Find the ratio of average kinetic energy of 4g of CH4 to 32g of SO2 at 227°C.

Steps to Solve

1. Determine Molar Masses
   Calculate the molar mass of each gas.
   For Methane ($CH_4$):

• Carbon (C) = 12 g/mol
• Hydrogen (H) = 1 g/mol
• Total = $12 + 4 \times 1 = 16 \text{ g/mol}$

For Sulfur Dioxide ($SO_2$):

• Sulfur (S) = 32 g/mol
• Oxygen (O) = 16 g/mol
• Total = $32 + 2 \times 16 = 64 \text{ g/mol}$

2. Calculate the Number of Moles
   Next, find the number of moles for each gas using the formula:
   $$ \text{Number of moles} = \frac{\text{Mass}}{\text{Molar mass}} $$

For Methane:
$$ n_{CH_4} = \frac{4 \text{ g}}{16 \text{ g/mol}} = 0.25 \text{ moles} $$

For Sulfur Dioxide:
$$ n_{SO_2} = \frac{32 \text{ g}}{64 \text{ g/mol}} = 0.5 \text{ moles} $$

3. Use Kinetic Energy Formula
   The average kinetic energy ($KE$) of a gas can be calculated using:
   $$ KE = \frac{3}{2} nRT $$
   Where:

• $n$ = number of moles
• $R$ = ideal gas constant (8.31 J/(mol·K))
• $T$ = temperature in Kelvin (for $227^\circ C$, $T = 227 + 273 = 500K$)

4. Calculate Kinetic Energy for Each Gas
   For Methane:
   $$ KE_{CH_4} = \frac{3}{2} \times 0.25 \times 8.31 \times 500 $$

For Sulfur Dioxide:
$$ KE_{SO_2} = \frac{3}{2} \times 0.5 \times 8.31 \times 500 $$

5. Find the Ratio of Kinetic Energies
   Now, find the ratio:
   $$ \text{Ratio} = \frac{KE_{CH_4}}{KE_{SO_2}} $$

6. Simplifying the Ratio
   By substituting the $KE$ formulas into the ratio:
   $$ \text{Ratio} = \frac{0.25}{0.5} = \frac{1}{2} $$