Calculate the kinetic energy per mole of carbondioxide at `27^@C`

Answer

To calculate the kinetic energy per mole of carbon dioxide (CO₂) at 27°C, we can follow these steps:

### Step 1: Convert the temperature from Celsius to Kelvin
To convert the temperature from Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]

Given:
\[ T(°C) = 27 \]
\[ T(K) = 27 + 273.15 = 300.15 \text{ K} \]

### Step 2: Use the formula for kinetic energy per mole
The kinetic energy (KE) per mole of an ideal gas can be calculated using the formula:
\[ KE = \frac{3}{2} nRT \]
where:
- \( n \) = number of moles (which is 1 for per mole calculation)
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin

### Step 3: Determine the value of the universal gas constant (R)
We can use the value of \( R \) in two different units:
1. In calories: \( R \approx 2 \, \text{cal/mol·K} \)
2. In joules: \( R \approx 8.314 \, \text{J/mol·K} \)

### Step 4: Calculate kinetic energy in calories
Using the value of \( R \) in calories:
\[ KE = \frac{3}{2} \times 1 \times 2 \, \text{cal/mol·K} \times 300.15 \, \text{K} \]
\[ KE \approx \frac{3}{2} \times 2 \times 300.15 \]
\[ KE \approx 3 \times 300.15 \]
\[ KE \approx 900.45 \, \text{cal} \]

### Step 5: Calculate kinetic energy in joules
Using the value of \( R \) in joules:
\[ KE = \frac{3}{2} \times 1 \times 8.314 \, \text{J/mol·K} \times 300.15 \, \text{K} \]
\[ KE \approx \frac{3}{2} \times 8.314 \times 300.15 \]
\[ KE \approx 3 \times 8.314 \times 150.075 \]
\[ KE \approx 3741.3 \, \text{J} \]

### Final Answer
The kinetic energy per mole of carbon dioxide at 27°C is approximately:
- **900.45 calories** or **3741.3 joules**.

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