The final pressure inside the hot spray can is [tex]\(\boxed{4.47\, \text{atm}}\)[/tex]. This calculation, based on Gay-Lussac's Law, reflects the increased pressure due to the significant rise in temperature when the can is exposed to fire.
The problem involves determining the final pressure inside a spray can when it is thrown into a fire. The initial conditions are a pressure of 1.77 atm at 23 °C, and the final condition is the can at a temperature of 475 °C. To solve this, we'll apply Gay-Lussac's Law from thermodynamics.
Gay-Lussac's Law states that for a given mass of gas at constant volume, the pressure of the gas is directly proportional to its temperature (in Kelvin). Mathematically, it is represented as:
[tex]$\frac{P_1}{T_1} = \frac{P_2}{T_2}[/tex]
where:
Initial temperature (°C to K):
[tex]$T_1 = 23 + 273.15 = 296.15\, \text{K}[/tex]
Final temperature (°C to K):
[tex]$T_2 = 475 + 273.15 = 748.15\, \text{K}[/tex]
Using the law, we calculate the final pressure [tex]\(P_2\)[/tex]:
[tex]$\begin{align*}P_2 &= P_1 \times \frac{T_2}{T_1}\\[1em]&= 1.77\, \text{atm} \times \frac{748.15\, \text{K}}{296.15\, \text{K}}\\[1em]&= \boxed{4.47\, \text{atm}}\end{align*}[/tex]
