Animated Solution for Chemistry - States of Matter: A car tyre is filled with nitrogen gas at $35\text{ psi}$ at $27^{\circ }\text{C}$. It will burst if pressure exceeds $40\text{ psi}$. The temperature in ${}^{\circ }\text{C}$ at which the car tyre will burst is ......... (Rounded-off to the nearest integer).

Imagine you are cruising down the highway on a scorching summer afternoon. The asphalt is radiating intense heat, and your car tyres are spinning at high speeds.

This rapid rotation generates even more thermal energy through continuous friction with the road. Have you ever wondered what happens to the gas trapped inside those tyres as they heat up?

This problem takes us straight into the heart of thermodynamics and the kinetic theory of gases, exploring a scenario that every driver should understand to prevent a dangerous blowout.

To truly grasp what is happening inside the tyre, we need to zoom in to the microscopic level.

Pressure is not just an abstract number on a gauge; it is the physical result of billions of gas molecules constantly colliding with the inner walls of the tyre.

According to the kinetic theory of gases, the temperature of a gas is a direct measure of the average kinetic energy of its molecules.

When the tyre heats up, the nitrogen molecules absorb that thermal energy and start moving much faster. Because they are moving faster, they strike the walls of the tyre more frequently and with much greater force. This increased bombardment is exactly what we read as an increase in pressure.

Let us break down the specific parameters given in our problem. We have a car tyre filled with nitrogen gas ($N_2$).

Initially, the pressure gauge reads $35\text{ psi}$ at a comfortable ambient temperature of $27^{\circ }\text{C}$.

The tyre has a critical structural limit: if the internal pressure exceeds $40\text{ psi}$, the material will fail, and the tyre will burst. We need to find the exact temperature at which this catastrophic failure occurs.

First, we must establish our physical constraints. A car tyre is essentially a rigid container. While the rubber might expand slightly under extreme pressure, for all practical purposes in ideal gas problems, we assume the volume remains constant.

Furthermore, the amount of gas—the number of moles—inside the tyre does not change because the tyre is completely sealed.

Gay-Lussac's Law

When the volume and the number of moles are held constant, the relationship between the pressure and the temperature of a gas is governed by Gay-Lussac's Law.

This fundamental principle states that the pressure of a fixed mass of gas is directly proportional to its absolute temperature.

Mathematically, this elegant relationship is expressed as:

Which allows us to set up a ratio between the initial and final states:

Before we rush into substituting our numbers, we must address one of the most common and fatal traps in thermodynamics.

Temperature must always be plugged into gas law equations in Kelvin, never in Celsius.

Why? Because the Celsius scale is relative; it is based on the freezing and boiling points of water. It does not start at absolute zero. If you were to use Celsius and the temperature was $0^{\circ }\text{C}$, the equation would mathematically blow up!

The Kelvin scale, however, is an absolute scale where zero represents the complete cessation of molecular motion. Since pressure is proportional to kinetic energy, we must use the absolute scale.

Let us convert our initial temperature to Kelvin:

For simplicity in standard calculations, we often use $273$ as the conversion factor, giving us exactly $300\text{ K}$.

Now that our units are perfectly aligned, let us set up our master equation with the known values.

We have our initial state: - Initial Pressure, $P_1=35\text{ psi}$ - Initial Temperature, $T_1=300\text{ K}$

And our critical final state: - Final Pressure (Bursting Point), $P_2=40\text{ psi}$ - Final Temperature, $T_2=?$

Plugging these into Gay-Lussac's Law, we get our raw setup:

The physics is complete; now it is time for the algebra. Let us isolate our unknown variable, $T_2$.

By cross-multiplying the equation, we get:

Executing this division gives us the bursting temperature in Kelvin:

We have found the critical temperature, but we are not quite done. The question specifically asks for the temperature in degrees Celsius. We must reverse our earlier conversion by subtracting $273.15$:

Finally, the problem instructs us to round off to the nearest integer. Looking at $69.7$, the decimal part is greater than $0.5$, so we round up to the next whole number.

Final Answer: $70^{\circ }\text{C}$

Why Nitrogen?

You might be wondering, why does the problem specifically mention nitrogen gas instead of just regular air?

While the air we breathe is already about 78% nitrogen, the remaining 21% oxygen and 1% other gases—especially water vapor—make a massive difference in high-performance scenarios.

Water vapor is highly sensitive to temperature changes. As a tyre heats up, any liquid water inside vaporizes, causing unpredictable and dangerous pressure spikes.

Pure nitrogen, on the other hand, is completely dry. It maintains a much more stable and predictable pressure profile as the tyre heats up. This is exactly why Formula 1 race cars, commercial aircraft, and heavy-duty vehicles exclusively use nitrogen in their tyres!