For diatomic carbon dioxide gas (CO2, molar mass 44.0 g/mol) at T = 300 K, calculate (a) the most probable speed v_mp;

Verified step by step guidance

1. First, understand that the most probable speed \( v_{mp} \) for a gas molecule is given by the formula \( v_{mp} = \sqrt{\frac{2kT}{m}} \), where \( k \) is the Boltzmann constant \( (1.38 \times 10^{-23} \text{ J/K}) \), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single molecule.
2. Convert the molar mass of CO2 from grams per mole to kilograms per molecule. The molar mass is 44.0 g/mol, which is equivalent to 0.044 kg/mol. To find the mass of a single molecule, divide this by Avogadro's number \( (6.022 \times 10^{23} \text{ molecules/mol}) \).
3. Substitute the values into the formula for \( v_{mp} \). Use \( T = 300 \text{ K} \) and the calculated mass of a single CO2 molecule from the previous step.
4. Calculate the expression \( \frac{2kT}{m} \) using the values for \( k \), \( T \), and \( m \). This will give you the value inside the square root.
5. Finally, take the square root of the result from the previous step to find the most probable speed \( v_{mp} \) for the CO2 molecules at 300 K.

Key Concepts

Kinetic Theory of Gases

The kinetic theory of gases describes a gas as a large number of small particles (atoms or molecules), all of which are in constant, random motion. This theory helps explain macroscopic properties of gases, such as pressure and temperature, by considering their molecular composition and motion. It is essential for understanding how temperature relates to the speed of gas molecules.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds among molecules in a gas. It shows that at a given temperature, there is a range of molecular speeds, with most molecules having a speed around the most probable speed. This concept is crucial for calculating the most probable speed of gas molecules at a specific temperature.

Most Probable Speed

The most probable speed (v_mp) is the speed at which the largest number of molecules in a gas sample are moving. For an ideal gas, it can be calculated using the formula v_mp = sqrt(2kT/m), where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of a single molecule. This speed is a key parameter in understanding the behavior of gas molecules at a given temperature.