Gas Laws: Pressure and Volume Changes in a Rising Weather Balloon

Gases

The Ideal Gas Law and Weather Balloons

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of a gas. It is commonly used to predict how gases behave under different conditions, such as when a weather balloon rises through the atmosphere.

• Ideal Gas Law Equation:

$PV = nRT$

• P: Pressure (in atmospheres, atm)

• V: Volume (in liters, L)

• n: Amount of gas (in moles, mol)

• R: Universal gas constant ($0.0821\ \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}$)

• T: Temperature (in Kelvin, K)

Pressure and Volume Changes as a Balloon Rises

As a weather balloon rises through the atmosphere:

• Atmospheric pressure decreases with altitude.

• If the temperature and amount of gas remain constant, the volume of the balloon increases as pressure decreases (Boyle's Law).

• If temperature also changes, the combined gas law is used to relate the changes in pressure, volume, and temperature.

Combined Gas Law

The Combined Gas Law is derived from the ideal gas law and is useful when the amount of gas is constant:

$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

• P1, V1, T1: Initial pressure, volume, and temperature

• P2, V2, T2: Final pressure, volume, and temperature

Example Calculation

Problem: A weather balloon has a volume of 2.0 L at a pressure of 1.0 atm and a temperature of 300 K at ground level. As it rises, the pressure drops to 0.50 atm and the temperature drops to 250 K. What is the new volume of the balloon?

• Given: $P_1 = 1.0\ \text{atm}$, $V_1 = 2.0\ \text{L}$, $T_1 = 300\ \text{K}$

• Final: $P_2 = 0.50\ \text{atm}$, $T_2 = 250\ \text{K}$, $V_2 = ?$

Using the combined gas law:

$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

Solving for $V_2$:

$V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}$

Plug in the values:

$V_2 = \frac{(1.0\ \text{atm})(2.0\ \text{L})(250\ \text{K})}{(0.50\ \text{atm})(300\ \text{K})}$

$V_2 = \frac{500}{150} = 3.33\ \text{L}$

• Result: The volume of the balloon increases to 3.33 L as it rises and the pressure and temperature decrease.

Summary Table: Effect of Altitude on Weather Balloon

Altitude	Pressure (atm)	Temperature (K)	Volume (L)
Ground Level	1.0	300	2.0
High Altitude	0.50	250	3.33

Additional info: In real atmospheric conditions, temperature and pressure can change non-linearly with altitude, but the combined gas law provides a useful approximation for many practical problems.