Question

A sealed tube that can withstand a pressure of 3 atmospheres is filled with air at $27{}^\circ C$ and 760 mm pressure. The temperature (in C) above which the tube will burst will be:

Answer 1

Gay Lussac’s law states that the pressure of a given amount of gas held at constant volume is directly proportional to the Kelvin temperature.
$P\propto T$
Replacing the proportionality by inserting a constant,
$\dfrac{P}{T}=C$ where P = pressure of gas ; T = temperature of gas (in K) ; k = constant
This clearly says that with an increase in temperature, the pressure will also go up.

Complete step by step answer:
-Similarly, initial and final pressures and temperatures under constant volume can be calculated as-
$\dfrac{{{P}_{1}}}{{{T}_{1}}}=\dfrac{{{P}_{2}}}{{{T}_{2}}}=\dfrac{{{P}_{3}}}{{{T}_{3}}}...$

-According to the question,
$\begin{aligned} & {{P}_{1}}=760\text{ mm Hg ; }{{\text{T}}_{1}}=273+37=310K \\\ & {{P}_{2}}=3\text{ atm = 2280 mm Hg ; }{{\text{T}}_{2}}=? \\\ \end{aligned}$

-Applying the above pressure-temperature relationship,
$\dfrac{{{P}_{1}}}{{{T}_{1}}}=\dfrac{{{P}_{2}}}{{{T}_{2}}}$
$\Rightarrow \dfrac{760}{310}=\dfrac{2280}{{{T}_{2}}}$
${{T}_{2}}=\dfrac{2280\times 310}{760}=930K$

Note: Gay-Lussac’s discovered the Pressure-Temperature law while building an ‘air thermometer’. This law holds true for gases because as the temperature increases, the kinetic energy of the gaseous molecules also increases due to which particles collide with walls more rapidly, thereby exerting increased pressure.

Let us consider some examples to understand this concept.
When a pressurized aerosol (such as a deodorant can or a sanitizer can) is heated, the resulting increase in the pressure exerted by the gaseous molecules on the container which if increased beyond the limit can cause an explosion. This is the reason why many pressurized containers have warning indications stating that the container must be kept away from the wire and stored in a cool environment.
Another example of Gay-Lussac’s law can be observed in pressure cooker cooking. When the cooker is heated, the pressure exerted by the steam from the food inside the container increases. The high temperature and pressure inside the container thus cause the food to cook faster.