To solve the problem, we can use the ideal gas law, which is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = ideal gas constant (approximately 0.0821 L·atm/(K·mol))
- \( T \) = temperature in Kelvin
Since the number of moles of gas does not change between the two states, we can write the relationship for the two conditions using \( P_1, V_1, T_1 \) for the initial conditions and \( P_2, V_2, T_2 \) for the final conditions:
\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]
Given:
- \( P_1 = 1 , \text{atm} \)
- \( V_1 = 0.3 , \text{L} \)
- \( T_1 = 300 , \text{K} \)
- \( P_2 = 0.5 , \text{atm} \)
- \( V_2 = 1 , \text{L} \)
Now we can substitute these values into the equation:
\[ \frac{1 \times 0.3}{300} = \frac{0.5 \times 1}{T_2} \]
Simplifying the left side:
\[ \frac{0.3}{300} = \frac{0.5}{T_2} \]
Cross-multiplying to solve for \( T_2 \):
\[ 0.3 T_2 = 150 \]
\[ T_2 = \frac{150}{0.3} = 500 , \text{K} \]
Thus, the final temperature \( T_2 \) is 500 K.