Question

A tire has a gauge pressure of \(300 . \mathrm{kPa}\) at \(15.0^{\circ} \mathrm{C} .\) What is the gauge pressure at \(45.0^{\circ} \mathrm{C}\) ? Assume that the change in volume of the tire is negligible.

Short answer

Answer: The tire's gauge pressure at a temperature of 45°C is approximately 330.9 kPa.

Step-by-step solution

Recall the ideal gas law

We can use the ideal gas law to find the relationship between initial and final pressure as follows: \[PV = nRT\] Where: - P: pressure of the gas - V: volume of the gas - n: number of moles of the gas - R: gas constant - T: temperature in Kelvin In this problem, the volume and the amount of gas are assumed to be constant. So we'll need to find the relationship between pressure and temperature by comparing the initial and final states of the tire.

Convert the temperatures to Kelvin

Before we proceed further, we need to convert the given temperatures in Celsius to Kelvin using the formula \[T(K) = T(^\circ C) + 273.15\] For the initial temperature: \[T_1(K) = 15.0 + 273.15 = 288.15 K\] For the final temperature: \[T_2(K) = 45.0 + 273.15 = 318.15 K\]

Compare the initial and final states of the tire using the ideal gas law

Since the volume and the number of moles remain constant, we can write the ideal gas law for the initial state \(P_1T_1\) and final state \(P_2T_2\) as follows: \[P_1V = nR T_1\] \[P_2V = nR T_2\] Since the volume of the tire remains constant, we can now write: \[\frac{P_1}{T_1} = \frac{P_2}{T_2}\] We can rearrange this equation to find \(P_2\) as follows: \[P_2 = P_1\frac{T_2}{T_1}\]

Calculate the final gauge pressure

Now, we can plug the known values of initial pressure, initial temperature, and final temperature into the equation to find the final gauge pressure: \[P_2 = 300\cdot\frac{318.15}{288.15} = 330.9 kPa\]

Finalize the answer

The tire's gauge pressure at a temperature of \(45.0 ^\circ C\) is approximately \(330.9 kPa\).