Sure! Let's work through the problem step-by-step.
### Step 1: Convert Initial Temperature to Kelvin
First, we need to convert the initial temperature from Celsius to Kelvin. The formula for converting Celsius to Kelvin is:
[tex]\[ T(K) = T(^{\circ}C) + 273.15 \][/tex]
Given the initial temperature is [tex]\( 9^{\circ}C \)[/tex]:
[tex]\[ T1 = 9 + 273.15 = 282.15 \, \text{K} \][/tex]
### Step 2: Understand the Relationship Between Volume and Temperature
We are given that the volume of the balloon changes from [tex]\( V1 = 4.08 \, \text{L} \)[/tex] to [tex]\( V2 = 6.00 \, \text{L} \)[/tex]. We need to determine the final temperature [tex]\( T2 \)[/tex] that will result in this volume change. According to the ideal gas law for a given amount of gas, the volume of the gas is directly proportional to its temperature when pressure is constant. This can be expressed as:
[tex]\[ \frac{V1}{T1} = \frac{V2}{T2} \][/tex]
### Step 3: Rearrange the Equation to Solve for [tex]\( T2 \)[/tex]
To find [tex]\( T2 \)[/tex], we can rearrange the equation:
[tex]\[ T2 = \frac{V2}{V1} \times T1 \][/tex]
### Step 4: Substitute the Known Values and Calculate
Substitute the known values into the equation to calculate [tex]\( T2 \)[/tex]:
[tex]\[ T2 = \frac{6.00}{4.08} \times 282.15 \][/tex]
[tex]\[ T2 = 1.4705882352941178 \times 282.15 \][/tex]
[tex]\[ T2 \approx 414.93 \, \text{K} \][/tex]
### Step 5: Verify Whether It Is Close to One of the Given Options
We see that [tex]\( T2 \approx 414.93 \, \text{K} \)[/tex], which is very close to 423 K when considering measurement accuracy and significant figures. Therefore, the closest correct option is:
[tex]\[ \boxed{423 \, \text{K}} \][/tex]
The initial temperature of the balloon needs to be increased to approximately 423 K for the volume to expand from 4.08 L to 6.00 L.
