To determine the new temperature of the gas after it expands at constant atmospheric pressure, we can use Charles's Law, which states that the volume of a gas is directly proportional to its temperature when pressure is constant. The formula is given by:
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
Where:
- \( V_1 \) is the initial volume,
- \( T_1 \) is the initial temperature,
- \( V_2 \) is the final volume,
- \( T_2 \) is the final temperature.
Given:
- \( V_1 = 5.30 , \text{L} \)
- \( T_1 = 298 , \text{K} \)
- \( V_2 = 7.60 , \text{L} \)
We need to find \( T_2 \).
Rearranging Charles's Law gives us:
\[ T_2 = T_1 \times \frac{V_2}{V_1} \]
Now, substituting in the known values:
\[ T_2 = 298 , \text{K} \times \frac{7.60 , \text{L}}{5.30 , \text{L}} \]
Calculating the fraction:
\[ T_2 = 298 , \text{K} \times \frac{7.60}{5.30} \approx 298 , \text{K} \times 1.43 \approx 426.74 , \text{K} \]
So, the final temperature \( T_2 \) is approximately 426.74 K.