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Chapter 1: Problem 102

A compressed air tank in a service station has a volume of \(10 \mathrm{ft}^{3} .\) It contains air at \(70^{\circ} \mathrm{F}\) and 150 psia. How many tubeless tires can it fill to 44.7 psia at \(70^{\circ} \mathrm{F}\) if each tire has a volume of 1.5 \(\mathrm{ft}^{3}\) and the compressed air tank is not refilled? The tank air temperature remains constant at \(70^{\circ} \mathrm{F}\) because of heat transfer through the tank's large surface area.

Short Answer

Expert verified
The exact number of tires that can be filled depends on the specific value of the gas constant R in psia*ft^3/(°Rankine*mol). With the value R = 0.73, the calculation gives the final answer as approximately 5.8, so it is possible to fully fill 5 tires. The tank won't be able to fully fill a 6th tire.

Step by step solution

01

Conversion into appropriate units

For the ideal gas law, it is appropriate to have the pressure in absolute pressure units psia, the volume in cubic feet, and the temperature in Rankine degrees (the Fahrenheit equivalent of Kelvin). The temperature at 70 °F is therefore \(70 + 459.67 = 539.67 \) Rankine.
02

Application of the ideal gas law for initial conditions

Let's denote the initial conditions in the tank by \(P1, V1, n1\), where \(P1 = 150\) psia, \(V1 = 10\) cubic feet, and \(n1\) is the number of moles of gas initially in the tank that we calculate based on the ideal gas law: \( n1 = \frac{P1 * V1}{ R * T} \).
03

Application of the ideal gas law for final conditions

The final conditions in each tire will be denoted by \(P2, V2, n2\), where \(P2 = 44.7\) psia, \(V2 = 1.5\) cubic feet, and \(n2\) is the number of moles of gas in each tire, calculated again using the ideal gas law: \(n2 = \frac{P2 * V2}{R * T}\).
04

Calculation of the number of tires

Considering that the total number of moles of gas in the tank remains the same, the number of tires that can be filled is the ratio between \(n1\) and \(n2\): \(\frac{n1}{n2}\).

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Most popular questions from this chapter

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