Sure, let's walk through the problem step-by-step.
1. Understanding the given information:
- Standard Temperature and Pressure (STP) conditions are given:
- Initial pressure, [tex]\( P_1 = 14.7 \)[/tex] psi (this is the atmospheric pressure at sea level).
- Initial temperature, [tex]\( T_1 = 273.15 \)[/tex] K (this is [tex]\(0^\circ C\)[/tex] in Kelvin).
- The gas is pressurized to a new pressure:
- Target pressure, [tex]\( P_2 = 39.8 \)[/tex] psi.
2. Using the ideal gas law relationship:
The ideal gas law states that for a given amount of gas, the ratio of pressure to temperature remains constant if the volume remains the same. Mathematically, it can be written as:
[tex]\[
\frac{P_1}{T_1} = \frac{P_2}{T_2}
\][/tex]
Here:
- [tex]\( P_1 \)[/tex] is the initial pressure.
- [tex]\( T_1 \)[/tex] is the initial temperature.
- [tex]\( P_2 \)[/tex] is the final pressure.
- [tex]\( T_2 \)[/tex] is the final temperature (which we need to find).
3. Rearranging the formula to solve for the final temperature:
We can rearrange the formula to solve for [tex]\( T_2 \)[/tex]:
[tex]\[
T_2 = \frac{P_2 \cdot T_1}{P_1}
\][/tex]
4. Plugging in the given values:
[tex]\[
T_2 = \frac{39.8 \text{ psi} \cdot 273.15 \text{ K}}{14.7 \text{ psi}}
\][/tex]
5. Calculating [tex]\( T_2 \)[/tex]:
[tex]\[
T_2 = \frac{39.8 \times 273.15}{14.7}
\][/tex]
6. Using the calculated value:
Upon performing the division, you get a result close to:
[tex]\[
T_2 \approx 739.999 \text{ K}
\][/tex]
7. Rounding to the nearest kelvin:
When rounding 739.999 to the nearest kelvin, we get:
[tex]\[
T_2 \approx 740 \text{ K}
\][/tex]
So, the temperature of the ideal gas when pressurized to 39.8 psi will be approximately 740 K.
