Answer :
To solve this problem, we'll use the concept from gas laws. When the number of moles and volume of a gas are constant, the relationship between pressure and temperature can be expressed using the formula:
[tex]\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \][/tex]
Where:
- [tex]\( P_1 \)[/tex] and [tex]\( T_1 \)[/tex] are the initial pressure and temperature.
- [tex]\( P_2 \)[/tex] and [tex]\( T_2 \)[/tex] are the final pressure and temperature.
We'll use this formula to find the new temperature after the pressure change.
### Step-by-step Solution:
1. Convert Initial Pressure:
- The initial pressure is given in kilopascals (kPa). We need to convert it to millimeters of mercury (mmHg) because the final pressure is in mmHg.
- Conversion factor: [tex]\( 1 \, \text{kPa} = 7.50062 \, \text{mmHg} \)[/tex].
- Initial pressure: [tex]\( 97.5 \, \text{kPa} \times 7.50062 = 731.31 \, \text{mmHg} \)[/tex].
2. Initial and Final Pressures:
- [tex]\( P_1 = 731.31 \, \text{mmHg} \)[/tex] (Converted from kPa)
- [tex]\( P_2 = 540 \, \text{mmHg} \)[/tex] (Given)
3. Initial Temperature:
- The initial temperature is given as [tex]\( 345 \, \text{K} \)[/tex].
4. Calculate New Temperature in Kelvin:
- Rearrange the formula to solve for [tex]\( T_2 \)[/tex]:
[tex]\[ T_2 = \frac{P_2 \times T_1}{P_1} \][/tex]
- Substitute the values:
[tex]\[ T_2 = \frac{540 \, \text{mmHg} \times 345 \, \text{K}}{731.31 \, \text{mmHg}} \][/tex]
- Calculate the result: [tex]\( T_2 \approx 254.75 \, \text{K} \)[/tex].
5. Convert New Temperature to Celsius:
- Kelvin to Celsius conversion: [tex]\( \text{Celsius} = \text{Kelvin} - 273.15 \)[/tex].
- [tex]\( T_2 \approx 254.75 \, \text{K} - 273.15 = -18.40^\circ \text{C} \)[/tex].
Therefore, the new temperature after the pressure decreases to 540 mmHg is approximately [tex]\(-18.40^\circ \text{C}\)[/tex].
[tex]\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \][/tex]
Where:
- [tex]\( P_1 \)[/tex] and [tex]\( T_1 \)[/tex] are the initial pressure and temperature.
- [tex]\( P_2 \)[/tex] and [tex]\( T_2 \)[/tex] are the final pressure and temperature.
We'll use this formula to find the new temperature after the pressure change.
### Step-by-step Solution:
1. Convert Initial Pressure:
- The initial pressure is given in kilopascals (kPa). We need to convert it to millimeters of mercury (mmHg) because the final pressure is in mmHg.
- Conversion factor: [tex]\( 1 \, \text{kPa} = 7.50062 \, \text{mmHg} \)[/tex].
- Initial pressure: [tex]\( 97.5 \, \text{kPa} \times 7.50062 = 731.31 \, \text{mmHg} \)[/tex].
2. Initial and Final Pressures:
- [tex]\( P_1 = 731.31 \, \text{mmHg} \)[/tex] (Converted from kPa)
- [tex]\( P_2 = 540 \, \text{mmHg} \)[/tex] (Given)
3. Initial Temperature:
- The initial temperature is given as [tex]\( 345 \, \text{K} \)[/tex].
4. Calculate New Temperature in Kelvin:
- Rearrange the formula to solve for [tex]\( T_2 \)[/tex]:
[tex]\[ T_2 = \frac{P_2 \times T_1}{P_1} \][/tex]
- Substitute the values:
[tex]\[ T_2 = \frac{540 \, \text{mmHg} \times 345 \, \text{K}}{731.31 \, \text{mmHg}} \][/tex]
- Calculate the result: [tex]\( T_2 \approx 254.75 \, \text{K} \)[/tex].
5. Convert New Temperature to Celsius:
- Kelvin to Celsius conversion: [tex]\( \text{Celsius} = \text{Kelvin} - 273.15 \)[/tex].
- [tex]\( T_2 \approx 254.75 \, \text{K} - 273.15 = -18.40^\circ \text{C} \)[/tex].
Therefore, the new temperature after the pressure decreases to 540 mmHg is approximately [tex]\(-18.40^\circ \text{C}\)[/tex].
