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Chapter 1: Problem 44

A regulation basketball is initially flat and is then inflated to a pressure of approximately \(24 \mathrm{lb} / \mathrm{in}^{2}\) absolute. Consider the air temperature to be constant at \(70^{\circ} \mathrm{F}\). Find the mass of air required to inflate the basketball. The basketball's inside radius is 4.67 in.

Short Answer

Expert verified
The mass of air required to inflate the basketball is approximately 0.031 kg.

Step by step solution

01

Calculate the Volume of the Basketball

First, calculate the volume of the basketball. The volume of a sphere is given by \(V = \frac{4}{3} \pi r^{3}\), where r is the radius. Substituting \(r = 4.67\) inches into the equation gives the volume in cubic inches. To convert this to liters (the unit used in the Ideal Gas Law), multiply by 0.016387064. The result is the volume in liters.
02

Convert Pressure Units

Next, convert the given pressure from pounds per square inch to atmospheres. One atmosphere is approximately 14.7 lb/in², so divide 24 by 14.7 to get the pressure in atmospheres.
03

Use the Ideal Gas Law to Find Moles of Air

Now use the Ideal Gas Law (\(PV=nRT\)) to find the number of moles of air needed to inflate the basketball. In this case, P is the pressure in atmospheres, V is the volume in liters, R is the gas constant (0.0821 L.atm/K.mol), and T is the temperature in Kelvin. To convert degrees Fahrenheit to Kelvin, use the formula \(K = \frac{5}{9}(F - 32) + 273.15\). Substitute \(P, V, R\), and \(T\) into the Ideal Gas Law and solve for \(n\), the number of moles.
04

Calculate Mass of Air

Finally, convert the number of moles to mass. The average molar mass of air is approximately 28.97 g/mol, so multiply the number of moles by this figure to get the mass of air in grams. To convert this to kilograms, simply divide by 1000.

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