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Chapter 1: Problem 48

Assume that the air volume in a small automobile tire is constant and equal to the volume between two concentric cylinders \(13 \mathrm{cm}\) high with diameters of \(33 \mathrm{cm}\) and \(52 \mathrm{cm}\). The air in the tire is initially at \(25^{\circ} \mathrm{C}\) and \(202 \mathrm{kPa}\). Immediately after air is pumped into the tire, the temperature is \(30^{\circ} \mathrm{C}\) and the pressure is 303 kPa. What mass of air was added to the tire? What would be the air pressure after the air has cooled to a temperature of \(0^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
The mass of air added to the tire will be calculated using the molar mass of air and the difference in number of moles before and after pumping. The new air pressure after cooling down will be calculated using the Ideal Gas Law equation with the new temperature and the final number of moles.

Step by step solution

01

Calculate the volume of the tire

We first need to calculate the volume of the tire. Since it's in the shape of a cylinder, we can use the formula for the volume of a cylinder which is \(V = \pi d^2 h / 4\) where \(d\) is the diameter and \(h\) is the height. The tire volume is the volume of the large cylinder minus the volume of the small cylinder. We convert all dimensions to meters to keep the units consistent.
02

Calculate the initial number of moles in the tire

To find the mass of air added to the tire, we need to calculate the number of moles of air initially present and then after the pumping. We can use the ideal gas law in the form \(n = PV / RT\), where \(n\) is the number of moles, \(P\) is the pressure in Pa, \(V\) is the volume in \(m^3\), \(R\) is the gas constant (\(8.314 J/(mol \cdot K)\)), and \(T\) is the temperature in K. The initial temperature in K can be calculated by adding 273.15 to the temperature in Celsius. Thus the initial number of moles will be calculated as \(n_i = P_i V / (R T_i)\).
03

Calculate the number of moles after pumping the air into the tire

Similar to step 2, we calculate the final number of moles using the same ideal gas law formula, but this time using the final temperature and pressure. \(n_f = P_f V_f / (R T_f)\).
04

Calculate the mass difference

The difference between the initial and final number of moles will give us the number of moles of air added to the tire. Therefore, \(n_{added} = n_f - n_i\). Since the molar mass of air is roughly 28.96 g / mol, we can now calculate the mass of the added air.
05

Calculate the air pressure at 0 Celsius

For the second part of the exercise, we use the same ideal gas law equation but rearranged to solve for pressure, \(P = nRT / V\), where \(n\) is the final mole of air in the tire, \(T\) is the new temperature in Kelvin (0 Celsius in this case), and \(R\) and \(V\) remain the same. We then calculate the new pressure.

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