Question

a container is divided in 2 parts by a piston p left part has one mole gas and right has spring connected when some heat is supplied piston displaces by x gas pressure is
​

Answer 1

Answer:

ANSWER

The system with heat capacity C=XR

Find x=?

Solution:-

Let, T  

1

​  

 be the initial temperature of the gas under the piston, and T  

2

​  

 the gas temperature after the amount of heat ΔQ has been supplied to the system. Since there is no friction and the vessel is thermally insulated, the entire amount of heat ΔQ is spent. On the change ΔW in the internal energy of the system

ΔQ=ΔW

The change in the internal energy of the system is the sum of the changes in the internal energy of the gas and in the potential energy of the compressed spring (Since we neglect the heat capacity of the vessel, piston and spring). The internal energy of a mole of an ideal monoatomic gas increases as a result of heating from T  

1

​  

 T−2 by

ΔW  

1

​  

=  

2

3

​  

R(T  

2

​  

−T  

1

​  

)→(1)

The potential energy of the compressed spring changes by

ΔW  

2

​  

=  

2

K

​  

(x  

2

2

​  

−x  

1

2

​  

)→(2)

Where, K us the rigidity of the spring, and x  

1

​  

 and x  

2

​  

 are the values of the absolute displacement (deformation) of the left end of the spring at temperature T−1 and T  

2

​  

 respectively.

Letus find the relation between the parameters of the gas under the piston and the deformation of the spring.

The equilibrium condition for the piston implies that

P=  

S

F

​  

=  

S

Kx

​  

,x=  

K

PS

​  

→(3)

Where P is the gas pressure and S is the area of the piston. According to the equation of state for an ideal gas , for one mole we have PV=RT for the deformation x of the spring, the volume of the gas under the piston is V=KS and pressure P=  

XS

RT

​  

 

Substituting this expression for P into equation (3 we obtain

x  

2

=  

K

RT

​  

→4

Thus the change in the potential energy of the compressed spring as result of heating of the system isΔW  

2

​  

=  

2

R

​  

(T  

2

​  

−T  

1

​  

)

The total charge in the internal energy of the system as a result of heating from T  

1

​  

 T  

2

​  

 is

ΔW=ΔW  

1

​  

+ΔW  

2

​  

 

=2R(T  

2

​  

−T  

1

​  

)

And the heat capacity of the system is

C=  

ΔT

ΔQ

​  

=  

T  

2

​  

−T  

1

​  

 

ΔU

​  

=2R

Now,

XR=2R

X=2

Explanation: