HC Verma - Concepts of Physics Part 2

Book: HC Verma - Concepts of Physics Part 2

Chapter: 27. Specific Heat Capacities of Gases

Subject: Physics - Class 12th

Q. No. 9 of Exercises

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9

An ideal gas is taken through a process in which the pressure and the volume are changed according to the equation p = kV. Show that the molar heat capacity of the gas for the process is given by .

Given:


P = kV …


Where,


P = pressure,


V = volume,


k = constant.


Formula used:


Equation of state of ideal gas:


PV = nRT = constant … (ii)


Where,


n is the number of moles of the gas,


R is the gas constant,


T is the temperature,


P = pressure, V2


T = temperature.


From (i), multiplying by dV on both sides:


PdV = kVdV.


Integrating from V = V1 to V2, we get



= with lower limit V1 and upper limit V2


=


Now, we know, PV = nRT - equation of state,


Where P = pressure, V = volume, n = number of moles, R = universal gas constant, T = temperature


Hence we can write, V1 = nRT1/P1. Since P1 = kV1, this becomes:


kV12 = nRT1. Similarly, KV22 = nRT2,


P1, V1, T1 - Pressure, volume, temperature of first gas


P2, V2, T2 - Pressure, volume, temperature of second gas


Therefore, subsituting, the above equation becomes:


= = … (iii)


Now, => (since P = kV) … (iv)


But Q = U + ഽPdV (first law of thermodynamics), where Q = heat, U = change in internal energy, W = total work done = ഽPdV


=> nCdT = nCvdT + (nR/2)dT


(since Q = nCdT and U = nCvdT)


=> C = Cv + nR/2 (proved),


Where


n = number of moles,


C = specific heat capacity,


Cv = specific heat capacity at constant volume,


R = universal gas constant,


dT = rise in temperature.


Chapter Exercises

Short Answer
Objective I
Objective II
Exercises

More Exercise Questions

1

A vessel containing one mole of a monatomic ideal gas (molecular weight = 20 g mol–1) is moving on a floor at a speed of 50 ms–1. The vessel is stopped suddenly. Assuming that the mechanical energy lost has gone into the internal energy of the gas, find the rise in its temperature.
2

5g of a gas is contained in a rigid container and is heated from 15°C to 25°C. Specific heat capacity of the gas at constant volume is 0.172 cal g–1°C–1 and the mechanical equivalent of heat is 4.2 J cal–1. Calculate the change in the internal energy of the gas.
3

Figure shows a cylindrical container containing oxygen (γ = 1.4) and closed by a 50 kg frictionless piston. The area of cross section is 100 cm2, atmospheric pressure is 100 kPa and g is 10 ms–2. The cylinder is slowly heated for some time. Find the amount of heat supplied to the gas if the piston moves out through a distance of 20 cm.
4

The specific heat capacities of hydrogen at constant volume and at constant pressure are 2.4 cal g−1 °C−1 and 3.4 cal g−1 °C−1 respectively. The molecular weight of hydrogen is 2 g mol−1 and the gas constant, R = 8.3 × 107 erg °C−1mol−1. Calculate the value of J
5

The ratio of the molar heat capacities of an ideal gas is CP/CV = 7/6. Calculate the change in internal energy of 1.0 mole of the gas when its temperature is raised by 50 K

(a) keeping the pressure constant,


(b) keeping the volume constant and


(c) adiabatically.
6

A sample of air weighing 1.18g occupies 1.0 × 103 cm3 when kept at 300K and 1.0 × 105 Pa. When 2.0 cal of heat is added to it at constant volume, its temperature increases by 1°C. Calculate the amount of heat needed to increase the temperature of air by 1°C at constant pressure if the mechanical equivalent of heat is 4.2 × 107 erg cal–1. Assume that air behaves as an ideal gas.
7

An ideal gas expands from 100 cm3 to 200 cm3 at a constant pressure of 2.0 × 105 Pa when 50J of heat is supplied to it. Calculate

(a) the change in internal energy of the gas.


(b) the number of moles in the gas if the initial temperature is 300K.


