Sure, let's go step-by-step to solve this problem.
### Step 1: Understanding the Problem
We have:
- Initial volume, [tex]\( V_1 = 2.30 \)[/tex] liters
- Final volume, [tex]\( V_2 = 6.50 \)[/tex] liters
- Initial temperature, [tex]\( T_1 = 40^\circ \text{C} \)[/tex]
We need to find the final temperature, [tex]\( T_2 \)[/tex].
### Step 2: Convert Initial Temperature to Kelvin
First, we convert the initial temperature from Celsius to Kelvin because gas law calculations typically use Kelvin.
[tex]\[ T_1(K) = T_1(°C) + 273.15 \][/tex]
[tex]\[ T_1(K) = 40 + 273.15 \][/tex]
[tex]\[ T_1(K) = 313.15 \, \text{K} \][/tex]
### Step 3: Apply Charles's Law
Charles's Law states that for a given mass of gas at constant pressure, the volume is directly proportional to its temperature (in Kelvin).
Mathematically, it can be written as:
[tex]\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \][/tex]
We need to find [tex]\( T_2 \)[/tex]. Rearranging the formula to solve for [tex]\( T_2 \)[/tex]:
[tex]\[ T_2 = \frac{V_2 \times T_1}{V_1} \][/tex]
### Step 4: Calculate the Final Temperature in Kelvin
Substitute the known values into the equation:
[tex]\[ T_2 = \frac{6.50 \, \text{L} \times 313.15 \, \text{K}}{2.30 \, \text{L}} \][/tex]
[tex]\[ T_2 = \frac{2035.475}{2.30} \][/tex]
[tex]\[ T_2 = 884.99 \, \text{K} \][/tex]
### Step 5: Convert the Final Temperature to Celsius
To convert the final temperature back to Celsius:
[tex]\[ T_2(°C) = T_2(K) - 273.15 \][/tex]
[tex]\[ T_2(°C) = 884.99 - 273.15 \][/tex]
[tex]\[ T_2(°C) = 611.84 \][/tex]
### Final Answer
The temperature should be raised to [tex]\( 611.84^\circ \text{C} \)[/tex] to occupy a volume of [tex]\( 6.50 \)[/tex] liters.
