(b) Carbon dioxide makes up approximately 0.04% of Earth's atmosphere. If you collect a 2.0-L sample from the atmosphere at sea level (101.33 kPa) on a warm day 127 °C2, how many CO2 molecules are in your sample?

Verified step by step guidance

1. Convert the temperature from Celsius to Kelvin using the formula: \( T(K) = T(°C) + 273.15 \).
2. Use the ideal gas law \( PV = nRT \) to find the number of moles of gas in the sample. Here, \( P \) is the pressure (101.33 kPa), \( V \) is the volume (2.0 L), \( R \) is the ideal gas constant (8.314 L·kPa/mol·K), and \( T \) is the temperature in Kelvin.
3. Calculate the total moles of gas in the sample using the ideal gas law.
4. Determine the moles of \( CO_2 \) by multiplying the total moles of gas by the percentage of \( CO_2 \) in the atmosphere (0.04%).
5. Convert the moles of \( CO_2 \) to molecules using Avogadro's number \( 6.022 \times 10^{23} \) molecules/mol.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for calculating the number of gas molecules in a given volume under specific conditions. In this case, it will help determine how many moles of CO2 are present in the 2.0-L sample at the given temperature and pressure.

Molar Volume of a Gas

At standard temperature and pressure (STP), one mole of an ideal gas occupies 22.4 liters. However, the conditions in this question are not at STP, so the molar volume must be adjusted based on the temperature and pressure using the Ideal Gas Law. Understanding how to convert between volume, moles, and molecules is crucial for solving the problem.

Avogadro's Number

Avogadro's Number, approximately 6.022 x 10^23, is the number of molecules in one mole of a substance. This concept is vital for converting the number of moles of CO2 calculated from the Ideal Gas Law into the actual number of CO2 molecules in the sample. It provides a bridge between the macroscopic measurements and the microscopic world of molecules.