(c) the molar heat capacity CP at constant pressure and


(d) the molar heat capacity CV at constant volume.
8

An amount Q of heat is added to a monatomic ideal gas in a process in which the gas performs a work Q/2 on its surrounding. Find the molar heat capacity for the process.
9

An ideal gas is taken through a process in which the pressure and the volume are changed according to the equation p = kV. Show that the molar heat capacity of the gas for the process is given by .
10

An ideal gas (CP/CV = γ) is taken through a process in which the pressure and the volume vary as p = αVb. Find the value of b for which the specific heat capacity in the process is zero.
11

Two ideal gases have the same value of CP/CV = γ. What will be the value of this ratio for a mixture of the two gases in the ratio 1: 2?
12

A mixture contains 1 mole of helium (CP = 2.5 R, CV = 1.5R) and 1 mole of hydrogen (CP = 3.5 R, CV = 2.5 R). Calculate the values of CP, CV and γ for the mixture.
13

Half mole of an ideal gas (γ =5/3) is taken through the cycle abcda as shown in figure. Take .


(a) Find the temperature of the gas in the states a, b, c and d.


(b) Find the amount of heat supplied in the processes ab and bc.


(c) Find the amount of heat liberated in the processes cd and da.
14

An ideal gas (γ = 1.67) is taken through the process abc shown in figure. The temperature at the point a is 300K. Calculate

(a) the temperature at b and c,


(b) the work done in the process,


(c) the amount of heat supplied in the path ab and in the path bc and


(d) the change in the internal energy of the gas in the process.
15

In Joly’s differential steam calorimeter, 3g of an ideal gas is contained in a rigid closed sphere at 20°C. The sphere is heated by steam at 100°C and it is found that an extra 0.095 g of steam has condensed into water as the temperature of the gas becomes constant. Calculate the specific heat capacity of the gas in J g–1 K–1. The latent heat of vaporization of water = 540 cal g–1.
16

The volume of an ideal gas (γ = 1.5) is changed adiabatically from 4.00 litres to 3.00 litres. Find the ratio of

(a) the final pressure to the initial pressure and


(b) the final temperature to the initial temperature.
17

An ideal gas at pressure 2.5 × 105 Pa and temperature 300K occupies 100 cc. It is adiabatically compressed to half its original volume. Calculate

(a) the final pressure,


(b) the final temperature and


(c) the work done by the gas in the process. Take γ = 1.5.
18

Air (γ = 1.4) is pumped at 2 atm pressure in a motor tyre at 20°C. If the tyre suddenly bursts, what would be the temperature of the air coming out of the tyre. Neglect any mixing with the atmospheric air.
19

A gas is enclosed in a cylindrical can fitted with a piston. The walls of the can and the piston are adiabatic. The initial pressure, volume and temperature of the gas are 100 kPa, 400 cm3 and 300 K respectively. The ratio of the specific heat capacities of the gas is CP/CV = 1.5. Find the pressure and the temperature of the gas if it is

(a) suddenly compressed


(b) slowly compressed to 100 cm3.
20

The initial pressure and volume of a given mass of a gas (CP/CV = γ) are P0 and V0. The gas can exchange heat with the surrounding.

(a) It is slowly compressed to a volume V0/2 and then suddenly compressed to V0/4. Find the final pressure.


(b) If the gas is suddenly compressed form the volume V0 to V0/2 and then slowly compressed to V0/4, what will be the final pressure?
21

Consider a given sample of an ideal gas (CP/CV = γ) having initial pressure P0 and volume V0.

(a) The gas is isothermally taken to a pressure P0/2 and from there adiabatically to a pressure P0/4. Find the final volume.


(b) The gas is brought back to its initial state. It is adiabatically taken to a pressure P0/2 and from there isothermally to a pressure P0/4. Find the final volume.
22

A sample of an ideal gas (γ = 1.5) is compressed adiabatically from a volume of 150 cm3 to 50 cm3. The initial pressure and the initial temperature are 150 kPa and 300 K. Find

(a) the number of moles of the gas in the sample,


(b) the molar heat capacity at constant volume,


(c) the final pressure and temperature,


(d) the work done by the gas in the process and


(e) the change in internal energy of the gas.
23

Three samples A, B and C of the same gas (γ = 1.5) have equal volumes and temperatures. The volume of each sample is doubled, the process being isothermal for A, adiabatic for B and isobaric for C. If the final pressure is equal for the three samples, find the ratio of the initial pressures.
24

Two samples A and B of the same gas have equal volumes and pressures. The gas in sample A is expanded isothermally to double its volume and the gas in B is expanded adiabatically to double its volume. If the work done by the gas is the same for the two cases, show that γ satisfies the equation 1 – 21– γ = (γ – 1) ln2.
25

1 litre of an ideal gas (γ = 1.5) at 300 K is suddenly compressed to half its original volume.

