(^220) Aptitude Test Problems in Physics
system. Since there is no friction and the vessel
is thermally insulated, the entire amount of heat
AQ is spent on the change A W in the internal en-
ergy of the system:
AQ = A W.
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 com-
pressed spring (since we neglect the heat capacity
of the vessel, piston, and spring).
The internal energy of a mole of an ideal mon-
atomic gas increases as a result of heating from
T 1 to T2 by
3
AW 1 = —
2
R (T 2 — T 1 ). (1)
The potential energy of the compressed spring
changes by
AW 2= 2 (x4 — (2)
where k is the rigidity of the spring, and x 1 and x 2
are the values of the absolute displacement (de-
formation) of the left end of the spring at temper-
atures T 1 and T2 respectively. Let us 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
F kx pSP = =S x k
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 = xS and the pressure
p = RT/(xS). Substituting this expression for p
into Eq. (3), we obtain
RT
= (4)
k •