What is the total translational kinetic energy of 1.0 mol of N₂ molecules at 25°C?

Verified step by step guidance

1. Step 1: Recall the formula for the translational kinetic energy of a single molecule in an ideal gas, which is given by \( KE = \frac{3}{2} k_B T \), where \( k_B \) is the Boltzmann constant \( (1.38 \times 10^{-23} \ \text{J/K}) \) and \( T \) is the absolute temperature in kelvins.
2. Step 2: Convert the given temperature from Celsius to kelvins using the formula \( T(K) = T(°C) + 273.15 \). For \( T = 25°C \), calculate \( T(K) = 25 + 273.15 \).
3. Step 3: To find the total translational kinetic energy for 1.0 mol of \( N_2 \) molecules, use the relationship between the Boltzmann constant and the gas constant \( R \). The total translational kinetic energy for \( n \) moles of gas is given by \( KE_{total} = \frac{3}{2} n R T \), where \( R = 8.314 \ \text{J/(mol·K)} \).
4. Step 4: Substitute \( n = 1.0 \ \text{mol} \), \( R = 8.314 \ \text{J/(mol·K)} \), and the temperature \( T \) in kelvins (calculated in Step 2) into the formula \( KE_{total} = \frac{3}{2} n R T \).
5. Step 5: Simplify the expression to calculate the total translational kinetic energy. Ensure the units are consistent throughout the calculation, and the result will be in joules (J).

Key Concepts

Translational Kinetic Energy

Translational kinetic energy refers to the energy possessed by an object due to its motion through space. For a gas, this energy can be calculated using the formula KE = (1/2)mv², where m is the mass and v is the velocity of the molecules. In the context of gases, the average translational kinetic energy per molecule is directly related to the temperature of the gas.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law helps in determining the behavior of gases under various conditions, which is essential for calculating kinetic energy.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). For nitrogen gas (N₂), the molar mass is approximately 28 g/mol. Understanding molar mass is crucial for converting between the number of moles and the mass of a substance, which is necessary for calculating the total translational kinetic energy of a given amount of gas.