Answer :
Sure! Let's use the ideal gas law to find the temperature of nitrogen gas when the pressure changes.
Step-by-Step Solution:
1. Identify what you know:
- Initial temperature ([tex]\( T_1 \)[/tex]): 298 K
- Initial pressure ([tex]\( P_1 \)[/tex]): 75.5 kPa
- Final pressure ([tex]\( P_2 \)[/tex]): 175 kPa
- We need to find the final temperature ([tex]\( T_2 \)[/tex]).
2. Understand the relationship between pressure and temperature:
- According to the ideal gas law, for a given amount of gas at constant volume, the pressure of a gas is directly proportional to its temperature. This can be expressed as:
[tex]\[
\frac{P_1}{T_1} = \frac{P_2}{T_2}
\][/tex]
- This formula indicates a direct relationship, meaning if pressure increases, temperature also increases, provided the volume remains constant.
3. Rearrange the formula to solve for the final temperature ([tex]\( T_2 \)[/tex]):
- Rearrange the formula to make [tex]\( T_2 \)[/tex] the subject:
[tex]\[
T_2 = \frac{P_2 \cdot T_1}{P_1}
\][/tex]
4. Substitute the known values into the rearranged formula:
[tex]\[
T_2 = \frac{175 \, \text{kPa} \times 298 \, \text{K}}{75.5 \, \text{kPa}}
\][/tex]
5. Calculate [tex]\( T_2 \)[/tex]:
- By performing the calculation, you will find:
[tex]\[
T_2 \approx 690.73 \, \text{K}
\][/tex]
Thus, the temperature of the nitrogen gas at 175 kPa is approximately 690.73 K.
Step-by-Step Solution:
1. Identify what you know:
- Initial temperature ([tex]\( T_1 \)[/tex]): 298 K
- Initial pressure ([tex]\( P_1 \)[/tex]): 75.5 kPa
- Final pressure ([tex]\( P_2 \)[/tex]): 175 kPa
- We need to find the final temperature ([tex]\( T_2 \)[/tex]).
2. Understand the relationship between pressure and temperature:
- According to the ideal gas law, for a given amount of gas at constant volume, the pressure of a gas is directly proportional to its temperature. This can be expressed as:
[tex]\[
\frac{P_1}{T_1} = \frac{P_2}{T_2}
\][/tex]
- This formula indicates a direct relationship, meaning if pressure increases, temperature also increases, provided the volume remains constant.
3. Rearrange the formula to solve for the final temperature ([tex]\( T_2 \)[/tex]):
- Rearrange the formula to make [tex]\( T_2 \)[/tex] the subject:
[tex]\[
T_2 = \frac{P_2 \cdot T_1}{P_1}
\][/tex]
4. Substitute the known values into the rearranged formula:
[tex]\[
T_2 = \frac{175 \, \text{kPa} \times 298 \, \text{K}}{75.5 \, \text{kPa}}
\][/tex]
5. Calculate [tex]\( T_2 \)[/tex]:
- By performing the calculation, you will find:
[tex]\[
T_2 \approx 690.73 \, \text{K}
\][/tex]
Thus, the temperature of the nitrogen gas at 175 kPa is approximately 690.73 K.
