What is the internal energy of 7.91 x 10²² gas molecules at a temperature of 300 K?

To determine the internal energy of a gas at a specific temperature, you can use the formula that relates internal energy to the number of molecules and temperature. The internal energy ( U ) for an ideal monatomic gas can be expressed using the equation:

[

U = \frac{3}{2} NkT

]

where ( N ) is the number of molecules, ( k ) is the Boltzmann constant, and ( T ) is the temperature in Kelvin.

1. The Boltzmann constant ( k ) is approximately ( 1.38 \times 10^{-23} , \text{J/K} ).

2. For this case, the number of molecules ( N ) is given as ( 7.91 \times 10^{22} ) and the temperature ( T ) is ( 300 , \text{K} ).

Substituting the values into the equation:

[

U = \frac{3}{2} \times (7.91 \times 10^{22}) \times (1.38 \times 10^{-23}) \times (300)

]

Calculating this step-by-step:

• First, calculate (