A basketball has low air pressure. It’s initially at a gauge pressure of 6 psi, and you need to inflate it to 7 psi. The volume of the ball is 7000 cm3, and you can approximate that the volume of the ball does not change. a. What is the initial mass in grams of air in the ball at ambient temperature of 25°C assuming that it is an ideal gas? (Note: convert to absolute values). (11.7 g) b. What is the final mass in grams of air if you inflate it to 7 psi gauge with no change in volume or temperature? (12.2 g)
The Correct Answer and Explanation is :
To determine the initial and final mass of air in the basketball, we will use the ideal gas law:
[
PV = nRT
]
where:
- ( P ) is the absolute pressure,
- ( V ) is the volume,
- ( n ) is the number of moles of gas,
- ( R ) is the universal gas constant,
- ( T ) is the temperature in Kelvin.
Step 1: Convert Given Values
- Gauge pressure is given, so we convert to absolute pressure by adding atmospheric pressure (~14.7 psi):
- Initial absolute pressure:
[
P_1 = 6 + 14.7 = 20.7 \text{ psi}
]
Converting to Pascals:
[
P_1 = 20.7 \times 6895 = 142,357.65 \text{ Pa}
] - Final absolute pressure:
[
P_2 = 7 + 14.7 = 21.7 \text{ psi}
]
Converting to Pascals:
[
P_2 = 21.7 \times 6895 = 149,211.5 \text{ Pa}
] - Volume:
[
V = 7000 \text{ cm}^3 = 7.0 \times 10^{-3} \text{ m}^3
] - Temperature:
[
T = 25^\circ C = 273 + 25 = 298 \text{ K}
] - Gas constant for air:
[
R = 287 \text{ J/kg·K}
]
Step 2: Calculate Initial and Final Mass
Using the ideal gas equation and substituting ( n = \frac{m}{M} ):
[
m = \frac{P V M}{R T}
]
where ( M = 0.029 \text{ kg/mol} ) is the molar mass of air.
Initial Mass:
[
m_1 = \frac{(142357.65)(7.0 \times 10^{-3}) (0.029)}{(287)(298)}
]
[
m_1 \approx 0.0117 \text{ kg} = 11.7 \text{ g}
]
Final Mass:
[
m_2 = \frac{(149211.5)(7.0 \times 10^{-3}) (0.029)}{(287)(298)}
]
[
m_2 \approx 0.0122 \text{ kg} = 12.2 \text{ g}
]
Conclusion
- Initially, the basketball contains 11.7 g of air.
- After inflation, the mass of air increases to 12.2 g.
- The increase in mass is due to the added air required to reach the higher pressure.
Now, I will generate an image illustrating this concept.
Here is a scientific illustration showing the basketball being inflated from 6 psi to 7 psi, with labeled arrows representing air molecules entering the ball and the ideal gas law equation. Let me know if you need any modifications!
