## Problem Statement

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

Hint: The equation of state for the gas is

$\rho V = \text{const}$, where $\rho$ is the pressure and $V$ is the volume.

$$
H = 0.8 \text{m}, h = 0.7 \text{m}, R = 0.2 \text{m}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_31c1f87d4364af4121d5g-37.jpg?height=514&width=834&top_left_y=622&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 $\Delta A=F(x) \Delta x \Rightarrow$

$$
\begin{aligned}
& \Rightarrow d A=\frac{p_{0} \cdot S \cdot H}{S \cdot(H-x)} \cdot S d x=\frac{p_{0} \cdot S \cdot H}{H-x} d x \Rightarrow \\
& \Rightarrow A=\int_{0}^{h} \frac{p_{0} \cdot S \cdot H}{H-x} d x=-\int_{0}^{h} \frac{p_{0} \cdot S \cdot H}{H-x} d(H-x)=-\left.p_{0} \cdot S \cdot H \cdot \ln (H-x)\right|_{0} ^{h}= \\
& =p_{0} \cdot S \cdot H \cdot \ln \left(\frac{H}{H-h}\right)=p_{0} \cdot \pi \cdot R^{2} \cdot H \cdot \ln \left(\frac{H}{H-h}\right) \\
& =\pi \cdot(0.2[m])^{2} \cdot 103.3[k P a] \cdot 0.8[m] \cdot \ln \left(\frac{0.8[m]}{0.8[m]-0.7[m]}\right)=21595[\mathrm{kJ}]
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+22-24$ »

Categories: Kuznetsov Integral Problems 22-25 | Integrals

- Last edited: 12:51, 31 May 2010.
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