Thermodynamic Processes and Indicator Diagrams

Author:Embibe Experts
JEE Main/Advance
IMPORTANT

Important Questions on Thermodynamic Processes and Indicator Diagrams

MEDIUM
IMPORTANT

Find the work done in the process is, 5.6 L of helium at STP is adiabatically compressed to 0.7 L. Taking the initial temperature to be T1
 

EASY
IMPORTANT

Two rigid boxes containing different ideal gases are placed on a table. The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of boxes). Box A contains one mole of nitrogen at temperature T0, while box B contains one mole of helium at temperature 73 T0. Then the final temperature of the gases, Tf in terms of T0 is

HARD
IMPORTANT

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure pi=105 Pa and volume Vi=10-3 m3 changes to a final state at pf=132×105 Pa and Vf=8×10-3 m3 in an adiabatic quasi-static process, such that p3V5=constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at pi followed by an isochoric (isovolumetric) process at volume Vf. The amount of heat supplied to the system in the two-step process is approximate,

EASY
IMPORTANT

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio CpCV for the gas is

EASY
IMPORTANT

When an ideal gas is compressed isothermally, then its pressure increases because

HARD
IMPORTANT

Cloud formation condition

Consider a simplified model of cloud formation. Hot air in contact with the earth’s surface contains water vapour. This air rises connectively till the water vapour content reaches its saturation pressure. When this happens, the water vapour starts condensing and droplets are formed. We shall estimate the height at which this happens. We assume that the atmosphere consists of the diatomic gases oxygen and nitrogen in the mass proportion 21:79 respectively. We further assume that the atmosphere is an ideal gas, g the acceleration due to gravity is constant and air processes are adiabatic. Under these assumptions one can show that the pressure is given by

p=p0T0-τZT0α

Here p0 and T0 is the pressure and temperature respectively at sea level (z = 0), τ is the lapse rate (magnitude of the change in temperature T with height z above the earth’s surface, i.e. τ > 0).

(a) Obtain an expression for the lapse rate Γ in terms of γ, R, g and ma. Here γ is the ratio of specific heat at constant pressure to specific heat at constant volume; R, the gas constant; and ma, the relevant molar mass. 

(b) Estimate the change in temperature when we ascend a height of one kilometre ?

(c) Show that pressure will depend on height as given by Eq. (1). Find an explicit expression for exponent α in terms of γ.

(d) According to this model what is the height to which the atmosphere extends? Take T0 = 300 K and p0 = 1 atm.

MEDIUM
IMPORTANT

A Carnot engine cycle is shown in the figure (2). The cycle runs between temperaturesTH=αT0 and TL=T0 (α > 1). Minimum and maximum volume at state 1 and state 3 are V0 and nV0, respectively. The cycle uses one mole of an ideal gas with CP CV=γ. Here CP and CV are the specific heats at constant pressure and volume respectively. You must express all answers in terms of the given parameters  α,n,T0,V0,? and universal gas constant R.

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(a) Find P, V, T for all the states

(b) Calculate the work done by the engine in each process W12, W23, W34, W41.

(c) Calculate Q, the heat absorbed in the cycle.

 

HARD
IMPORTANT

Two samples A and B of the same gas have equal volumes and pressures. The gas in sample A is expanded isothermally to four times of its initial volume and the gas in B is expanded adiabatically to double of its volume. If works done in isothermal process is twice that of adiabatic process, then show that γ satisfies the equation 1  21γ=(γ1)In2.

MEDIUM
IMPORTANT

An ideal gas CpCv=γ having initial pressure P0 and volume V0.

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

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

HARD
IMPORTANT

Two containers A and B of equal volume V02 each are connected by a narrow tube which can be closed by a valve. The containers are fitted with pistons which can be moved to change the volumes. Initially, the valve is open and the containers contain an ideal gas CpCv=γ at atmospheric pressure P0 and atmospheric temperature 2T0. The walls of the containers A are highly conducting and of B are non-conducting. The valve is now closed and the pistons are slowly pulled out to increase the volumes of the containers to double the original value. (a) Calculate the temperatures and pressures in the two containers. (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.

MEDIUM
IMPORTANT

A weightless piston divides a thermally insulated cylinder into two parts of volumes V and 3V.2 moles of an ideal gas at pressure P = 2 atmosphere are confined to the part with volume V = 1 liter. The remainder of the cylinder is evacuated. The piston is now released and the gas expands to fill the entire space of the cylinder. The piston is then pressed back to the initial position. Find the increase of internal energy in the process and final temperature of the gas. The ratio of the specific heats of the gas, γ=1.5.

HARD
IMPORTANT

Two moles of a monatomic gas, initially at pressure P1 and volume V1, undergo an adiabatic compression until its volume becomes V2. Then the gas is given heat Q at constant volume V2.

(a) Sketch the complete process on a P-V diagram.

(b) Find the total work done by the gas, the total change in its internal energy and the final temperature of the gas.

HARD
IMPORTANT

One mole of a diatomic ideal gas (γ = 1.4) is taken through a cyclic process starting from point A. The process A  B is an adiabatic compression. B  C is isobaric expansion. C  D an adiabatic expansion and D  A is isochoric as shown in P-V diagram. The volume ratios are VAVB=16 & VCVB=2 and the temperature at A is TA = 300 K. Calculate the temperature of the gas at the points B and D and find the efficiency of the cycle.

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EASY
IMPORTANT

At 27º C two moles of an ideal monoatomic gas occupy a volume V. The gas expands adiabatically to a volume 2V. Calculate:

(i) the final temperature of the gas,

(ii) change in its internal energy and 

(iii) the work done by the gas during the process.

MEDIUM
IMPORTANT

A vessel of volume V is evacuated by means of a piston air pump. In one stroke the piston is pulled out to make the volume of gas V+V then V volume from this is taken out leaving volume V in the cylinder. How many strokes are needed to reduce the pressure in the vessel to 1η times the initial pressure? The process is assumed to be isothermal, and the gas is an ideal.

HARD
IMPORTANT

Consider a spherical shell of radius R at temperature T. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume u=UVT4 and pressure P=13UV. If the shell now undergoes an adiabatic expansion the relation between T and R is

MEDIUM
IMPORTANT

The above p-v diagram represents the thermodynamic cycle of an engine, operating with an ideal mono atomic gas. The amount of heat, extracted from the source in a single cycle is 

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MEDIUM
IMPORTANT

One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100 K and the universal gas constant 8.0J mol1K1 , the decrease in its internal energy, in Joule, is__________.

MEDIUM
IMPORTANT

A diatomic ideal gas is compressed adiabatically to 132 of its initial volume. If the initial temperature of the gas is Ti (in Kelvin) and the final temperature is aTi, the value of a is :

HARD
IMPORTANT

The figure shows the P-V plot of an ideal gas taken through a cycle ABCDA. The part ABC is a semi-circle and CDA is half of an ellipse. Then,

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