Based on the Ideal Gas Law, we know that for a fixed amount of gas in a closed container, the pressure and temperature of the gas are directly proportional as long as the volume doesn't change. Mathematically, this relationship can be represented by the combined gas law, which for a constant volume and amount of gas, simplifies to:
[tex]\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \][/tex]
Where:
- [tex]\( P_1 \)[/tex] is the initial pressure of the gas.
- [tex]\( T_1 \)[/tex] is the initial temperature of the gas (in Kelvin).
- [tex]\( P_2 \)[/tex] is the final pressure of the gas.
- [tex]\( T_2 \)[/tex] is the final temperature of the gas (in Kelvin).
In this scenario, we're given the following information:
- Initial pressure [tex]\( P_1 \)[/tex] = 20.0 atm.
- Maximum pressure [tex]\( P_2 \)[/tex] = 35.0 atm.
- Initial temperature [tex]\( T_1 \)[/tex] = 293 K.
We need to solve for [tex]\( T_2 \)[/tex], which is the maximum temperature the tank can reach before exploding.
Now, let's rearrange the formula to solve for [tex]\( T_2 \)[/tex]:
[tex]\[ T_2 = \frac{P_2 \times T_1}{P_1} \][/tex]
We can plug the values into this equation to find [tex]\( T_2 \)[/tex]:
[tex]\[ T_2 = \frac{35.0 \text{ atm} \times 293 \text{ K}}{20.0 \text{ atm}} \][/tex]
[tex]\[ T_2 = \frac{10255 \text{ atm}\cdot\text{K}}{20.0 \text{ atm}} \][/tex]
[tex]\[ T_2 = 512.75 \text{ K} \][/tex]
So, the maximum temperature the tank can reach before exploding is approximately 513 K.
