HC Verma solutions for Concepts of Physics Vol. 2 [English] Class 11 and 12 chapter 5 - Specific Heat Capacities of Gases [Latest edition]

Does a gas have just two specific heat capacities or more than two? Is the number of specific heat capacities of a gas countable?

Can we define specific heat capacity at constant temperature?

Can we define specific heat capacity for an adiabatic process?

Does a solid also have two kinds of molar heat capacities C_p and C_v? If yes, is C_p > C_v? Or is C_p − C_v = R?

Can a process on an ideal gas be both adiabatic and isothermal?

Show that the slope of the p−V diagram is greater for an adiabatic process compared to an isothermal process.

Is a slow process always isothermal? Is a quick process always adiabatic?

Can two states of an ideal gas be connected by an isothermal process as well as an adiabatic process?

The ratio C_p / C_v for a gas is 1.29. What is the degree of freedom of the molecules of this gas?

Work done by a sample of an ideal gas in a process A is double the work done in another process B. The temperature rises through the same amount in the two processes. If C_Aand C_B be the molar heat capacities for the two processes,

For a solid with a small expansion coefficient,

  The value of C_p − C_v is 1.00 R for a gas sample in state A and 1.08 R in state B. Let p_Aand p_B denote the pressures and T_A and T_B denote the temperatures of the states A and B, respectively. It is most likely that

Let C_v and C_p denote the molar heat capacities of an ideal gas at constant volume and constant pressure respectively. Which of the following is a universal constant?

70 calories of heat are required to raise the temperature of 2 mole of an ideal gas at constant pressure from 30° C to 35° C. The amount of heat required to raise the temperature of the same gas through the same range at constant volume is

The figure shows a process on a gas in which pressure and volume both change. The molar heat capacity for this process is C.

The molar heat capacity for the process shown in the figure is

In an isothermal process on an ideal gas, the pressure increases by 0.5%. The volume decreases by about

In an adiabatic process on a gas with γ = 1.4, the pressure is increased by 0.5%. The volume decreases by about

Two samples A and B are initially kept in the same state. Sample A is expanded through an adiabatic process and the sample B through an isothermal process. The final volumes of the samples are the same. The final pressures in A and B are p_A and p_Brespectively.

Let T_a and T_b be the final temperatures of the samples A and B, respectively, in the previous question.

Let ∆W_a and ∆W_b be the work done by the systems A and B, respectively, in the previous question.

The molar heat capacity of oxygen gas at STP is nearly 2.5 R. As the temperature is increased, it gradually increases and approaches 3.5 R. The most appropriate reason for this behaviour is that at high temperatures

A gas kept in a container of finite conductivity is suddenly compressed. The process
(a) must be very nearly adiabatic
(b) must be very nearly isothermal
(c) may be very nearly adiabatic
(d) may be very nearly isothermal

Let Q and W denote the amount of heat given to an ideal gas and the work done by it in an isothermal process.

Let Q and W denote the amount of heat given to an ideal gas and the work done by it in an adiabatic process.
(a) Q = 0
(b) W = 0
(c) Q = W
(d) Q ≠ W

Consider the processes A and B shown in the figure. It is possible that

Three identical adiabatic containers A, B and C contain helium, neon and oxygen, respectively, at equal pressure. The gases are pushed to half their original volumes.
(a) The final temperatures in the three containers will be the same.
(b) The final pressures in the three containers will be the same.
(c) The pressures of helium and neon will be the same but that of oxygen will be different.
(d) The temperatures of helium and neon will be the same but that of oxygen will be different.

A rigid container of negligible heat capacity contains one mole of an ideal gas. The temperature of the gas increases by 1° C if 3.0 cal of heat is added to it. The gas may be
(a) helium
(b) argon
(c) oxygen
(d) carbon dioxide

Four cylinders contain equal number of moles of argon, hydrogen, nitrogen and carbon dioxide at the same temperature. The energy is minimum in

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 m s⁻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.

5 g 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

The 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 m s⁻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.

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⁻¹ mol⁻¹. Calculate the value of J.

The ratio of the molar heat capacities of an ideal gas is C_p/C_v = 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) adiaba

A sample of air weighing 1.18 g occupies 1.0 × 10³ cm³ when kept at 300 K 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.

An ideal gas expands from 100 cm³ to 200 cm³ at a constant pressure of 2.0 × 10⁵ Pa when 50 J 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 300 K (c) the molar heat capacity C_p at constant pressure and (d) the molar heat capacity C_v at constant volume.

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

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 `"C" ="C"_"v" +"R"/2.`

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

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

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

Half mole of an ideal gas (γ = 5/3) is taken through the cycle abcda, as shown in the figure. Take  `"R" = 25/3"J""K"^-1 "mol"^-1 `. (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.

An ideal gas (γ = 1.67) is taken through the process abc shown in the figure. The temperature at point a is 300 K. Calculate (a) the temperatures 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 bcand (d) the change in the internal energy of the gas in the process.

In Joly's differential steam calorimeter, 3 g 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 vaporisation of water = 540 cal g⁻¹ 

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.

An ideal gas at pressure 2.5 × 10⁵ Pa and temperature 300 K 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

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.

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, C_p / C_v = 1.5. Find the pressure and the temperature of the gas if it is (a) suddenly compressed (b) slowly compressed to 100 cm³.

The initial pressure and volume of a given mass of a gas (C_p/C_v = 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 from the volume V₀ to V₀/2 and then slowly compressed to V₀/4, what will be the final pressure?

Consider a given sample of an ideal gas (C_p/C_v = 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.

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.

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 pressures are equal for the three samples, find the ratio of the initial pressures.

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 − 2^1−γ = (γ − 1) ln2.

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.

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 sides 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.

Figure shows two rigid vessels A and B, each of volume 200 cm³, containing an ideal gas (C_v = 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 vessel A and 10 J to the gas in vessel B. Assuming there's 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.

The figure shows two vessels with adiabatic walls, one containing 0.1 g of helium (γ = 1.67, M = 4 g mol⁻¹)  and the other containing some amount of hydrogen (γ = 1.4, M = 2 g 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.

Two vessels A and B of equal volume V₀ are connected by a narrow tube that can be closed by a valve. The vessels are fitted with pistons that can be moved to change the volumes. Initially, the valve is open and the vessels contain an ideal gas (C_p/C_v = γ) at atmospheric pressure p₀ and atmospheric temperature T₀. The walls of 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 pressure.

The 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. C_p/C_v = 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 and (c) the final common pressure of the gases.

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 0.03 g 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 of the gas falls to 0.355 atm. Find the speed of sound in the gas at atmospheric temperature.

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 C_p and C_v of hydrogen.

An ideal gas of density 1.7 × 10⁻³ g cm⁻³ at a pressure of 1.5 × 10⁵ Pa is filled in a Kundt's 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 C_p and C_v of the gas.

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 C_p and C_v of the gas.