(a) Find the ratio of the final pressure to the initial pressure.


(b) If the original pressure is 100 kPa, find the work done by the gas in the process.


(c) What is the change in internal energy?


(d) What is the final temperature?


(e) The gas is now cooled to 300 K keeping its pressure constant.


Calculate the work done during the process.


(f) The gas is now expanded isothermally to achieve its original volume of 1 litre. Calculate the work done by the gas.


(g) Calculate the total work done in the cycle.
26

Figure shows a cylindrical tube with adiabatic walls and fitted with an adiabatic separator. The separator can be slid into the tube by an external mechanism. An ideal gas (γ= 1.5) is injected in the two aides at equal pressures and temperatures. The separator remains in equilibrium at the middle. It is now slid to a position where it divides the tube in the ratio 1 : 3. Find the ratio of the temperatures in the two parts of the vessel.
27

Figure shows two rigid vessels A and B, each of volume 200 cm3 containing an ideal gas (CV = 12.5 J K–1 mol–1). The vessels are connected to a manometer tube containing mercury. The pressure in both the vessels is 75 cm of mercury and the temperature is 300 K.

(a) Find the number of moles of the gas in each vessel.


(b) 5.0 J of heat is supplied to the gas in the vessel A and 10 J to the gas in the vessel B. Assuming no appreciable transfer of heat from A to B calculate the difference in the heights of mercury in the two sides of the manometer. Gas constant R = 8.3 J K–1 mol–1.
28

Figure shows two vessels with adiabatic walls, one containing 0.1g of helium (γ = 1.67, M = 4 g mol–1) and the other containing some amount of hydrogen (γ= 1.4, M = 2g mol–1). Initially, the temperatures of the two gases are equal. The gases are electrically heated for some time during which equal amounts of heat are given to the two gases. It is found that the temperatures rise through the same amount in the two vessels. Calculate the mass of hydrogen.
29

Two vessels A and B of equal volume V0 are connected by a narrow tube which can be closed by a valve. The vessels are fitted with pistons which can be moved to change the volumes. Initially, the valve is open and the vessels contain an ideal gas (CP/CV = γ) at atmospheric pressure p0 and atmospheric temperature T0. The walls of the vessel A are diathermic and those of B are adiabatic. The valve is now closed and the pistons are slowly pulled out to increase the volumes of the vessels to double the original value.

(a) Find the temperatures and pressures in the two vessels.


(b) The valve is now opened for sufficient time so that the gases acquire a common temperature and pressure. Find the new values of the temperature and the pressure.
30

Figure shows an adiabatic cylindrical tube of volume V0 divided in two parts by a frictionless adiabatic separator. Initially, the separator is kept in the middle, an ideal gas at pressure p1 and temperature T1 is injected into the left part and another ideal gas at pressure p2 and temperature T2 is injected into the right part. CP/CV = γ is the same for both the gases. The separator is slid slowly and is released at a position where it can stay in equilibrium. Find

(a) the volumes of the two parts,


(b) the heat given to the gas in the left part


(c) the final common pressure of the gases.
31

An adiabatic cylindrical tube of cross-sectional area 1 cm2 is closed at one end and fitted with a piston at the other end. The tube contains 0.03g of an ideal gas. At 1 atm pressure and at the temperature of the surrounding, the length of the gas column is 40 cm. The piston is suddenly pulled out to double the length of the column. The pressure f the gas falls to 0.355 atm. Find the speed of sound in the gas at atmospheric temperature.
32

The speed of sound in hydrogen at 0°C is 1280 m s–1. The density of hydrogen at STP is 0.089 kg m–3. Calculate the molar heat capacities CP and CV of hydrogen.
33

4.0 g of helium occupies 22400 cm3 at STP. The specific heat capacity of helium at constant pressure is 5.0 cal K–1 mol–1. Calculate the speed of sound in helium at STP.
34

An ideal gas having density 1.7 × 10–3 g cm–3 at a pressure 1.5 × 105 Pa is filled in a Kundt tube. When the gas is resonated at a frequency of 3.0 kHz, nodes are formed at a separation of 6.0 cm. Calculate the molar heat capacities CP and CV of the gas.
35

Standing waves of frequency 5.0 kHz are produced in a tube filled with oxygen at 300 K. The separation between the consecutive nodes is 3.3 cm. Calculate the specific heat capacities CP and CV of the gas

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