To solve the problem of how temperature affects the volume of a gas, we can utilize the ideal gas law, which is expressed as PV = nRT. In this scenario, we are given two volumes and one temperature, indicating that we need to derive a new formula based on the ideal gas law.
Since the pressure remains constant, we can focus on the relationship between volume and temperature. The relevant variables are the initial volume V_1 and temperature T_1, as well as the final volume V_2 and the unknown final temperature T_2. The derived formula can be expressed as:
\(\frac{V_1}{T_1}\) = \(\frac{V_2}{T_2}\)
In this case, we have:
- V_1 = 8.30 \, \(\text{liters}\)
- V_2 = 5.25 \, \(\text{liters}\)
- T_1 = 202 \, \(\text{°C}\) = 202 + 273.15 = 475.15 \, \(\text{K}\)
- T_2 = ?
Next, we substitute the known values into the derived formula:
\(\frac{8.30}{475.15}\) = \(\frac{5.25}{T_2}\)
Cross-multiplying gives us:
8.30 \(\cdot\) T_2 = 475.15 \(\cdot\) 5.25
Solving for T_2 involves dividing both sides by 8.30:
T_2 = \(\frac{475.15 \cdot 5.25}{8.30}\) \(\approx\) 300.55 \, \(\text{K}\)
Finally, to convert T_2 back to degrees Celsius, we subtract 273.15:
T_2 \(\approx\) 300.55 - 273.15 \(\approx\) 27.40 \, \(\text{°C}\)
Thus, the temperature needed to decrease the volume of sulfur hexachloride gas to 5.25 liters is approximately 27.40 degrees Celsius.
