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 P_{0}. 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
C.
FBD of piston at equilibrium
⇒ P_{atm} A + mg = P_{0}A
FBD of piston when piston is pushed down a distance x
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%
12.5%
A.
15.4%
The efficiency of a process is defined as the ratio of work done to energy supplied.
Here,
Where C_{p} and C_{v} are two heat capacities (molar)
A solid body of constant heat capacity 1 J/°C is being heated by keeping it in contact with reservoirs in two ways:
(i) Sequentially keeping in contact with 2 reservoirs such that each reservoir supplies the same amount of heat.
(ii) Sequentially keeping in contact with 8 reservoirs such that each reservoir supplies the same amount of heat. In both the cases body is brought from an initial temperature 100°C to final temperature 200°C. Entropy change of the body in the two cases respectively is:
ln2,4ln2
ln2,ln2
ln2,2ln2
2ln2,8ln2
B.
ln2,ln2
Since entropy is a state function, therefore a change in entropy in both the processes must be same .
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 and pressure. If the shell now undergoes an adiabatic expansion the relation between T and R is
T ∝ e^{-R}
T ∝ e-^{3R}
T ∝ (1/R)
T ∝(1/R^{3})
C.
T ∝ (1/R)
According to given equation,
Three perfect gases at absolute temperature T_{1}, T_{2} and T_{3} are mixed. The masses of molecules are m_{1},m_{2} and m_{3} and the number of molecules is n_{1},n_{2} and n_{3} respectively.Assuming no loss of energy, the final temperature of the mixture is
A.
For adiabatic process i.e., no heat change
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
600 K
C.
750 K
Efficiency
A Carnot engine operating between temperatures T_{1} and T_{2} has efficiency 1/6. When T_{2} is lowered by 62 K, its efficiency increases to 1/3. Then T_{1} and T_{2} are, respectively
372 K and 330 K
330 K and 268 K
310 K and 248 K
372 K and 310 K
D.
372 K and 310 K
The efficiency is given by,
One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in the figure. The process BC is adiabatic. The temperatures at A, B and C are 400 K, 800 K and 600 K respectively. Choose the correct statement:
The change in internal energy in the process AB is -350 R.
The change in internal energy in the process BC is -500 R.
The change in internal energy in the whole cyclic process is 250 R.
The change in internal energy in the process CA is 700 R.
D.
The change in internal energy in the process CA is 700 R.
According to first law of thermodynamics,
(i) change in internal energy from A to B i.e,
Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion the average time of collision between molecules increases as V^{q} , where V is the volume of the gas. The value of q is:
C.
For an adiabatic process TV^{γ-1} = constant
We know that average time of collision between molecules
Where n= number of molecules per unit volume
v_{rms} = rms velocity of molecules
Thus, we can write
n =K_{1}V^{-1} and V_{rms} = K_{2}T^{1/2}
Where K_{1} and K_{2} are constants.
For adiabatic process TV^{γ-1} = constant. Thus we can write
The above p-v diagram represents the thermodynamic cycle of an engine, operating with an ideal monoatomic gas. The amount of heat extracted from the source in a single cycle is
p_{o}v_{o}
4p_{o}v_{o}
B.
Heat is extracted from the source in path DA and AB is