## Task Condition

A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston moving inside the cylinder by $h$ meters (see figure).

Hint: The equation of state of the gas

$\rho V=$ const, where $\rho$ - pressure, $V$ - volume.

$$
H=1.6 \text{ m}, h=1.2 \text{ m}, R=0.3 \text{ m.}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_31c1f87d4364af4121d5g-45.jpg?height=517&width=834&top_left_y=621&top_left_x=1088)
## Solution

Piston area: $S=\pi R^{2}$

Volume of gas during compression: $V(x)=S \cdot(H-x) ; 0 \leq x \leq h$

Pressure of gas during compression: $p(x)=\frac{p_{0} \cdot S \cdot H}{V(x)}$

Force of pressure on the piston: $F(x)=p(x) \cdot S$

By definition, the elementary work $\Delta A=F(x) \Delta x \Rightarrow$

$$
\begin{aligned}
& \Rightarrow d A=\frac{p_{0} \cdot S \cdot H}{S \cdot(H-x)} \cdot S d x=\frac{p_{0} \cdot S \cdot H}{H-x} d x \Rightarrow \\
& \Rightarrow A=\int_{0}^{h} \frac{p_{0} \cdot S \cdot H}{H-x} d x=-\int_{0}^{h} \frac{p_{0} \cdot S \cdot H}{H-x} d(H-x)=-\left.p_{0} \cdot S \cdot H \cdot \ln (H-x)\right|_{0} ^{h}= \\
& =p_{0} \cdot S \cdot H \cdot \ln \left(\frac{H}{H-h}\right)=p_{0} \cdot \pi \cdot R^{2} \cdot H \cdot \ln \left(\frac{H}{H-h}\right) \\
& =\pi \cdot(0.3[m])^{2} \cdot 103.3[k P a] \cdot 1.6[m] \cdot \ln \left(\frac{1.6[m]}{1.6[m]-1.2[m]}\right)=64784 \approx 64800[ \\
& \text { kJ] }
\end{aligned}
$$

Source — "http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+22-28$ "

Categories: Kuznetsov Integral Problems 22-29 | Integrals

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