Calculate the temperature at which kinetic energy of 0.5 mole of `Cl_(2)` gas is 2.182 kJ.

Explanation

To solve the problem of calculating the temperature at which the kinetic energy of 0.5 moles of Cl₂ gas is 2.182 kJ, we can follow these steps:

### Step 1: Understand the formula for kinetic energy of a gas
The kinetic energy (KE) of an ideal gas can be expressed using the formula:
\[ KE = \frac{3}{2} nRT \]
where:
- \( KE \) is the kinetic energy,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant (8.314 J/mol·K),
- \( T \) is the temperature in Kelvin.

### Step 2: Convert the given kinetic energy to Joules
The kinetic energy is given as 2.182 kJ. To convert this to Joules:
\[ KE = 2.182 \, \text{kJ} = 2.182 \times 10^3 \, \text{J} \]

### Step 3: Substitute the known values into the formula
We know:
- \( KE = 2.182 \times 10^3 \, \text{J} \)
- \( n = 0.5 \, \text{moles} \)
- \( R = 8.314 \, \text{J/mol·K} \)

Substituting these values into the kinetic energy formula:
\[ 2.182 \times 10^3 = \frac{3}{2} \times 0.5 \times 8.314 \times T \]

### Step 4: Simplify the equation
First, calculate the right side:
\[ \frac{3}{2} \times 0.5 = \frac{3}{4} \]
Now, substituting this back into the equation gives:
\[ 2.182 \times 10^3 = \frac{3}{4} \times 8.314 \times T \]

### Step 5: Solve for T
Now, isolate \( T \):
\[ T = \frac{2.182 \times 10^3}{\frac{3}{4} \times 8.314} \]

Calculating the denominator:
\[ \frac{3}{4} \times 8.314 = 6.2355 \]

Now substitute this value back into the equation:
\[ T = \frac{2.182 \times 10^3}{6.2355} \]

Calculating \( T \):
\[ T \approx 349.9 \, \text{K} \]

### Step 6: Convert Kelvin to Celsius (if needed)
To convert Kelvin to Celsius:
\[ T_{C} = T_{K} - 273 \]
\[ T_{C} = 349.9 - 273 \approx 76.9 \, \text{°C} \]

### Final Answer
The temperature at which the kinetic energy of 0.5 moles of Cl₂ gas is 2.182 kJ is approximately **349.9 K** or **76.9 °C**.

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