Question

A sample of gas expands from an initial pressure and volume of$\mathbf{10}\mathbf{}\mathbf{\text{Pa}}\mathbf{}\mathbf{\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}}\mathbf{1}\mathbf{.0}\mathbf{}{\mathbf{\text{\hspace{0.17em}m}}}^{\mathbf{\text{3}}}$to a final volume of$\mathbf{2}\mathbf{.0}\mathbf{}{\mathbf{\text{\hspace{0.17em}m}}}^{\mathbf{\text{3}}}$. During the expansion, the pressure and volume are related by the equation,$\mathit{p}\mathbf{=}\mathbf{}\mathit{a}{\mathit{V}}^{\mathbf{2}}$ where. $\mathit{a}\mathbf{=}\mathbf{}\mathbf{10}\mathbf{}{\mathbf{\text{N/m}}}^{\mathbf{\text{8}}}$Determine the work done by the gas during this expansion.

Short answer

Work done by the gas during expansion is $23.33\text{\hspace{0.17em}J}$.

Step-by-step solution

Stating the given data

1. Initial pressure is$P=10\text{\hspace{0.17em}Pa}$
2. Initial volume is$V_i=1.0{\text{\hspace{0.17em}m}}^{\text{3}}$
3. Final volume is$V_f=2.0{\text{\hspace{0.17em}m}}^{\text{3}}$
4. Equation of the pressure-volume relation,$P=a{V}^2$
5. Value of a is $a=10{\text{\hspace{0.17em}N/m}}^{\text{8}}$.

Understanding the concept of work done

By expanding or contracting against a steady external pressure, gases can perform work. We use the concept of work done by gas. Using the equation of work done we can integrate it from initial volume to final volume.

Formula:

The work done by the system in thermodynamic cycle $W=\int PdV$,. …(i)

Calculation of the work done by the gas

We can write the equation of work done by the gas during expansion using equation (i) and the given equation as

$\begin{array}{c}W=\underset{V_i}{\overset{V_f}{}}a{V}^{2}dV\\ =a{\left|\frac{V^3}{3}\right|}_{V_i}^{V_f}\\ =\frac{a}{3}(V_f^3-V_i^3)\end{array}$

Plugging the values in the above equation, we get the work done by the gas as

$\begin{array}{c}W=\frac{10{\text{\hspace{0.17em}N/m}}^{\text{8}}}{3}({2.0}^3-{1.0}^3){\text{m}}^9\\ =23.33\text{\hspace{0.17em}J}\end{array}$

Hence, the value of the work done by the gas is$23.33\text{\hspace{0.17em}J}$