Solutions 29
Thus, by the moment of displacement of the
piston by x = 1/2, the total work done by the gas
will be equal to the sum of the potential energies
stored in the springs:
A 2 f 1 \ 2
(^2) k 2)
If an amount of heat Q is supplied to the gas
in the right part of the vessel, and the gas in the
left part transfers an amount of heat Q' to the
thermostat, the total amount of heat supplied
to the system is Q — Q', and we can write (the
first law of thermodynamics)
Q—Q,=2 k 1
2 k 2
)
2 i-AU, (4)
where A U is the change in the internal energy of
the gas. Since the piston does not conduct heat,
the temperature of the gas in the left part does not
change, and the change A U in the internal energy
of the gas is due to the heating of the gas in the
right part by AT. For n moles of the ideal gas, we
have AU = n (3/2)R AT. Tho temperature incre-
ment AT can be found from the condition of equi-
librium at the end of the process.
In accordance with the equation of state, the
pressure of the gas in the right part of the vessel is
p = nR (T + AT)/[S (1 + 112)]. On the other hand,
it must be equal to the sum of the gas pressure
p' = nRT/[S (1 — 1/2)] in the left part and the
pressure p" = 2kI1(2S) created by the springs, i.e.
2nR (T AT) 2nRT kl
3S1 SI S
Hence we can find that AT = 2T + 3kl^2 /(2nR).
Using Eq. (4), we finally obtain
5
Q' =Q-3nRT --
2
k12.
2.13. Let T 1 be the initial temperature of the gas
under the piston, and T2 the gas temperature after
the amount of heat AQ has been supplied to the