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Chapter 8: Problem 56

A bicycle tire is filled with air to a pressure of \(75\) psi at a temperature of \(19^{\circ} \mathrm{C}\). Riding the bike on asphalt on a hot day increases the temperature of the tire to \(58^{\circ} \mathrm{C}\). The volume of the tire increases by \(4.0 \% .\) What is the new pressure in the bicycle tire?

Short Answer

Expert verified
The new pressure in the bicycle tire is approximately \(79.52\) psi.

Step by step solution

01

Convert temperatures to Kelvin

First, we need to convert the given temperatures from Celsius to Kelvin, as the ideal gas law requires temperatures to be in Kelvin. We can do this using the formula \(T(K) = T(^\circ C) + 273.15\). Initial temperature in Kelvin: \(T_1 = 19 + 273.15 = 292.15 K\) Final temperature in Kelvin: \(T_2 = 58 + 273.15 = 331.15 K\)
02

Determine the final volume

The volume of the tire increases by \(4.0\%\). Denoting the initial volume as \(V_1\), the final volume can be calculated as \(V_2 = V_1 + 0.04 V_1 = 1.04V_1\).
03

Apply the ideal gas law formula

We will now apply the ideal gas law formula \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\) to determine the final pressure. We can rearrange the formula to solve for \(P_2\): \(P_2 = \frac{P_1V_1T_2}{T_1V_2}\)
04

Substitute values and solve for the final pressure

Substitute given values, \(P_1 = 75 \text{ psi}\), \(T_1 = 292.15 K\), \(T_2= 331.15 K\), and \(V_2 = 1.04V_1\): \(P_2 = \frac{75 \times V_1 \times 331.15}{292.15 \times 1.04V_1}\) You can see that the volume terms \(V_1\) cancel out: \(P_2 = \frac{75 \times 331.15}{292.15 \times 1.04}\) Now, calculating the final pressure, \(P_2\): \(P_2 \approx 79.52 \text{ psi}\) The new pressure in the bicycle tire is approximately \(79.52\) psi.

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Most popular questions from this chapter

A mixture of chromium and zinc weighing \(0.362 \mathrm{g}\) was reacted with an excess of hydrochloric acid. After all the metals in the mixture reacted, \(225 \mathrm{mL}\) dry of hydrogen gas was collected at \(27^{\circ} \mathrm{C}\) and \(750 .\) torr. Determine the mass percent of \(\mathrm{Zn}\) in the metal sample. [Zinc reacts with hydrochloric acid to produce zinc chloride and hydrogen gas; chromium reacts with hydrochloric acid to produce chromium(III) chloride and hydrogen gas.]

A steel cylinder contains \(5.00\) moles of graphite (pure carbon) and \(5.00\) moles of \(\mathrm{O}_{2}\). The mixture is ignited and all the graphite reacts. Combustion produces a mixture of \(\mathrm{CO}\) gas and \(\mathrm{CO}_{2}\) gas. After the cylinder has cooled to its original temperature, it is found that the pressure of the cylinder has increased by \(17.0 \% .\) Calculate the mole fractions of \(\mathrm{CO}, \mathrm{CO}_{2},\) and \(\mathrm{O}_{2}\) in the final gaseous mixture.

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Equal moles of sulfur dioxide gas and oxygen gas are mixed in a flexible reaction vessel and then sparked to initiate the formation of gaseous sulfur trioxide. Assuming that the reaction goes to completion, what is the ratio of the final volume of the gas mixture to the initial volume of the gas mixture if both volumes are measured at the same temperature and pressure?
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