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

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Understand the Problem

The question is asking to find the ratio of the average kinetic energy of methane (CH4) to sulfur dioxide (SO2) at a given temperature, based on the provided masses of each gas. This involves understanding the relationship between the kinetic energy of gases and their molecular weights, particularly under the same conditions of temperature and pressure.

Answer

The ratio is $1:2$.
Answer for screen readers

The ratio of average kinetic energy of 4 g of $CH_4$ to 32 g of $SO_2$ at 227°C is $1:2$.

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 = $32 + 2 \times 16 = 64 \text{ g/mol}$
  1. 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:
$$ n_{SO_2} = \frac{32 \text{ g}}{64 \text{ g/mol}} = 0.5 \text{ moles} $$

  1. 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 $227^\circ C$, $T = 227 + 273 = 500K$)
  1. 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:
$$ KE_{SO_2} = \frac{3}{2} \times 0.5 \times 8.31 \times 500 $$

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

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

The ratio of average kinetic energy of 4 g of $CH_4$ to 32 g of $SO_2$ at 227°C is $1:2$.

More Information

This result illustrates that at the same temperature, differences in molecular masses of the gases affect their kinetic energies. Despite having different masses, the number of moles relative to kinetic energy makes a significant impact on the ratio.

Tips

  • Confusing mass with moles; remember to convert mass to moles using molar mass.
  • Forgetting to convert temperature to Kelvin when applying gas laws.
  • Misapplying the kinetic energy formula; ensure it's used correctly based on moles and temperature.

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