Answer :
To solve this problem, we can use the relationship between pressure and temperature for an ideal gas, given by the formula:
[tex]\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \][/tex]
Where:
- \(P_1\) is the initial pressure,
- \(T_1\) is the initial temperature,
- \(P_2\) is the final pressure,
- \(T_2\) is the final temperature.
Let's denote the given values:
- Initial pressure, \(P_1 = 0.96 \, \text{atm}\)
- Initial temperature, \(T_1 = 25^\circ \text{C} + 273.15 = 298.15 \, \text{K}\) (converted from Celsius to Kelvin)
- Final pressure, \(P_2 = 1.25 \, \text{atm}\)
We need to determine the final temperature \(T_2\) in Kelvin and then convert it back to Celsius.
Firstly, rearrange the equation to solve for \(T_2\):
[tex]\[ T_2 = \frac{P_2 \cdot T_1}{P_1} \][/tex]
Substitute the known values into the equation:
[tex]\[ T_2 = \frac{1.25 \, \text{atm} \cdot 298.15 \, \text{K}}{0.96 \, \text{atm}} \][/tex]
Perform the calculation:
[tex]\[ T_2 \approx 388.216 \, \text{K} \][/tex]
To convert \(T_2\) from Kelvin to Celsius, use the formula:
[tex]\[ T_2^{\circ \text{C}} = T_2 - 273.15 \][/tex]
[tex]\[ T_2^{\circ \text{C}} \approx 388.216 - 273.15 \approx 115.066 \, ^{\circ} \text{C} \][/tex]
Therefore, the new temperature of the gas, after the temperature change, is approximately \(115.066^\circ \text{C}\).
Given the choices we have:
- -44.2°C
- 32.6°C
- 115°C
- 388°C
The closest match to our calculated temperature is:
[tex]\[ \boxed{115^\circ C} \][/tex]
[tex]\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \][/tex]
Where:
- \(P_1\) is the initial pressure,
- \(T_1\) is the initial temperature,
- \(P_2\) is the final pressure,
- \(T_2\) is the final temperature.
Let's denote the given values:
- Initial pressure, \(P_1 = 0.96 \, \text{atm}\)
- Initial temperature, \(T_1 = 25^\circ \text{C} + 273.15 = 298.15 \, \text{K}\) (converted from Celsius to Kelvin)
- Final pressure, \(P_2 = 1.25 \, \text{atm}\)
We need to determine the final temperature \(T_2\) in Kelvin and then convert it back to Celsius.
Firstly, rearrange the equation to solve for \(T_2\):
[tex]\[ T_2 = \frac{P_2 \cdot T_1}{P_1} \][/tex]
Substitute the known values into the equation:
[tex]\[ T_2 = \frac{1.25 \, \text{atm} \cdot 298.15 \, \text{K}}{0.96 \, \text{atm}} \][/tex]
Perform the calculation:
[tex]\[ T_2 \approx 388.216 \, \text{K} \][/tex]
To convert \(T_2\) from Kelvin to Celsius, use the formula:
[tex]\[ T_2^{\circ \text{C}} = T_2 - 273.15 \][/tex]
[tex]\[ T_2^{\circ \text{C}} \approx 388.216 - 273.15 \approx 115.066 \, ^{\circ} \text{C} \][/tex]
Therefore, the new temperature of the gas, after the temperature change, is approximately \(115.066^\circ \text{C}\).
Given the choices we have:
- -44.2°C
- 32.6°C
- 115°C
- 388°C
The closest match to our calculated temperature is:
[tex]\[ \boxed{115^\circ C} \][/tex]
