# Question 1a490

Jun 25, 2015

U_("tot")=49.16"kJ"

#### Explanation:

In an ideal gas we can consider the internal energy to be due to the kinetic energy of the particles.

There are 3 degrees of freedom for a monotomic gas: they can move in the x, y, and z directions.

Each degree of freedom contributes $\frac{1}{2} k T$ of energy which makes$\frac{3}{2} k T$ in total.

So total internal energy $U$ is given by:

$U = \frac{3}{2} k T$

$k$ is the gas constant per mole and is The Boltzmann Constant.

$k = 1.38 \times {10}^{- 23} {m}^{2} . k g . {s}^{- 2} . {K}^{- 1}$

$T$ is the absolute temperature.

A diatomic gas like nitrogen ${N}_{2}$ has 5 degrees of freedom. 3 for translational movement (x,y and z) and 2 for rotation.

This makes 5 in all. So for a diatomic gas:

$U = \frac{5}{2} k T$

In our example we have 4.5 moles of gas so:

${U}_{\text{tot}} = \frac{5}{2} \times L \times 4.5 \times k T$

$L$ is the Avogadro Constant = $6.02 \times {10}^{23} m o {l}^{- 1}$

${U}_{\text{tot}} = \frac{5}{2} \times 6.02 \times \cancel{{10}^{23}} \times 4.5 \times 1.38 \times \cancel{{10}^{- 23}} \times 526$

U_("tot")=49160.2"J"#

$= 49.16 \text{kJ}$