Advertisement

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M. The piston and cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is V0 and its pressure is P0. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency

  • fraction numerator 1 over denominator 2 straight pi end fraction fraction numerator straight A subscript straight gamma straight p subscript 0 over denominator straight V subscript 0 straight M end fraction
  • fraction numerator 1 over denominator 2 straight pi end fraction fraction numerator straight V subscript 0 Mp subscript 0 over denominator straight A squared straight gamma end fraction
  • fraction numerator 1 over denominator 2 straight pi end fraction square root of fraction numerator straight A squared γp subscript 0 over denominator MV subscript 0 end fraction end root
  • fraction numerator 1 over denominator 2 straight pi end fraction square root of fraction numerator straight A squared γp subscript 0 over denominator MV subscript 0 end fraction end root


C.

fraction numerator 1 over denominator 2 straight pi end fraction square root of fraction numerator straight A squared γp subscript 0 over denominator MV subscript 0 end fraction end root


FBD of piston at equilibrium
⇒ Patm A + mg = P0A

FBD of piston when piston is pushed down a distance x

401 Views

Advertisement

Three perfect gases at absolute temperature T1, T2 and T3 are mixed. The masses of molecules are m1,m2 and m3 and the number of molecules is n1,n2 and n3 respectively.Assuming no loss of energy, the final temperature of the mixture is

  • fraction numerator straight n subscript 1 straight T subscript 1 plus straight n subscript 2 straight T subscript 2 space plus straight n subscript 3 straight T subscript 3 over denominator straight n subscript 1 plus space straight n subscript 2 plus space straight n subscript 3 end fraction
  • fraction numerator straight n subscript 1 straight T subscript 1 superscript 2 space plus straight n subscript 2 straight T subscript 2 superscript 2 space plus straight n subscript 3 straight T subscript 3 superscript 2 over denominator straight n subscript 1 straight T subscript 1 space plus straight n subscript 2 straight T subscript 2 space plus straight n subscript 3 straight T subscript 3 end fraction
  • fraction numerator straight n squared subscript 1 straight T subscript 1 superscript 2 space plus straight n squared subscript 2 straight T subscript 2 superscript 2 space plus straight n squared subscript 3 straight T subscript 3 superscript 2 over denominator straight n subscript 1 straight T subscript 1 space plus straight n subscript 2 straight T subscript 2 space plus straight n subscript 3 straight T subscript 3 end fraction
  • fraction numerator straight n squared subscript 1 straight T subscript 1 superscript 2 space plus straight n squared subscript 2 straight T subscript 2 superscript 2 space plus straight n squared subscript 3 straight T subscript 3 superscript 2 over denominator straight n subscript 1 straight T subscript 1 space plus straight n subscript 2 straight T subscript 2 space plus straight n subscript 3 straight T subscript 3 end fraction

A.

fraction numerator straight n subscript 1 straight T subscript 1 plus straight n subscript 2 straight T subscript 2 space plus straight n subscript 3 straight T subscript 3 over denominator straight n subscript 1 plus space straight n subscript 2 plus space straight n subscript 3 end fraction

For adiabatic process i.e., no heat change

straight F over 2 straight n subscript 1 straight k subscript 1 straight T subscript 1 space plus fraction numerator begin display style straight F end style over denominator begin display style 2 end style end fraction straight n subscript 2 kT subscript 2 space plus fraction numerator begin display style straight F end style over denominator begin display style 2 end style end fraction straight n subscript 3 kT subscript 3 space
equals fraction numerator begin display style straight F end style over denominator begin display style 2 end style end fraction left parenthesis straight n subscript 1 plus straight n subscript 2 plus straight n subscript 3 right parenthesis kT
rightwards double arrow space straight T space equals space fraction numerator straight n subscript 1 straight T subscript 1 space plus straight n subscript 2 straight T subscript 2 plus straight n subscript 3 straight T subscript 3 over denominator space straight n subscript 1 plus straight n subscript 2 plus straight n subscript 3 end fraction

911 Views

A Carnot engine operating between temperatures T1 and T2 has efficiency 1/6. When T2 is lowered by 62 K, its efficiency increases to 1/3. Then T1 and T2 are, respectively

  • 372 K and 330 K

  • 330 K and 268 K

  • 310 K and 248 K

  • 310 K and 248 K


D.

310 K and 248 K

The efficiency  is given by,

straight eta subscript 1 space equals space 1 space minus straight T subscript 2 over straight T subscript 1
rightwards double arrow space 1 over 6 space equals space 1 space minus space straight T subscript 2 over straight T subscript 1
rightwards double arrow space straight T subscript 2 over straight T subscript 1 space equals space 5 over 6
straight eta subscript 2 space equals space 1 minus fraction numerator straight T subscript 2 minus 62 over denominator straight T subscript 1 end fraction space space.... space left parenthesis straight i right parenthesis
1 third space equals space 1 minus fraction numerator straight T subscript 2 minus 62 over denominator straight T subscript 1 end fraction space... space left parenthesis ii right parenthesis
On space solving space Eqs space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis
straight T subscript 1 space equals space 372 space straight K space and space straight T subscript 2 space equals space 310 space straight K

763 Views

Helium gas goes through a cycle ABCDA (consisting of two isochoric and two isobaric lines) as shown in the figure. Efficiency of this cycle is nearly:(Assume the gas to be close to ideal gas)

  • 15.4%

  • 9.1%

  • 10.5%

  • 10.5%


A.

15.4%

The efficiency of a process is defined as the ratio of work done to energy supplied.
Here,
Where Cp and Cv are two heat capacities (molar)

1344 Views

A Carnot engine, whose efficiency is 40%, takes in heat from a source maintained at a temperature of 500 K It is desired to have an engine of efficiency 60%. Then, the intake temperature for the same exhaust (sink) temperature must be

  • the efficiency of Carnot engine cannot be made larger than 50%

  • 1200 K

  • 750 K 

  • 750 K 


C.

750 K 

Efficiency

690 Views

Advertisement