Question

A car tire is filled with nitrogen gas at 35 psi at 27\[^\circ C\] . It will burst if pressure exceeds 40 psi. The temperature in \[^\circ C\] at which the car tire will burst is?

Answer 1

Hint:Temperature as we know imparts kinetic energy to the molecules of the gas and increases their randomness. This affects the volume of gas and increases the volume of gas in a direct proportion, i.e. with an increase in temperature, the volume of confined gas (isolated in a closed container) increases and vice-versa. With this increase in temperature as the volume of the gas increases, the pressure exerted by the molecules of the gas on the sides of the container also increases. This can cause the tire to burst in the above-given case in the question.

Formula used:
\[\dfrac{{{P_1}}}{{{T_1}}} = \dfrac{{{P_2}}}{{{T_2}}}\]
Where,
\[{T_1}\] and \[{T_2}\] are the initial and final temperatures.
\[{P_1}\] and \[{P_2}\] are the initial and final pressure

Complete step by step solution:
Gay-Lussac’s law is a gas law that states that the pressure exerted by a gas (of a specific mass, kept at a constant volume) varies directly with the absolute temperature of the gas. In other words, while the mass is fixed and the volume is constant, the pressure a gas exerts is proportional to the temperature of the gas.

Given, \[{P_1} = 35{\text{ }}psi{\text{ and}}\,\,{{\text{T}}_1} = 300K\]
We need to find, \[{T_2}\].
Now, substituting these values in the equation, we get,
\[\dfrac{{35}}{{300}} = \dfrac{{40}}{{{T_2}}} \\ \Rightarrow {T_2} = \dfrac{{40 \times 300}}{{35}} \\ \Rightarrow {T_2} = 342.86K \\ \therefore {T_2} \approx {70^0}C \\ \]
Therefore, at a temperature nearly equal to 700C, the pressure will exceed the 40psi mark and the tire will burst.

Note: Law of Gaseous Volumes was proposed by Joseph Louis Gay-Lussac in 1808. The ratio of the reacting gas volumes is a tiny whole number when measured at the same temperature and pressure, according to this law. It's possible to think of this as an alternative application of the law of fixed proportions. While the law of definite proportion governs mass, this law governs volume.