Question

An automobile tire is inflated to a pressure of 3.05 atm on a rather warm day when the temperature is \(40 .{ }^{\circ} \mathrm{C}\). The car is then driven to the mountains and parked overnight. The morning temperature is \(-5.0{ }^{\circ} \mathrm{C}\). Calculate the gas pressure in the tire. Assume the volume of the tire doesn't change.

Short answer

The gas pressure in the tire in the morning is 2.61 atm.

Step-by-step solution

Convert Temperatures to Kelvin

First, we need to convert the given temperatures from Celsius to Kelvin since gas laws require temperatures in Kelvin. The conversion formula is: \[ T(K) = T(°C) + 273.15 \]So, for the initial temperature (\(40°\ C\)):\[ T_1 = 40 + 273.15 = 313.15\ K \]And for the final temperature (\(-5°\ C\)):\[ T_2 = -5 + 273.15 = 268.15\ K \]

Use the Ideal Gas Law (Gay-Lussac's Law)

Since the volume of the tire is constant, we can apply Gay-Lussac's Law, which relates pressure and temperature: \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]We know the initial pressure \(P_1 = 3.05\ atm\) and the temperatures \(T_1 = 313.15\ K\) and \(T_2 = 268.15\ K\). We need to find \(P_2\), the final pressure.

Solve for Final Pressure

Rearrange the formula to solve for \(P_2\):\[ P_2 = P_1 \times \frac{T_2}{T_1} \]Substitute in the known values:\[ P_2 = 3.05\ \text{atm} \times \frac{268.15\ \text{K}}{313.15\ \text{K}} \]Calculate \(P_2\):\[ P_2 \approx 3.05 \times 0.856 \approx 2.61208\ \text{atm} \]Therefore, the pressure in the tire in the morning is approximately \(2.61\ atm\).

Key concepts

Pressure and Temperature Relationship

Did you ever notice how a balloon shrinks when you put it in the freezer? This is because of the relationship between pressure and temperature in gases. Gases behave in specific ways, and one fundamental aspect of this behavior is how their pressure and temperature are related.

When you have a gas in a closed container, if you change the temperature, the pressure will change too, as long as the volume remains constant. This is because gas particles move faster at higher temperatures and slower at lower ones. Faster moving particles collide more frequently with the walls of their container, causing an increase in pressure. Conversely, slower-moving particles reduce the pressure.

Understanding this relationship is crucial in many real-world contexts, like in our automobile tire example. Here, the tire is a closed system where the gas pressure depends directly on the temperature of the air within the tire.

Temperature Conversion

One common mistake people make is forgetting to convert temperatures to the correct unit when working with gases. Most gas laws, like Gay-Lussac's Law, require temperatures to be in Kelvin rather than Celsius or Fahrenheit. The Kelvin scale is an absolute temperature scale, which starts at absolute zero. This is very important for calculations, because it avoids negative temperatures which can lead to incorrect results in gas calculations.

To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example:

• For a temperature of 40°C, the conversion is:
  (40 + 273.15) = 313.15 K
• For a temperature of -5°C, convert by adding: (-5 + 273.15) = 268.15 K

Always remember, this conversion step is essential whenever you are applying gas laws, to ensure that your calculations are accurate.

Gay-Lussac's Law

Gay-Lussac's Law is a handy tool when dealing with gases in closed containers, especially when the volume doesn't change. This law shows the direct proportional relationship between the pressure of a gas and its absolute temperature, provided the volume remains constant.

The formula used for this law is \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). Here, \(P_1\) and \(P_2\) represent the initial and final pressures, while \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.

In our exercise, Gay-Lussac's Law allowed us to calculate the change in tire pressure when the temperature dropped from 40°C to -5°C. We maintained the volume of the tire constant and used the converted Kelvin temperatures to find the final pressure. This law makes predictions about what happens to the pressure as temperature changes, which can help in understanding and predicting real-world scenarios, like the behavior of tire pressure during temperature shifts.

Remember, keeping the conditions clear - constant volume, using Kelvin - is key to applying this formula correctly.