Problem 26 - Chemistry The Molecular Science

Question

A balloon is filled with helium to a volume of \(1.05 \times 10^{3} \mathrm{~L}\) on the ground, where the pressure is \(745 \mathrm{mmHg}\) and the temperature is \(20 .{ }^{\circ} \mathrm{C}\). (a) Calculate the amount (mol) of helium in the balloon. (b) Calculate the volume of helium when the balloon ascends to a height of 2 miles, where the pressure is only \(600, \mathrm{mmHg}\) and the temperature is \(-33{ }^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

(a) 42.35 mol (b) 1117.7 L

Step 1: Convert Temperature Units

Convert the temperatures from Celsius to Kelvin using the formula \( T(K) = T(°C) + 273.15 \). Initially, \( T_1 = 20 + 273.15 = 293.15 \, K \). At height, \( T_2 = -33 + 273.15 = 240.15 \, K \).

Step 2: Use Ideal Gas Law to Find Moles on Ground

The ideal gas law is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant \( (0.0821 \, \text{L atm/mol K}) \), and \( T \) is temperature. Convert pressure from mmHg to atm by dividing by 760: \( P_1 = \frac{745}{760} = 0.9803 \, \text{atm} \). Calculate moles: \( n = \frac{P_1 V_1}{R T_1} = \frac{0.9803 \times 1050}{0.0821 \times 293.15} \approx 42.35 \, \text{mol} \).

Step 3: Use Combined Gas Law to Find Volume at Height

The combined gas law \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \) relates initial and final states. Convert \( P_2 = \frac{600}{760} = 0.7895 \, \text{atm} \). Solve for \( V_2 \): \( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} = \frac{0.9803 \times 1050 \times 240.15}{0.7895 \times 293.15} \approx 1117.7 \, \text{L} \).

Key Concepts

Temperature Conversion

In many scientific problems, especially those involving gases, temperatures must be converted to the Kelvin scale. This scale is directly related to absolute energy since it starts at absolute zero, the point where all kinetic motion ceases. To convert from Celsius to Kelvin, simply use the formula:

• \( T(K) = T(^\circ C) + 273.15 \)

This formula adds 273.15 to the Celsius temperature, converting it to Kelvin, which is necessary for gas law calculations.
For example, in our problem, the temperature on the ground is initially given as 20°C. We convert it to Kelvin like this:

• 20°C + 273.15 = 293.15 K

Similarly, at a height where the temperature is -33°C, it becomes:

• -33°C + 273.15 = 240.15 K

Always remember to convert your temperature to Kelvin when dealing with gas laws for accurate calculations.

Combined Gas Law

The Combined Gas Law is essential for understanding how different state variables relate in a gaseous system. It combines three simple gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law.
The equation is expressed as:

• \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \)

Here:

• \( P_1 \) and \( P_2 \) are the initial and final pressures.
• \( V_1 \) and \( V_2 \) are the initial and final volumes.
• \( T_1 \) and \( T_2 \) are the initial and final temperatures (in Kelvin).

Before using this formula, ensure all pressures are in consistent units (usually atm if using the R constant 0.0821). Also, temperatures must be in Kelvin.
By relating these variables, we can calculate how a change in one state (like pressure) affects another (like volume), keeping temperature effects in mind. For instance, when the helium balloon rises to a higher altitude, both pressure and temperature decrease. Using the known values in the problem, the equation helps to calculate the new volume of the gas at the higher altitude.

Moles of Gas Calculation

Calculating moles of a gas at a given condition requires the Ideal Gas Law, fundamental in understanding relationships between pressure, volume, temperature, and the amount of gas.
The Ideal Gas Law is given by:

• \( PV = nRT \)

In this equation:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) represents the number of moles.
• \( R \) is the ideal gas constant, typically 0.0821 L atm/mol K.
• \( T \) is the temperature in Kelvin.

To solve for the number of moles, rearrange the formula to:

• \( n = \frac{PV}{RT} \)

This rearranged form is useful when you know the other parameters.
In our example, converting pressure from mmHg to atm (by dividing by 760) and using the given volume and temperature, we can easily find the amount of helium gas in moles using the Ideal Gas Law. This method is a cornerstone of chemistry that allows us to quantify gas molecules under various conditions.