A meteorological balloon filled with 0.50\text{ kg} of helium gas is launched at sea level… — JAMB UTME

A meteorological balloon filled with $0.50\text{ kg}$ of helium gas is launched at sea level, where the atmospheric pressure is $1.01 \times 10^5\text{ Pa}$ and the ambient temperature is $27^\circ\text{C}$ . The balloon ascends to a high altitude where the external ambient pressure decreases to $4.00 \times 10^4\text{ Pa}$ and the ambient temperature drops to $-23^\circ\text{C}$ . As the balloon expands, its elastic membrane exerts an additional pressure, causing the internal gas pressure to be $20\%$ higher than the surrounding ambient pressure. Assuming helium behaves as an ideal gas with a molar mass of $4.0\text{ g/mol}$ and the molar gas constant $R = 8.31\text{ J mol}^{-1}\text{K}^{-1}$ , calculate the final volume of helium gas inside the balloon at this altitude in $\text{m}^3$ .

Answer: 5.41 m^3

Answer

The final volume of helium gas inside the balloon at altitude is

5.41\text{ m}^3

The ideal gas equation

PV = nRT

relates state variables. By determining

n = 125\text{ moles}

from mass and molar mass, absolute temperature

T_2 = 250\text{ K}

, and total internal pressure

P_2 = 4.80 \times 10^4\text{ Pa}

, the final volume

V_2

evaluates to

5.41\text{ m}^3

Step-by-Step Solution

Convert the mass of helium into moles

n = 125\text{ moles}

Mass

m = 0.50\text{ kg} = 500\text{ g}

divided by molar mass

M = 4.0\text{ g/mol}

gives

n = \frac{500}{4.0} = 125\text{ moles}

Convert final temperature to Kelvin

T_2 = 250\text{ K}

Gas laws strictly require thermodynamic temperature:

T_2 = -23 + 273 = 250\text{ K}

Calculate final internal pressure of the gas

P_2 = 4.80 \times 10^4\text{ Pa}

The gas pressure inside the balloon is

20\%

higher than external ambient pressure:

P_2 = 1.20 \times (4.00 \times 10^4) = 4.80 \times 10^4\text{ Pa}

Solve for the final volume using the ideal gas equation

V_2 = 5.41\text{ m}^3

Rearranging

P_2 V_2 = n R T_2

yields

V_2 = \frac{n R T_2}{P_2} = \frac{125 \times 8.31 \times 250}{4.80 \times 10^4} = 5.41015...\text{ m}^3 \approx 5.41\text{ m}^3

Key Concept

Ideal Gas Equation (

PV = nRT

)

Estimated Time:3m 0s

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