Question

Question: An ideal monatomic gas is contained in a tall cylindrical jar of cross-sectional area $\mathbf{0}\mathbf{.080}{\mathbf{m}}^{\mathbf{2}}$ fitted with an airtight frictionless 0.15-kg movable piston. When the gas is heated (at constant pressure) from 25°C to 55°C, the piston rises 1.0 cm. How much heat was required for this process? Assume atmospheric pressure outside. (Hint: See Section 14–2.)

Short answer

The heat required for this process is $202\mathrm{J}$.

Step-by-step solution

Determination of pressure

The pressure can be obtained by dividing the force exerted on an object by the cross-sectional area of the object. Its value is inversely related to the value of the cross-sectional area of the object.

Given information

The cross-sectional area of the cylinder is $A=0.080{\mathrm{m}}^2$.

The atmospheric pressure or outside pressure is $P_{\mathrm{o}}=1.01\times {10}^5\mathrm{P}\mathrm{a}$.

The mass of the piston is $m=0.15\mathrm{k}\mathrm{g}$.

The displacement of the piston is $\mathrm{\Delta }y=1.0\mathrm{c}\mathrm{m}$.

Evaluation of the pressure inside the cylindrical jar

The pressure inside the cylindrical jar is calculated below:

$\begin{array}{rl}c{P}_{\mathrm{i}}& =P_{\mathrm{o}}+\frac{mg}{A}\\ P_{\mathrm{i}}& =\left(1.01\times {10}^5\mathrm{P}\mathrm{a}\right)+\frac{\left(0.15\mathrm{k}\mathrm{g}\right)\left(9.8\mathrm{m}/\phantom{\mathrm{m}{\mathrm{s}}^2}{\mathrm{s}}^2\right)}{\left(0.080{\mathrm{m}}^2\right)}\\ P_{\mathrm{i}}& =1.0102\times {10}^5\mathrm{P}\mathrm{a}\approx 1.01\times 10^5\mathrm{P}\mathrm{a}\end{array}$

The change in the pressure is negligible; therefore, the system can be treated as a constant pressure process.

Evaluation of the heat required for the process

The heat required for the process is calculated below:

$\begin{array}{rl}lQ& =\mathrm{\Delta }U+W\\ Q& =\frac{3}{2}P\mathrm{\Delta }V+P\mathrm{\Delta }V\\ Q& =\frac{5}{2}P\mathrm{\Delta }V\\ Q& =\frac{5}{2}PA\mathrm{\Delta }y\end{array}$

Substitute the values in the above equation.

$\begin{array}{rl}cQ& =\frac{5}{2}\left(1.01\times {10}^5\mathrm{P}\mathrm{a}\right)\left(0.080{\mathrm{m}}^2\right)\left(1.0\mathrm{c}\mathrm{m}\right)\left(\frac{10^{-2}\mathrm{m}}{1\mathrm{c}\mathrm{m}}\right)\\ Q& =202\mathrm{J}\end{array}$

Thus, the heat required for this process is $202\mathrm{J}$.