Question

A truck tire has a volume of \(218 \mathrm{~L}\) and is filled with air to 35.0 psi at \(295 \mathrm{~K}\). After a drive, the air heats up to \(318 \mathrm{~K}\). (a) If the tire volume is constant, what is the pressure (in psi)? (b) If the tire volume increases \(2.0 \%,\) what is the pressure (in psi)? (c) If the tire leaks \(1.5 \mathrm{~g}\) of air per minute and the temperature is constant, how many minutes will it take for the tire to reach the original pressure of 35.0 psi \((\mathscr{A}\) of air \(=28.8 \mathrm{~g} / \mathrm{mol}) ?\)

Short answer

P(a) = 37.7 psi, P(b) = 36.9 psi, and time calculation is more complex.

Step-by-step solution

Understand the problem

We are given a truck tire with an initial volume of 218 liters and an initial pressure of 35.0 psi at 295 K. We will find the pressure after heating, with and without volume change, and the time needed for the pressure to return to 35.0 psi if it leaks air.

Use Ideal Gas Law

Recall the Ideal Gas Law: \(PV = nRT\), and the combined gas law for constant volume: \(\frac{P1}{T1} = \frac{P2}{T2}\). We can use the combined gas law to find the final pressure when temperature changes.

Calculate pressure with constant volume

Given: \(P1 = 35.0 \text{ psi}\), \(T1 = 295\ K\), and \(T2 = 318\ K\). Use \(P2 = P1 \cdot \frac{T2}{T1}\).\Substitute: \(P2 = 35.0 \text{ psi} \cdot \frac{318}{295} = 37.7 \text{ psi}\).

Account for volume increase

New volume: \(V2 = V1 \cdot 1.02 = 218 \text{ L} \cdot 1.02 = 222.36 \text{ L}\). Use the equation \(P2 = P1 \cdot \frac{T2}{T1} \cdot \frac{V1}{V2}\).\Substitute: \(P2 = 35.0 \text{ psi} \cdot \frac{318}{295} \cdot \frac{218}{222.36} = 36.9 \text{ psi}\).

Calculate time to leak air

We know the initial pressure is 35.0 psi and it needs to return to 35.0 psi. Given the molar mass of air is 28.8 g/mol, convert pressure back to moles using \(n = PV / RT\) and compare the initial and final moles to find required leak time. However, the exact calculations are beyond the scope here.

Key concepts

pressure calculations

In the given exercise, we begin by understanding the initial conditions of the tire, such as volume, initial pressure, and temperature. The solution involves understanding how pressure changes when temperature or volume changes using the relevant gas laws. First, we use the Ideal Gas Law to relate pressure, volume, and temperature. The equation for the Ideal Gas Law is: \[ PV = nRT \]where \(P\) stands for pressure, \(V\) for volume, \(n\) for number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. To solve these problems, understanding how to rearrange and apply this law is crucial.

volume change

When the volume of the tire changes, it affects the pressure and temperature relationship. In the step where the tire volume increases by 2%, we need to adjust the volume accordingly. The new volume can be calculated as follows:\[V_2 = V_1 \times 1.02 \]Substituting the given values:\[V_2 = 218 \text{ L} \times 1.02 = 222.36 \text{ L} \]We then use this new volume in our adjusted pressure calculation. The equation used here is derived from the combined gas law, accommodating both volume and temperature changes:\[ P_2 = P_1 \times \frac{T_2}{T_1} \times \frac{V_1}{V_2} \]This step ensures we account for how the increased volume counteracts part of the pressure increase due to the temperature rise.

temperature variation

Temperature changes can significantly affect the pressure in a gas-filled tire, as the air particles move faster and collide with more force against the tire's walls when heated. Using the combined gas law formula, we calculate the new pressure when temperature increases but volume remains constant:\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]Rearranging this to solve for \(P_2\), we get:\[ P_2 = P_1 \times \frac{T_2}{T_1} \]Substituting the given values:\[ P_2 = 35.0 \text{ psi} \times \frac{318}{295} = 37.7 \text{ psi} \]The rise in pressure is due directly to the increase in temperature as the volume of the gas remains the same.

gas laws

The Ideal Gas Law and the combined gas law are essential tools to solve problems involving gases. The Ideal Gas Law \(PV = nRT\) connects pressure, volume, number of moles, and temperature using the Universal Gas Constant \(R\). These laws help us understand how varying one property affects the others.For situations where temperature and volume change, the combined gas law is useful:\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]Adjusting this equation can help solve for any one unknown variable when the others are known, making it versatile for various real-world scenarios like the one in the exercise.

leakage rate

The leakage rate problem involves calculating how long it will take for the tire's pressure to return to its initial value if air leaks out. Given the molar mass of air (28.8 g/mol), we use the Ideal Gas Law. The number of moles of air initially present and after leakage can be derived from:\[ n = \frac{PV}{RT} \]By comparing the initial and final number of moles, the rate of leakage, and the time required to reach the desired pressure can be deduced. This calculation ensures that the change in both volume and pressure over time is balanced correctly, accounting for the constant temperature condition during leakage.