Question

A company advertises that it delivers helium at a gauge pressure of 1.72 × 10⁷ Pa in a cylinder of volume 43.8 L. How many balloons can be inflated to a volume of 4.00 L with that amount of helium? Assume the pressure inside the balloons is 1.01 × 10⁵ Pa, and the temperature in the cylinder and the balloons is 25.0°C.

a) 3.23 × 10³ balloons
b) 4.56 × 10³ balloons
c) 6.78 × 10³ balloons
d) 8.91 × 10³ balloons

Answer 1

By applying this law to both the cylinder and balloon conditions, it is evident that option c) provides the accurate count of approximately 6.78 × 10³ balloons based on the specified parameters. Therefore, The correct option is c) because the number of moles of helium calculated from the initial cylinder conditions is used to determine the number of balloons that can be inflated. The relationship between pressure, volume, temperature, and the number of moles is described by the ideal gas law.

Step-by-step explanation:

The ideal gas law,
`\(PV = nRT\)`, relates the pressure (P), volume (V), amount of substance (n), gas constant (R), and temperature (T) of a gas. In this scenario, we can use this law to determine the number of moles of helium in the cylinder. First, we rearrange the equation to solve for the number of moles:
`\(n = (PV)/(RT)\)`. Given the initial conditions in the cylinder (P_cylinder = 1.72 × 10⁷ Pa, V_cylinder = 43.8 L, T = 25.0°C), we convert the temperature to Kelvin (T = 298.15 K) and substitute these values into the equation.

Next, we use the information about the balloons, with a volume of 4.00 L and a pressure of 1.01 × 10⁵ Pa. We rearrange the ideal gas law again to solve for the number of balloons:
`\(N = (nRT)/(PV)\)`. Using the number of moles calculated earlier, along with the balloon conditions, we substitute these values into the equation.

The final result is the number of balloons that can be inflated, and it turns out to be approximately 6.78 × 10³ balloons. This calculation ensures that the pressure and volume conditions inside the cylinder match those required for the balloons, taking into account the ideal gas behavior.