Question

An ideal gas at NTP is enclosed in a adiabatic vertical cylinder having area of cross section $A=27c{m}^2$, between two light movable piston as shown in the figure. Spring with force constant k = 3700 N/m is in a relaxed state initially. Now the lower piston is moved upward at a height h/2, h being the initial length of gas column. It is observed that the upper piston moves up by a distance h/16.

Find h taking $\gamma$ for the gas to be 1.5. Also find the final temperature of the gas.

Answer 1

NTP: $T_o=0^{o}C=273K{P}_o=1bar={10}^{5}Pa$

Final volume, $V_f=[h+\frac{h}{16}-\frac{h}{2}]A=\frac{9}{16}hA=\frac{9}{16}{V}_o$

Using $P{V}^{\gamma }=K$

$\frac{P_f}{P_o}=(\frac{V_o}{\frac{9}{16}{V}_o}{)}^{1.5}$

$⟹P_f=2.37P_o$

Now $P_f=P_o+\frac{Kh}{16A}$ where $k=3700N/mA=27\times {10}^{-4}{m}^2$

$2.37P_o=P_o+\frac{3700\left(h\right)}{16(27\times {10}^{-4})}$

$⟹h=1.6m$

Also, $T{V}^{\gamma -1}=constant$

$\frac{T_f}{273}=(\frac{V_o}{\frac{9}{16}{V}_o}{)}^{.5}$

$⟹T_f=364K$