Question

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains$\mathbf{499}\mathbf{}\mathit{c}{\mathit{m}}^{\mathbf{3}}$of air at atmospheric pressure$\mathbf{1}\mathbf{.}\mathbf{01}\mathbf{*}{\mathbf{10}}^{\mathbf{5}}\mathit{P}\mathit{a}$and a temperature of$\mathbf{27}\mathbf{.}\mathbf{0}\mathbf{°}\mathit{C}$. At the end of the stroke, the air has been compressed to a volume of$46.2c{m}^3$and the gauge pressure has increased to$\mathbf{2}\mathbf{.}\mathbf{72}\mathbf{}\mathbf{*}\mathbf{}{\mathbf{10}}^{\mathbf{6}}\mathit{P}\mathit{a}$. Compute the final temperature.

Short answer

The final temperature after compression is $475.{23}^{o}C$.

Step-by-step solution

The ideal gas equation and expression for final temperature

The ideal gas equation isPV=nRT.

The expression for temperature is thus$T=\frac{PV}{nR}$ where P is the pressure, V is the volume, T is the temperature, R is the universal gas constant = $8.314J/mol.K$.

There exist two set of conditions , that of ($\left(P_1,V_1,T_1\right)$) and ($\left(P_2,V_2,T_2\right)$), then $T_1=\frac{P_1{V}_1}{nR}$and $T_2=\frac{P_2{V}_2}{nR}$. Combining which gives $\frac{T_1}{T_2}=\frac{\frac{P_1{V}_1}{nR}}{\frac{P_2{V}_2}{nR}}$.

Since the number of moles and R are constant,$\frac{T_1}{T_2}=\frac{P_1{V}_1}{P_2{V}_2}$ where$P_1,V_1$ are the initial pressure and volume respectively, androle="math" localid="1668169464494" $P_2,V_2$ are the final pressure and volume respectively.

Then, expression for final temperature is derived from the above relation as $T_2=\frac{P_2{V_{2}T}_1}{P_1{V}_1}$.

Calculating the value of final temperature

Substituting the values of $P_1=1.01*{10}^{5}Pa$, $V_1=499c{m}^3$, $T_1={27}^{o}C=27+273K=300K$, $P_2=2.72*{10}^{6}Pa$,$V_2=46.2c{m}^3$ in $T_2=\frac{P_2{V}_2{T}_1}{P_1{V}_1}$,

$\begin{array}{r}{T}_2=\frac{{2.72}^{\ast }{10}^{6\ast }{46.2}^{\ast }300.15}{{1.01}^{\ast }{10}^{5\ast }499}\\ T_2=748.38\mathrm{K}\\ T_2=(748.38-273.15{)}^{\circ }\mathrm{C}\\ T_2={475.23}^{\circ }\mathrm{C}\end{array}$

Thus, the final temperature after compression is ${475.23}^{\circ }\mathrm{C}$.