Certainly! Let's solve this problem step-by-step.
### Step 1: Understand the Given Values
- Initial volume ([tex]\( V_1 \)[/tex]) = 7474 mL
- Initial pressure ([tex]\( P_1 \)[/tex]) = 110 kPa
- Final pressure ([tex]\( P_2 \)[/tex]) = 0.403 atm
### Step 2: Convert Initial Pressure from kPa to atm
We need to use consistent units, so let's convert the initial pressure from kilopascals (kPa) to atmospheres (atm). The conversion factor between kPa and atm is:
[tex]\[ 1 \text{kPa} = 0.00986923 \text{atm} \][/tex]
So, the initial pressure in atmospheres ([tex]\( P_1 \)[/tex]) is:
[tex]\[ P_1 = 110 \text{kPa} \times 0.00986923 \text{atm/kPa} \][/tex]
[tex]\[ P_1 \approx 1.0856153 \text{atm} \][/tex]
### Step 3: Apply Boyle's Law
Boyle's Law states that for a given mass of an ideal gas kept at a constant temperature, the product of pressure and volume is constant. Mathematically, it’s written as:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
We need to find the final volume ([tex]\( V_2 \)[/tex]) in mL:
[tex]\[ V_2 = \frac{P_1 \times V_1}{P_2} \][/tex]
Substitute the known values into the equation:
[tex]\[ V_2 = \frac{1.0856153 \text{atm} \times 7474 \text{mL}}{0.403 \text{atm}} \][/tex]
[tex]\[ V_2 \approx 2040049.627791563 \text{mL} \][/tex]
### Step 4: Convert the Final Volume from mL to Liters
We need the volume in liters. The conversion factor is:
[tex]\[ 1 \text{L} = 1000 \text{mL} \][/tex]
Thus, the final volume in liters ([tex]\( V_2 \)[/tex]) is:
[tex]\[ V_2 = \frac{2040049.627791563 \text{mL}}{1000} \][/tex]
[tex]\[ V_2 \approx 2040.0496277915631 \text{L} \][/tex]
### Conclusion
When the inflatable toy is taken up a mountain where the pressure is 0.403 atm, its volume will be approximately 2040.05 liters.
