Calculate the average kinetic energy in joules of the molecules in 8.0 g of methane at `27^@C`.

Explanation

To calculate the average kinetic energy of the molecules in 8.0 g of methane (CH₄) at 27°C, we can follow these steps:

### Step 1: Convert Temperature to Kelvin
The temperature must be in Kelvin for the calculations. To convert from Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Given \( T = 27°C \):
\[ T = 27 + 273 = 300 \, K \]

### Step 2: Calculate the Number of Moles of Methane
To find the number of moles (n) of methane, we use the formula:
\[ n = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
The molar mass of methane (CH₄) is 16 g/mol. Given the mass is 8 g:
\[ n = \frac{8 \, g}{16 \, g/mol} = 0.5 \, moles \]

### Step 3: Use the Average Kinetic Energy Formula
The average kinetic energy (KE) for n moles of a gas is given by the formula:
\[ KE = \frac{3}{2} nRT \]
Where:
- R = 8.314 J/(mol·K) (the gas constant)
- T = 300 K (temperature in Kelvin)
- n = 0.5 moles (calculated in Step 2)

### Step 4: Substitute Values into the Kinetic Energy Formula
Now we can substitute the values into the kinetic energy formula:
\[ KE = \frac{3}{2} \times 0.5 \, moles \times 8.314 \, J/(mol·K) \times 300 \, K \]

### Step 5: Calculate the Kinetic Energy
Calculating the above expression:
\[ KE = \frac{3}{2} \times 0.5 \times 8.314 \times 300 \]
\[ KE = \frac{3}{2} \times 0.5 \times 2494.2 \]
\[ KE = \frac{3}{2} \times 1247.1 \]
\[ KE = 1870.65 \, J \]

### Final Answer
The average kinetic energy of the molecules in 8.0 g of methane at 27°C is approximately **1870.65 Joules**.

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