An inflated bicycle inner tube is 2.2 cm in diameter and 200 cm in circumference. A small leak causes the gauge pressure to decrease from 110 psi to 80 psi on a day when the temperature is 20°C. What mass of air is lost? Assume the air is pure nitrogen.

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Determine the initial and final pressures in absolute terms by adding atmospheric pressure (approximately 14.7 psi) to the gauge pressures. Convert these pressures to Pascals (Pa) using the conversion factor: 1 psi = 6894.76 Pa.

Use the ideal gas law, \( PV = nRT \), to relate the initial and final states of the gas. Here, \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (8.314 J/(mol·K)), and \( T \) is temperature in Kelvin. Convert the temperature from Celsius to Kelvin using \( T(K) = T(°C) + 273.15 \).

Calculate the volume of the inner tube. Treat the tube as a cylindrical shell. The volume \( V \) can be approximated as \( V = \pi r^2 h \), where \( r \) is the radius (half the diameter) and \( h \) is the circumference. Convert the dimensions to meters before calculating.

Using the ideal gas law, calculate the initial number of moles \( n_1 \) and the final number of moles \( n_2 \) of nitrogen in the inner tube. Rearrange the ideal gas law to solve for \( n \): \( n = \frac{PV}{RT} \). Use the initial and final pressures, the calculated volume, and the temperature in Kelvin.

Find the mass of air lost by calculating the difference in moles \( \Delta n = n_1 - n_2 \) and multiplying by the molar mass of nitrogen (28.02 g/mol). Convert the result to kilograms by dividing by 1000.

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주요 개념

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Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding how changes in pressure and temperature affect the volume and amount of gas present. In this scenario, it helps determine how the decrease in pressure due to the leak correlates with the loss of air mass.

Gauge Pressure

Gauge pressure is the pressure of a system above atmospheric pressure. It is measured relative to the ambient atmospheric pressure and is crucial for understanding the pressure changes in the inner tube. In this question, the initial and final gauge pressures indicate how much air has escaped due to the leak.

Density of Nitrogen

The density of nitrogen at standard conditions is approximately 1.25 kg/m³. Knowing the density is vital for calculating the mass of the air lost when the pressure changes. By using the Ideal Gas Law and the change in volume due to the pressure drop, one can find the volume of nitrogen lost and subsequently its mass.