A motorist discovers that her auto tire measures 29.2 psi rather than the recommended 35 psi. If the tire's volume is assumed constant at 11.6 L and the temperature is 45F at the filling station, how many moles of air will the motorist need to add to bring the tire to 35 psi? (assume 1.00 atm = 14.7 psi)

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Quizplus · Accepted Answer

To find the number of moles of air needed to increase the tire pressure from 29.2 psi to 35 psi, we can use the ideal gas law, $PV = nRT$, where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the ideal gas constant, and $T$ is temperature. Since the volume and temperature are constant, the change in the number of moles ($\Delta n$) needed to increase the pressure can be calculated by comparing the initial and final states.First, convert the temperature to Kelvin: $T(K) = (45°F - 32) \times \frac{5}{9} + 273.15 = 280.372K$.The ideal gas constant $R$ in terms of L·atm/(mol·K) is approximately 0.0821.The pressure needs to be in atm for the calculation, so convert psi to atm:- Initial pressure: $29.2 \, \text{psi} \times \frac{1 \, \text{atm}}{14.7 \, \text{psi}} = 1.986 \, \text{atm}$- Final pressure: $35 \, \text{psi} \times \frac{1 \, \text{atm}}{14.7 \, \text{psi}} = 2.381 \, \text{atm}$The change in moles can be found from the difference in pressure since $V$, $R$, and $T$ are constant:$\Delta n = V \times (\frac{P_{final} - P_{initial}}{RT})$Substituting the values:$\Delta n = 11.6 \times (\frac{2.381 - 1.986}{0.0821 \times 280.372})$$\Delta n \approx 1.2 \, \text{moles}$Therefore, the motorist needs to add approximately 1.2 moles of air to the tire to increase the pressure to the recommended 35 psi.

A Motorist Discovers That Her Auto Tire Measures 29