The volume is kept constant

1. By considering the expansion of an ideal gas contained in a cylinder and enclosed by a piston, show that the work done in a small expansion is equal to the pressure times the volume change.
2. An ideal gas, at a temperature of 290 K and a pressure of 1.0 , occupies a volume of 1.0 . Its density conditions is 0.30 .

3. The gas is now compressed isothermally to its original volume.

7.1 .

A litre of air, initially at 20 and at 760mmHg pressure, is heated at constant pressure until its volume is doubled. Find

2. The external work done by the air in expanding
3. The quantity of heat supplied.

Assume that the density of air at S.T.P is 1.293 and that the specific heat capacity of air at constant volume is 714.

(b) If the specific heat capacity of air at constant pressure is 1013 and the density at S.T.P is 1.29, estimate a value for the specific heat capacity of air at constant volume.

Problem 137

(a) What is the importance of the ratio of the specific heat capacities of an ideal gas? (b) A mass of air occupying initially a volume 2 at a pressure of

760mmHg and a temperature 20 is expanded adiabatically and reversibly to twice its volume, and then compressed isothermally and reversibly to a volume of 3

. Find the final temperature and pressure, assuming the ratio of the

Problem 138

Air initially at 27 and at 750mmHg pressure is compressed isothermally until its volume is halved. It is then expanded adiabatically until its original volume is recovered. Assuming the changes to be reversible find the final pressure and temperature take = 1.40

Problem 139

When water at 100 and pressure of 101 kPa changes to steam under the same conditions, its volume increases by a factor of 1670 given the density of water is 960 and 101 kPa, and its specific latent heat of vaporization is 2.26

1. The heat supplied to convert 1 kg of water at 100 to steam at the same temperature.
2. The work done when 1 kg of water turns to steam at 101kPa pressure.
3. The increase of internal energy.

Problem 140

A fixed mass of ideal gas is contained in a cylinder. The cylinder volume can be varied by moving a piston in or out. The gas has an initial volume 0.01 at 100 kPa pressure and its temperature is initially 300K. The gas is cooled at constant pressure until its volume is 0.006
. Sketch a pressure against volume graph to show the change.

1. The final temperature of the gas.
2. The work done on the gas.
3. The number of moles of gas.
4. The change of internal energy of the gas.
5. The heat transfer from the gas

A motor car tyres has a pressure of four atmospheres at a room temperature of 27. If the Tyre suddenly bursts, calculate the temperature of the escaping air. Value of for air is 1.4

A molecule of a gas at 27 expands isothermally until its volume is doubled. Find the amount of wo
rk done and heat absorbed.

A cylinder fitted with a movable piston contains 3 moles of hydrogen at standard temperature are made and pressure. The walls of the cylinder are late 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? Given = 1.4 Problem 145

A quantity of air is compressed

1. Slowly and
2. Suddenly to one third of its volume. Find the change in temperature in each case.

A Tyre pumped to a pressure of 6 a.t.m suddenly burst. The room temperature is 15. Calculate the temperature of escaping air.

Take = 1.4

A litre of air, initially at 20 and at 760mmHg pressure, is heated at constant pressure until its volume is doubled.

1. The final temperature
2. The external work done by the air in expanding

(iii)The quantity of heat supplied.Assume that the density of air at N.T.P is 1.293 =

714

A gas is suddenly compressed to one-half of its volume. Calculate the rise in temperature, the original temperature being 27 . Take = 1.5

Calculate the final pressure and temperature in each case. Take = 1.4

10 moles of hydrogen gas at NTP are compressed adiabatically so that its temperature becomes 400. How much work is done by the gas? Also find the increase in internal energy of the gas.

Given R = 8.4 and = 1.4

Calculate the work done when one mole of a perfect gas is compressed diabatically. The initial pressure and volume of the gas are and 6 litres respectively. The final volume of the gas is 2 litres. Molar specific heat of the gas at constant volume is

A cylinder contains 1 mole of oxygen at a temperature of 27. The cylinder is provided with a frictionless piston maintains a constant pressure of 1 a.t.m on the gas. The gas is heated until its temperature rises to 127.

Given that = 7.03 cal and

R = 1.99 cal

Problem 153

Two moles of helium gas are initially at a temperature 27 and occupy a volume of 20 litres. The gas is first expanded at constant pressure until the volume is

(iii) What is the work done by the gas? Gas constant R = 8.3

Problem 154

Given that R = 8.3J

A cylinder contains 3 moles of oxygen at a temperature of 27. The cylinder is provided with a frictionless piston which maintains a constant pressure of 1 atmosphere on the gas. The gas is heated unless its temperature rises to 127.

