Calculate the root mean square speed and most probable speed for co2 at25°c

To calculate the root mean square (rms) speed and most probable speed for CO2 at 25°C, we will use the following formulas:

1. Root mean square speed (vrms):
vrms = √(3RT / M), where R is the ideal gas constant, T is the temperature in Kelvin, and M is the molar mass of CO2.

2. Most probable speed (vmp):
vmp = √(2RT / M)

Now, let's calculate it step by step.

Step 1: Convert the temperature to Kelvin
Given temperature in °C = 25°C
To convert it to Kelvin, we use the formula:
Temperature in Kelvin = (25°C + 273.15)
Temperature in Kelvin = 298.15 K

Step 2: Calculate the molar mass of CO2.
CO2 consists of one carbon (C) atom and two oxygen (O) atoms.
The atomic mass of carbon (C) is approximately 12.01 g/mol, and the atomic mass of oxygen (O) is approximately 16.00 g/mol.
Therefore, the molar mass of CO2 = (12.01 g/mol) + 2 * (16.00 g/mol)
Molar mass of CO2 = 44.01 g/mol

Step 3: Calculate the root mean square (rms) speed (vrms).
Using the formula:
vrms = √(3RT / M)
where R is the ideal gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin (298.15 K), and M is the molar mass of CO2 (44.01 g/mol).
Now substitute the values:
vrms = √(3 * (8.314 J/(mol·K)) * (298.15 K) / 44.01 g/mol)

Step 4: Convert the units to the required units.
The given units for speed are not specified, so we can leave it in m/s.

Step 5: Calculate the most probable speed (vmp).
Using the formula:
vmp = √(2RT / M)
where R is the ideal gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin (298.15 K), and M is the molar mass of CO2 (44.01 g/mol).
Now substitute the values:
vmp = √(2 * (8.314 J/(mol·K)) * (298.15 K) / 44.01 g/mol)

Step 6: Convert the units to the required units.
The given units for speed are not specified, so we can leave it in m/s.

Finally, calculate the values using a calculator or software:

vrms ≈ 455 m/s
vmp ≈ 398 m/s

Therefore, the root mean square (rms) speed of CO2 at 25°C is approximately 455 m/s, and the most probable speed is approximately 398 m/s.