To find the missing temperature in degrees Celsius for the second state of the gas, we can use the Ideal Gas Law, specifically the combined gas law, which relates the initial and final states of pressure (P), volume (V), and temperature (T):
[tex]\[
\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
\][/tex]
where:
- [tex]\(P_1\)[/tex] is the initial pressure
- [tex]\(V_1\)[/tex] is the initial volume
- [tex]\(T_1\)[/tex] is the initial temperature in Kelvin (since the Ideal Gas Law requires temperature in Kelvin)
- [tex]\(P_2\)[/tex] is the final pressure
- [tex]\(V_2\)[/tex] is the final volume
- [tex]\(T_2\)[/tex] is the final temperature in Kelvin
Let's break it down step by step:
1. Convert Initial Temperature to Kelvin:
The initial temperature [tex]\(T_1\)[/tex] is given in degrees Celsius:
[tex]\[
T_1 = 441.5^\circ\text{C}
\][/tex]
To convert this to Kelvin, use the formula:
[tex]\[
T_1 (\text{K}) = T_1 (\text{°C}) + 273.15
\][/tex]
[tex]\[
T_1 (\text{K}) = 441.5 + 273.15 = 714.65 \text{ K}
\][/tex]
2. Write Down the Given Values:
Initial state:
[tex]\[
P_1 = 173.5 \text{ Torr}, \quad V_1 = 25.7 \text{ L}, \quad T_1 = 714.65 \text{ K}
\][/tex]
Final state:
[tex]\[
P_2 = 246.7 \text{ Torr}, \quad V_2 = 46.4 \text{ L}, \quad T_2 = \text{?}
\][/tex]
3. Apply the Combined Gas Law:
[tex]\[
\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
\][/tex]
Rearrange to solve for [tex]\(T_2\)[/tex]:
[tex]\[
T_2 = \frac{P_2 V_2 T_1}{P_1 V_1}
\][/tex]
4. Substitute the Known Values:
[tex]\[
T_2 = \frac{246.7 \text{ Torr} \times 46.4 \text{ L} \times 714.65 \text{ K}}{173.5 \text{ Torr} \times 25.7 \text{ L}}
\][/tex]
Simplifying this expression yields [tex]\(T_2 = 1834.63 \text{ K}\)[/tex].
5. Convert Final Temperature to Degrees Celsius:
To convert from Kelvin back to Celsius:
[tex]\[
T_2 (\text{°C}) = T_2 (\text{K}) - 273.15
\][/tex]
[tex]\[
T_2 (\text{°C}) = 1834.63 - 273.15 = 1561.48^\circ\text{C}
\][/tex]
Therefore, the missing temperature in the final state is:
[tex]\[
1561.48^\circ\text{C}
\][/tex]
