Question

Calculate the total kinetic energy, in joules, of $155\mathrm{g}{\mathrm{N}}_2(\mathrm{g})$ at ${25}^{\circ }\mathrm{C}$ and 1.00 atm. $[\text{Hint}:$ First calculate the average kinetic energy, ${\overline{e}}_k$.

Short answer

Therefore, the total kinetic energy of $155g$ of Nitrogen gas at ${25}^{\circ }C$ and $1.00atm$ is $2.05\times {10}^{4}J$.

Step-by-step solution

Find Temperature in Kelvin

Convert the given temperature from Celsius to Kelvin using the relation: $T(K)=T{(}^{\circ }C)+273.15$. So, for $T={25}^{\circ }C$, the temperature in Kelvin is $T=25+273.15=298.15K$.

Calculate Average Kinetic Energy

Use the formula for the average kinetic energy, which is ${\overline{e}}_k=\frac{3}{2}kT$. Substituting Boltzmann's constant $k=1.38\times {10}^{-23}J/K$ and calculated temperature $T=298.15K$, you find ${\overline{e}}_k=\frac{3}{2}\times 1.38\times {10}^{-23}J/K\times 298.15K=6.17\times {10}^{-21}J$.

Find the Number of Molecules

To find the number of nitrogen molecules, first convert the given mass from grams to moles using the molar mass of nitrogen gas ($28.02g/mol$). Then, multiply by Avogadro's number to get the total number of molecules. So, for $155g$ of nitrogen gas, the total number of molecules is $(155g/28.02g/mol)\times 6.022\times {10}^{23}mo{l}^{-1}=3.33\times {10}^{24}$.

Find the Total Kinetic Energy

Now multiply the average kinetic energy per molecule by the total number of molecules to find the total kinetic energy, which is $3.33\times {10}^{24}\times 6.17\times {10}^{-21}J=2.05\times {10}^{4}J$.

Key concepts

Nitrogen Gas

Nitrogen gas, represented chemically as $ext{N}_2$, is a diatomic molecule consisting of two nitrogen atoms. It is abundant in Earth's atmosphere, making up about 78% by volume. In its gaseous form, nitrogen is colorless, odorless, and inert under standard conditions.

This inertness is due to the strong triple bond between the nitrogen atoms, making $ext{N}_2$ highly stable. In chemical reactions at room temperature, nitrogen gas typically does not participate, unless under specific conditions. It becomes important in various natural and industrial processes, such as in the production of ammonia through the Haber process.

When you encounter nitrogen gas in exercises like the one we're discussing, you are dealing with this nonreactive but critical component of our atmosphere. Its molecular mass is around $28.02\text{ g/mol}$, which is essential when converting between grams and moles.

Temperature Conversion

Temperature conversion is key to many scientific calculations, especially when dealing with gases. The most common conversions involve Celsius and Kelvin. The Kelvin scale is an absolute temperature scale used in scientific settings because it starts at absolute zero, the point where all molecular motion stops.

Converting temperatures from Celsius to Kelvin is straightforward: simply add $273.15$ to the degrees Celsius. For example, to convert ${25}^{\circ }C$ into Kelvin, you calculate:

$$T(K)=25+273.15=298.15\text{ K}$$

Kelvin provides uniformity when using formulas based on thermodynamics, like calculating average kinetic energy, as it allows scientists and students to apply the same concepts universally without negative values, which do not work in these equations.

Moles and Molecules

The concept of moles and molecules forms the backbone of stoichiometry in chemistry. A mole represents $6.022\times {10}^{23}$ entities, be it atoms, molecules, or ions. Avogadro's number is the constant used to define this quantity.

In practical terms, a mole bridges the microscopic world of atoms and molecules with the macroscopic quantities we observe and measure. For instance, if you have $155\text{ g}$ of nitrogen gas and its molar mass is $28.02\text{ g/mol}$, you can calculate the number of moles as follows:

$$\text{moles of }{\text{N}}_2=\frac{155\text{ g}}{28.02\text{ g/mol}}=5.53\text{ moles}$$

To find the number of molecules, multiply the moles by Avogadro's number:

$$\text{molecules}=5.53\times 6.022\times {10}^{23}=3.33\times {10}^{24}\text{ molecules}$$

This conversion is crucial for understanding the scale and quantity at a molecular level.

Kinetic Theory

The kinetic theory of gases provides insight into the behavior of gases in terms of their molecular motion. It essentially breaks down the concept that gas particles are in constant, random motion, colliding with each other and the walls of their container.

One aspect of kinetic theory involves calculating kinetic energy, proportional to temperature. The average kinetic energy of a molecule is calculated using:

$${\overline{e}}_k=\frac{3}{2}kT$$

where $k$ is Boltzmann's constant $(1.38\times {10}^{-23}\text{ J/K})$, and $T$ is the temperature in Kelvin. As we increase temperature, the average kinetic energy and, consequently, the velocity of gas particles increase. This formula is key when you need to find the energy associated with the motion of particles at a specific temperature.

In our exercise, you apply this principle to find the energy for a known number of nitrogen gas molecules:

Taking the average kinetic energy per molecule and multiplying it by the number of molecules results in the total kinetic energy of the gas collection. The kinetic theory is a brilliant tool for simplifying and understanding complex gas behaviors.