1. How much heat is supplied to the gas?
2. What is the change in internal energy of the gas?
3. How much work is done by the gas in the process? Given that = 7.03 cal

An ideal gas having initial pressure P, volume V and temperature T is allowed to expand adiabatically until its volume becomes 5.66 V while its temperature falls to

i) What is the value of for the gas?

As a result of isobaric heating by = 72 K, one mole of a certain ideal gas obtains an amount of heat Q = 1.60 KJ.

1. The work done by the gas
2. The increment in its internal energy
3. The value of

Two different adiabatic paths for the same gas intersect two isothermal at as shown

in the P – V diagram below. How does compare with

At 27 two moles of an ideal mono atomic gas occupies a volume V. The gas expands adiabatically to a volume 2V. Find:

Given that R = 8.31 J

A gas volume and at a pressure of 4 is compressed adiabatically to a volume 0.5 . Find its new pressure. Compare it with the pressure obtained if compression were isothermal. Calculate the work done in each process.

1. Explain what is meant by temperature gradient.
2. An ideally lagged compound bar 25 cm long consists of a copper bar 15 cm long joined to an aluminum bar 10 cm long and of equal cross-sectional area. The free end of the copper is maintained at 100and the free end of the aluminium at 0. Calculate the temperature gradient in each bar when steady state conditions have been reached. (Thermal conductivity of copper =

390 W . Thermal conductivity of aluminium = 210 W .)

Problem 166

1. If a copper kettle has a base of thickness 2.0mm and area3.0 x , estimate the steady difference in temperature between inner and outer surfaces of the base which must be maintained to enable enough heat to through so that the temperature of 1.00 kg of water rises at the rate of 0.25 . Assume that there are no heat losses, the thermal conductivity of copper =

3.8 W and the specific heat capacity of water = 4.2 .

2. After reaching the temperature of 373 K the water in (a) above is allowed to boil under the same conditions for 120 seconds and the mass of water remaining in the kettle is 0.948 kg. Deduce a value for the specific latent heat of vaporization of water ( neglecting condensation of the steam in the kettle

Problem 167

A cubical container full of hot water at a temperature of 90 is completely lagged with an insulating material of thermal conductivity 6.4 W . The edges of the container are 1.0m. Estimate the rate of flow of heat through the lagging if the external temperature of the lagging is 40. Mention any assumptions you make in deriving your result.

Problem 168

A thin-walled hot-water tank having a total surface area 5, contains 0.8 of water at a temperature of 350 K. It is lagged with a 50mm thick layer of material of thermal conductivity 4

. The temperature of the outside surface of the lagging is 290 K. What

electrical power must be supplied to an immersion heater to maintain the temperature of the water at 350 K? Assume the thickness of the copper walls of the tank to be negligible) What is the justification for the assumption that the thickness of the copper walls of the tank may be neglected? (Thermal conductivity of copper = 400 W .

(Density of water 1000 kg specific heat capacity of water = 4170 J .)

1. Sketch graphs to illustrate the temperature distribution along a metal bar heated to one end when the bar is (a) lagged, and (b) unlagged. In each case assume the temperature equilibrium has been reached. Explain the difference between the two graphs.
2. A window pane consists of a sheet of glass of area 2.0 and thickness 5.0mm. if the surface temperatures are maintained at 0 and 20 , calculate the rate of flow of heat through the pane assuming a steady state is maintained. The window is now double glazed by adding a similar sheet of glass so that a layer of air 10mm thick is trapped between the two panes. Assuming that the air is still calculate the ratio of the rate of flow of heat through the window in the first case to that in the second.

(Conductivity of glass = 0.80 W , conductivity of air = 0.025 W .)

An iron pan containing water boiling steadily at 100 stands on a hot-plate and heat conducted through the base of the pan evaporates 0.090 kg of water per minute. If the base of the pan has an area of 0.04 and a uniform thickness of 2.0 m, calculate the surface temperature of the pan.

(Thermal conductivity of iron = 66 W . Specific latent heat of vaporization of water at

100)

1. A sheet a glass has an area of 2.0 and a thickness 8.0 x m. The glass has a thermal conductivity of 0.80 W . Calculate the rate of heat transfer through the glass when there is a temperature difference of 20 K between its faces.
2. A room in a house is heated to a temperature 20 K above that outside. The room has

2 of windows of glass similar to the type used in (a) above. Suggest why the rate of heat transfer through glass is much less than the value calculated above.

1. Explain why two sheets of similar glass each 4mm thick separated by a 10mm layer of air. Assuming the thermal conductivity of glass to be 50 times greater than that of air calculate the ratio.

2. A double-glazed window consists of two panes of glass each 4mm thick separated by a 10mm layer of air. Assuming the thermal conductivity of glass to be 50 times greater than that of air calculate the ratio.

