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.

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 (C_P/C_V = 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.

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.

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.

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.

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.

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 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.

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.

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.

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

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.