1. Outline an experiment to measure the thermal conductivity of a solid which is a poor conductor, showing how the result is calculated from the measurements.
2. Calculate the theoretical percentage change in heat loose by conduction achieved by replacing a single glass window by a double window consisting of
   two sheets of glass separated by 10mm of air.

Problem 174

(The Stefan constant, = 6 ).

Problem 175

1. Explain what is meant by black body radiation
2. A blackened metal sphere of diameter 10mm is placed at the focus of a concave mirror of diameter 0.5m directed towards the sun. If the solar power incident on the mirror is 1600 W . Calculate the maximum temperature in which the sphere can attain. State the assumptions you have estimated.

(The Stefan‘s constant, = 6 ).

Problem 176

=

Where is the Stephan constant and R is the radius of the Earth‘s orbit around the sun.

An unlagged thin-walled copper pipe of diameter 2.0 cm carries water at a temperature of 40 K above that the surrounding air. Estimate the power loss per unit length of the pipe if the temperature of the surroundings is 300K and the Stefan constant, , is 5.67 ).

The solar radiation falling normally on the surface of the Earth has an intensity 1.40 k W. If this radiation fell normally on one side of a thin, freely suspended blackened metal plate and the temperature of the surroundings was 300 K, calculate the equilibrium temperature of the plate. Assume that all heat interchange is by radiation.

(The Stefan constant = 5.67 ).

Problem 179

A steel rod has length 1.5m and radius 1 cm. One end of the rod is maintained at 100 and the other end is at 0. Find the quantity of heat conducted through the rod in 2 minutes. The thermal conductivity of steel is 50.4 W/m K.

Problem 180

A glass window pane of a room has dimensions 2m 0.5 m x 0.002m. The temperature on its two sides are 300 K and 295 K respectively. Find the quantity of heat conducted out of the room in 10 minutes if the room has two windows, each having two such panes.

Problem 181

In Searle‘s method. A metal rod of length 50cm and area of cross-section 8 is used. The flow of water through the tube is adjusted at 20 grams per minute. The stead temperature of 65 and 55 respectively are shown by the two thermometers instead in the rod. The separation between the thermometers is 4 cm. The out flowing water shows a rise of 6. Find the thermal conductivity of the metal.

Problem 182

In Searle‘s experiment for the measurement of thermal conductivity of a metal, a road having a cross-sectional area of 10 is used. The flow of water through the cooling tube is adjusted at 150gm/minute. When a steady state is reached, two thermometers, inserted in the road at a distance of 5cm from each other, record temperature of 60 and 50 respectively. If the rise in temperature of the water flowing through the cooling tube is 5 , find the thermal conductivity of metal.

Problem 183

The temperature inside an air-conditioned room is maintained at 20 when the outside temperature is 30 . Calculate the quantity of heat conducted per minute through a glass window pane of area 0.25 and thickness 5mm if the thermal conductivity of glass is 0.84 W/mK.

Problem 184

The temperature inside the room is 15 and that of outside is 5. How much heat will be lost by conduction per hour through one square meter of the wall if its thickness is 25cm [K for the material of the wall = 2.5 w/mK]

Problem 185

Calculate the amount of heat conducted per minute through a glass window pane of length 50cm, breadth 20cm and thickness 0.5 cm, if there is a steady temperature difference of 10 on its two sides (Thermal conductivity of glass = 0.002 CGS units)

Problem 186

One end of a copper rod 20 cm long and 5 cm in diameter is maintained at 50 while the other end is kept at a constant temperature of 20. Calculate the quantity of heat conducted through the rod in 10 seconds if the thermal conductivity of copper is 0.92 CGS units

Problem 187

A large glass window has an area of 10 and thickness of 3mm. If the temperature in side and outside the room is 20 and -10 respectively, calculate the quantity of heat flowing per second through the window. Thermal conductivity of glass = 1.5 MKS units)

Problem 188

In Searle‘s method, rod of length 30cm and cross-sectional area 5 is used and flow of water is adjusted at 60 grams per minute. Steady temperature
of 60 and 50 respectively are shown by two thermometers inserted in the rod 8cm apart. If the water coming out of the spiral shows 5 rise in temperature, calculate the thermal conductivity of the metal.

Problem 189

The thermal conductivity of brass is 0.26 cal/s cm. In Searle‘s experiment a brass rod having a cross-sectional area of 10 is used. When the steady state is reached, the temperature recorded by the two thermometers, inserted in the rod at a distance of 4 cm from each other, differ by 5. If the rate of flow of water through the cooling tube is 0.5 gm/s. find the rise in temperature of water.

Problem190

A hollow cube of metal having mean length 10cm and thickness 0.25cm is filled with ice at 0 and is surrounded by water at 80. How much ice will melt in ten minutes?